**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 13.
Heuristic Models of Nanoscale Systems
(A nanobio-technológia fizikai alapjai )
(Heurisztikus modellek nanoméretű rendszerekhez)
Árpád I. CSURGAY,
Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ
Table of Contents
13. Heuristic Models of Nanoscale Systems 1. The Heuristic Model
2. Dynamics of an Individual Isolated Nanoparticle 3. Nanoparticle in Dissipative Environment
4. Quantum Interference Devices
5. Phase Modulation by Electric and Magnetic Fields
1. The Heuristic Model
In this context the term ‘nanoparticle’ shall cover single
molecules, molecular aggregates, supra-molecular complexes embedded in an environment, composed of the neighboring nanoparticles and a thermal bath. We assume that this complex is subject of external electromagnetic field.
Stand-alone molecule
Thermal Bath, T neighbor
External field
Dissipation
neighbor
An exact treatment is impossible, thus we shall construct simplified heuristic models.
We start with a time-independent Hamiltonian operator specifying a ‘molecule’, we separate the nuclear and electronic motion
according to the BornOppenheimer separation (large mass- difference between nuclei and electrons).
We solve the Schrödinger equation for the electronic states alone, while the positions of the nuclei are fixed. Calculating the
electronic energy spectrum for different positions of the nuclei, we obtain potential energy surfaces which govern the motion of the nuclei.
The effect of the neighboring nanoparticles will be simplified by the assumption that there is no overlap between the electronic wave-functions of the neighbors, thus the interaction is
classical. Its effect can be taken into account by adding an external potential to the molecular Hamiltonian.
The external electromagnetic field only perturbs the Hamiltonian also (it could be time-independent or time-dependent
perturbation).
On the other hand, the real environment, the reservoir (thermal bath) has a very large degree of freedom (practically infinite).
Therefore our nanoparticle together with its neighbors and
electromagnetic illumination, should be considered to be an
‘open’ quantum system, which is in interaction with an unknown reservoir.
With a number of approximations, the theory of open quantum systems help us to construct the
Quantum Master Equations
of the reduced system including the effect of an interaction with the thermal bath as well.
2. Dynamics of an Individual Isolated Nanoparticle
In case of a two-state molecule, , with one nuclear
degree of freedom R, the pure state Hamiltonian matrix H of an isolated molecule depends on R
1 , 2
11 12
12 22
1
2
H H
H H
E R G R
G R E R
H
E
E20
E10
1
E R
2
E R
G R
R10 R20 R
In this case the nanodevice is an isolated quantum system, its
dynamics can be described by the time-dependent Schrödinger equation
The density matrix is
The time evolution of (t)
1
1 2
j t t , c t c t .
t
H
1 2 1 21
1 2 2
2 2 1 2
, c c c .
c c c
c c c c
t t t ,
Liouvillevon Neumann equation
In a 2-dimensional Hilbert space the number of independent real variables representing a Hermitean matrix is n2 = 4, because between the 2n2 real numbers there are n2 relations. If the trace of the matrix is 1, this is an additional relation, thus the number of independent real variables is 3. The density matrix depends on time, and its time dependence can be represented by s = 3 real-valued time-functions
t j
t t
t
H H
1( ),t 2( ),t 3( ).t
Any 22 Hermitian time-varying matrix, A(t), with three real time-varying functions as
The trace of the density matrix is equal to 1, thus
1 Tr
1
1
1 2
2 3
3
.2 2
t t t t t
A A 1 T T T
3 1 21 2 3
1 ( ) ( ) j ( )
1 .
( ) j ( ) 1 ( ) 2
t t t
t t t t
Substituting the density and Hamiltonian matrices into the
Liouvillevon Neumann equation, and expressing the time- derivative of the coherence vector we get the Bloch equation describing the dynamics of the coherence vector
d ,
d
t t
t
λ Ω λ
22 11 12 12 0 2
22 11 12 12 0 1
2 1
12 12 12 12
0 j 0
0 0 .
j 0 0
H H H H
H H H H
H H H H
Note that in a pure state the determinant of the density matrix is zero, i.e. the coherence vector’s length is one, and , thus the length of the coherence vector does not change in time. This means that as long as the Hamilton matrix is Hermitian, the
dynamics remain in pure state and in a three-dimensional space the coherence vector points to the surface of a unit sphere.
