SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

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PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY Development of Complex Curricula for Molecular

CATHOLIC UNIVERSITY Consortium members SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER The Project has

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

1 PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY 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 *** **Molekuláris bionika és Infobionika Szakok tananyagának komplex fejlesztése konzorciumi keretben ***A projekt az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg. 1

2 Peter Pazmany Catholic University Faculty of Information Technology PHYSICS FOR NANOBIO-TECHNOLOGY Principles of Physics for Bionic Engineering (A nanobio-technológia fizikai alapjai ) Chapter 13. Heuristic Models of Nanoscale Systems (Heurisztikus modellek nanoméretű rendszerekhez) Árpád I. CSURGAY, Ádám FEKETE, Kristóf TAHY, Ildikó CSURGAYNÉ

3 Table of Contents 13. Heuristic Models of Nanoscale Systems 1. The Heuristic Model. Dynamics of an Individual Isolated Nanoparticle 3. Nanoparticle in Dissipative Environment 4. Quantum Interference Devices 5. Phase Modulation by Electric and Magnetic Fields 3

supra-molecular complexes embedded in an

4 1. The Heuristic Model Physics for Nanobio-technology: 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. neighbor Dissipation Stand-alone molecule Thermal Bath, T External field neighbor 4

5 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 massdifference 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. 5

6 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). 6

7 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. 7

8 . Dynamics of an Individual Isolated Nanoparticle In case of a two-state molecule, 1,, with one nuclear degree of freedom R, the pure state Hamiltonian matrix H of an isolated molecule depends on R H H H H H E1 R G R G R E R E E 0 E 10 E E1 R R G R R10 R0 R 8

9 In this case the nanodevice is an isolated quantum system, its dynamics can be described by the time-dependent Schrödinger equation j t t, c1 t 1 c t. t H The density matrix is c c1 c1 c 1 c1, c. c c c1 c The time evolution of (t) t t t, 9

10 Liouvillevon Neumann equation t j t Ht t H In a -dimensional Hilbert space the number of independent real variables representing a Hermitean matrix is n = 4, because between the n real numbers there are n 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), ( t), ( t)

11 Any Hermitian time-varying matrix, A(t), with three real time-varying functions as 1 1 At Tr At1 1 t T1 t T 3 t T3. The trace of the density matrix is equal to 1, thus t 1 1 ( t) ( t) j ( t) ( t) j ( t) 1 3 ( t). 11

12 Substituting the density and Hamiltonian matrices into the Liouvillevon Neumann equation, and expressing the timederivative of the coherence vector we get the Bloch equation describing the dynamics of the coherence vector dλ t dt Ωλ t, 0 H H11 j H1 H H H11 0 H1 H j H 1 H1 H1 H

13 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: 13

14 d 1 R( t) P( t), dt M Physics for Nanobio-technology: d Pt ( ) ( Vnn Ven ) Vnn Ven, dt R R R where R and P are the position and momentum of the vibration, M is the vibrating mass, and V nn, V en 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. 14

15 In case of a two state stand-alone molecule the coupled quantumclassical state equations are as follows t d d 1 t, R( t) P( t), dt dt M d 1 E1( R) 1 E( R) Pt ( ) (1 3) (1 3). dt R R 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. 15

16 3. Nanoparticle in Dissipative Environment Our model assumed so far that there is no environment around the molecule, thus this model describes a lossless quantumclassical 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. 16

17 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 t dλ d 1 Ωλ t Rλ t k, R( t) P( t), dt dt M d 1 E1( R) 1 E( R) P( t) (1 3) (1 3) P, dt R R 17

18 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 R 1 k1, k k, 3 k 3 18

19 with the constraint that the matrix A composed of the elements of R and k jk3 jk 1 A jk3 1 3 jk1 1 jk jk1 1 3 should be positive semidefinit. 19

20 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 h m E eff. 0

21 (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-freepath 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 : 1

22 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 k B T, where k B 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 π n,

23 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. 3

24 The electronic devices communicate with the waveguides, and through them with the contacts. It is assumed that the potentials inside the device are given V pot (r) and A(r) or can be determined by a self-consistent procedure, and the oneelectron state satisfies the time-dependent effective mass Schrödinger equation 1 r, t j e, j. t A V r t pot m 4

25 The propagation of electrons with a fixed energy E along any one of the waveguides can be described by a finite number of incident a 1, a, a 3 and reflected waves b 1, b, b 3 the dimensions of them are M 1, M and M 3. Contact 1 Contact a 1 b 1 a DEVICE Vpot r Ar b a 3 b 3 Contact 3 Contact 3 5

26 The relation between the incident and reflected waves can be described by a scattering matrix as E E E E E E E E E b1 S11 S1 S13 a1 b S S S a. 1 3 b3 S31 S3 S33 a3 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. 6

27 Special case: there are only two contacts and the potential difference between the contacts is U 1. E S E E S E b1 S11 1 a1. b S1 a In this case the current is the ensemble average of the one-electron currents with different energies (E) M1 M e mn I f E f E eu1 S1 E d E, π 0 m1 n1 where f(e) is the Fermi-function f E 1 k 1 e B E EF T. 7

28 For small U 1 voltages f f E f E eu1 e U1, E and for low temperature f E EF E, M1 M e mn thus the current is I S E U π m1 n1 1 F 1. The conductance is given by the Landauer formula I G S E S E U M1 M M1 M e 1 mn 1 F mn 1 F 1 π m1 n1 RQ m1 n1. 8

29 For small U 1 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 π RQ 13 k. e 9

30 Conductance (1/R Q ) Physics for Nanobio-technology: 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 The experimental evidence of the Landauer formula has been demonstrated in 1988 by B. Van Wees, and later by many others ,7-1,5-1,3-1,1 Split gate voltage [V] 30

31 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). U G Channel 1 Channel Source Gate Drain Channel 1 Channel 31

32 If the electric field perpendicular to the wave propagation in channel 1 is different from the field in channel, 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 1 E pot m 0 z z0 e Ez, Ax Bz, Ay 0, Az 0, where ω 0 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 z 0 = z 1, for channel z 0 = z. 3

33 Thus Physics for Nanobio-technology: If E = 0 and B = 0 this equation is reduced to the z-directional harmonic oscillator problem, thus the eigenvalues are 1 1 j ebz m 0 z z0 eez m x y z jkxxk yy E x, y, z z e. energy d 1 1 k y kx ebz m 0 z z0 eez z m d z m m E. energy z 1 E n k k m n 0 x y. 33

34 Modulation by electric field Let B = 0, thus the equation reduces to d 1 k x ky m 0z z0 e Ez z Eenergy z. m d z m This can be rewritten as d 1 ee m 0 z z0 z m d z m 0 e E Eenergy kx k y e Ez0 z. m m 0 34

35 Comparing this equation with the harmonic oscillator s eigenvalue problem, it can be seen that the eigenfunctions resemble the harmonic oscillator eigenfunctions: ee 1 e E n z un z z0, E n n 0 e Ez0. m 0 m 0 If we have two channels centered at z 0 = z 1, and z 0 = z, then the wavefunctions with the same energy E, and the same k y, must have different values of k x, say k and k x, so that there occurs a 1 x phase shift, which is given by ee z 1 z kx k, 1 x L vx i.e. the electric field modulates the phase-difference. 35

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER

SEMMELWEIS UNIVERSITY, DIALOG CAMPUS PUBLISHER PETER PAZMANY CATHOLIC UNIVERSITY SEMMELWEIS UNIVERSITY Development of Complex Curricula for Molecular Bionics and Infobionics Programs within a consortial* framework** Consortium leader PETER PAZMANY