The nuclear vibrations change the Hamiltonian in time,
nevertheless, its Hermitian character remains untouched. If the nuclear vibration has only one degree of freedom (an example is a diatomic molecule) it can be characterized by two classical dynamic equations:
where R and P are the position and momentum of the vibration, M is the vibrating mass, and Vnn, Ven are the nucleusnucleus and electronnucleus potential energies respectively. They are determined by the potential energy surfaces (PES). The
expectation values depend on the density operator, thus on the coherence vector. The Hamilton operator depends on R. Thus the Bloch equation together with the classical dynamical equations describe the pure-state dynamics of a stand-alone molecule.
nn en nn en
d 1
( ) ( ) , d
d ( ) ( ) ,
d
R t P t
t M
t P t R R R
V V V V
In case of a two state stand-alone molecule the coupled quantum- classical state equations are as follows
The perturbations caused by the neighbors and the external electric field do not change the Hermitian character of the
Hamilton operator. The molecule’s state changes in time, but at any instant it can be characterized by a wave function being a linear combination of the two stationary states.
1 2
3 3
d d 1
, ( ) ( ) ,
d d
( ) ( )
d 1 1
( ) (1 ) (1 ) .
d 2 2
t t R t P t
t t M
E R E R
t P t R R
3. Nanoparticle in Dissipative Environment
Our model assumed so far that there is no environment around the molecule, thus this model describes a ‘lossless’ quantum-
classical system without any relaxation or dissipation. Pure quantum-state dynamics of the electrons is coupled to lossless classical vibrations of the nuclei.
The isolated molecule represents a ‘closed’ quantum system. The real nanodevice is always subject to the effect of its
environment. It interacts locally with its own thermal bath, and the interaction of the molecular array with the macroscopic
instruments setting the inputs and performing measurements should be taken into account as well.
The master equation describing the evolution is a generalization of the reversible state equation. For quantum Markovian dynamics the general mathematical form of the Bloch equation has been given as
1 2
3 3
d d 1
, ( ) ( ) ,
d d
( ) ( )
d 1 1
( ) (1 ) (1 ) ,
d 2 2
t t t R t P t
t t M
E R E R
P t P
t R R
λ Ωλ Rλ k
where is the Bloch matrix of the conservative system, R and k are the damping matrix and damping vector, respectively, and
characterizes the nuclear relaxation. The value of is zero or positive, the damping matrix and vector have been studied and determined for various damping channels.
Their general forms are
1 1
2 2
3 3
, ,
k k k
R k
with the constraint that the matrix A composed of the elements of R and k
should be positive semidefinit.
1 2 3 3 2
3 1 2 3 1
2 1 1 2 3
1 j j
2
j 1 j
2
j j 1
2
k k
k k
k k
A
4. Quantum Interference Devices
If scattering and thermal effects could be ignored, quantum effects due to the wave properties of electrons must be considered in the following two situations:
(i) Electrons are confined in a region of a size that is comparable with the wavelength of electrons. In this case the spacing
between two allowed energy levels is in the same order of magnitude as the kinetic energy itself. The wavelength of an electron () is related to its kinetic energy (E) by
eff
h .
2 2m E
(ii) When the thickness of a potential barrier is comparable with or less than the imaginary wavelength of an electron inside the barrier, then the electron has a good chance of tunneling
through the barrier.
Both scattering and thermal effects tend to destroy quantum effects. Scattering will randomize the phase of electron waves In order to observe a quantum effect, electrons should not be
scattered in the region of interest, and therefore the mean-free- path of electrons has to be longer than the size where the
quantum effect occurs. An estimate of the mean-free-path ℓ can be given as a function of mobility :
where n is electron concentration per unit area, e is the
magnitude of electron charge. (The estimate is based on the assumptions that in a two-dimensional electron gas the
electrons travel at the Fermi velocity, and there is only one electron valley.)
Thermal effects will cause the initial distribution of electron energy to have a width of kBT, where kB is the Boltzmann constant, T is the temperature. If this energy is larger than a quantum level spacing, then the quantum effect will not be observable.
/ e
2πn ,
In order to observe a quantum interference effect, the spacing between two neighboring quantum levels must be larger than the energy broadening caused by scattering and thermal effects.
Let us assume that devices are built from quantum dots, quantum wires and quantum wells, as well as from electronic devices.
Quantum wires could be called ‘electron waveguides’, in which waves propagate freely without any scattering, i.e. ideal
interference phenomena can take place. These waveguides are contacted to the environment by pieces of the second type, in which inelastic scattering (or other phase breaking processes) and dissipation can take place.
The electronic devices communicate with the waveguides, and
through them with the contacts. It is assumed that the potentials inside the device are given Vpot(r) and A(r) or can be
determined by a self-consistent procedure , and the one- electron state satisfies the time-dependent effective mass Schrödinger equation
2
pot
1 ,
j e , j .
2
V t t
m t
A r r
The propagation of electrons with a fixed energy E along any one of the waveguides can be described by a finite number of
incident a1, a2, a3 and reflected waves b1, b2, b3 the dimensions of them are M1, M2 and M3.
Contact 1 Contact 2
DEVICE
pot
V r A r
a1
b1
b2
a2
a3
b3
Contact 3
The relation between the incident and reflected waves can be described by a scattering matrix as
The scattering matrix can be calculated if the interior of the device is free of phase-breaking or scattering phenomena, and are
known. In this case the solution of the effective mass equation provides the one-electron scattering matrix for any number of contacts.
1 11 12 13 1
2 21 22 23 2
3 31 32 33 3
.
E E E
E E E
E E E
b S S S a
b S S S a
b S S S a
Special case: there are only two contacts and the potential difference between the contacts is U12.
In this case the current is the ensemble average of the one-electron currents with different energies (E)
where f(E) is the Fermi-function
11 12
1 1
21 22
2 2
E E .
E E
S S
b a
S S
b a
12
1 2 21
21 1
0
e e d ,
π
M M
mn
m n
I f E f E U S E E
FkB
1 .
1 e
E E T
f E
For small U12 voltages
and for low temperature
thus the current is
The conductance is given by the Landauer formula
e 12
f e 12 ,f E f E U U
E
F
,f E E
E
1 2
2 2
21 F 12
1 1
e .
π
M M
mn m n
I S E U
1 2 1 2
2 2 2
21 F 21 F
1 1 1 1
12 Q
e 1
π .
M M M M
mn mn
m n m n
G I S E S E
U R
For small U12 voltages and for low temperature the circuit model of a two-contact device is a conductance, which can be
calculated by solving a scattering problem.
The coefficient is a universal constant
Q 2
R π 13 k .
e
The channel of electron flow between source and drain can be constricted with a repelling electrical field, thus the number of propagating electron-modes can be controlled. The measured conductance between source and drain turned to be ‘quantized’, depending the number of modes.
Drain Split gates Drain
-1,7 -1,5 -1,3 -1,1 Split gate voltage [V]
0 1 2 3 4
Conductance (1/R Q)
The experimental evidence of the Landauer formula has been demonstrated in 1988 by B. Van Wees, and later by many others.
5. Phase Modulation by Electric and Magnetic Fields
Quantum interference can be observed if the electron wave can take one of two alternative paths between two leads, and there is a phase difference between them. It has been shown that in symmetric structures the phase can be modulated by an electric or magnetic field (Aharonov–Bohm effect).
Channel 1 Channel 2
Source Gate Drain
UG
Channel 1
Channel 2
If the electric field perpendicular to the wave propagation in channel 1 is different from the field in channel 2, a phase
difference takes place between the two channels. The magnetic vector potential A can play the same role.
If we can approximate the confining potential in each of the two channels, the phase-shift can be calculated exactly. Let us start with the effective mass Schrödinger equation, assuming that where ω0 is the parameter describing the parabolic confining potential, E is the electric field in the z-direction, B is the
magnetic field in the y-direction, and the center of the quantum well for channel 1 is z0 = z1, for channel 2 z0 = z2.
22
pot 0 0
1 e , , 0, 0,
2 x y z
E m z z Ez A Bz A A
Thus
If E = 0 and B = 0 this equation is reduced to the z-directional harmonic oscillator problem, thus the eigenvalues are
2 2 2
2 2 2 2
0 0
2 2
j energy
1 1
j e e
2 2
, , e k x k yx y .
Bz m z z Ez
m x y z
E x y z z
2 2 2
2 2
2 2
0 0
2
energy
d 1 1
e e
2 d 2 2 2
.
y x
k k Bz m z z Ez z
m z m m
E z
2
2 2
0
1 .
2 2
n x y
E n k k
m
Modulation by electric field
Let B = 0, thus the equation reduces to
This can be rewritten as
2 2 2
2 2 2 2
0 0 energy
2
d 1
e .
2 d 2 kx ky 2 m z z Ez z E z
m z m
2 2 2
2
0 0
2 2
0
2 2 2
2 2
energy 0 2
0
d 1 e
2 d 2
e e .
2 x y 2
m z z E z
m z m
E k k Ez E z
m m
Comparing this equation with the harmonic oscillator’s eigenvalue problem, it can be seen that the eigenfunctions resemble the harmonic oscillator eigenfunctions:
If we have two channels centered at z0 = z1, and z0 = z2, then the wavefunctions with the same energy E, and the same ky, must have different values of kx, say kx
1 and kx
2, so that there occurs a phase shift , which is given by
i.e. the electric field modulates the phase-difference.
0 2 0 0 2 2 20 0
e 1 e
, e .
2 2 2
n n n
E E
z u z z E n Ez
m m
1 2
1 2
e ,
x x
x
E z z
k k L
v
