**UFRJ**

# Physics of high technology devices

PHYS1180 -Physics of High Technology Devices

Lecture 24 –The Quantum & Nano-worlds: Quantum confinement and resonators

Dr. Geoff Pryde Room 1.12 Physics Annex 3365 1026 pryde@physics.uq.edu.au

Key points & learning objectives

1.What is quantum confinement?

2.How does confinement affect the energy level structure?

3.What are quantum harmonic oscillators and how do they interact with the environment?

Physics –quantum physics

Review: the Bohr model –stable orbits

Bohr postulated that conventional EM theory did not apply on the atomic scale –a different Physics was needed!

He constructed a theory in which the electrons were confined to a series of stable orbits of fixed radii (and hence energy and orbital angular mo mentum).

Radiation is emitted (or absorbed) in “quanta”when electrons make transitions between these fixed states.

Theory successfully predicted the hydrogen spectra.

n=1 n=2 n=3 ke E n=1 (ground state) n>1 (excited state) n=infinity (ionised)

Review: Quantisation of orbital angular momentum

“The quantisation of angular momentum is a postulate un-derivable from any deeper laws. It’s validity depends simply upon the agreement of the Bohr model with experiment”.

i.e. Bohr had no real physical reason for stating that the orbits were fixed, and that the electrons would have fixed angular momenta and energies

Another problem –why should electrons of different initial energies all “collapse”into the same set of fixed energies?

Summary & conclusions -last lecture

1.Quantum physics is needed to describe the behaviour of nano-scale devices

2.Quantum physics is a probabilistic science, and correctly predicts / explains atomic structure and the behaviour of tiny particles

3.Particles can behave like waves, and their behaviours are described by the associated wavefunctions and quantum numbers

4.Schrodingers wave equation is a pivotal theory in quantum physics, and can be used to predict the wavefunction at any subsequent time if some initial conditions are known

5.Classical physics is just a limiting form of quantum physics where the number of particles is very large

6.Quantum physics will be at the heart of many new high technology innovations

Lecture Outline

Key points & learning objectives

Time independent Schrödinger equation Confinement: particle in an infinite square well

Real life quantum confinement

The quantum harmonic oscillator

Nanom echanical resonators

Summary & conclusions

SE –Time Independent Form

If U(x)is truly only spatially dependent (for example for standing wave problems), then separation of variables is often used to solve SE:

SE can be reduced to the time independent formwhich is useful for predicting lots of systems ( E =energy):

What are the analogies to the classical wave equation?

Schrödinger used this approach to solve: Particle in 1D box

Harmonic oscillator

1D Hydrogen atom

Particle in a Box (Infinite Square Well)

Classical analogy: a ball bouncing elastically off the inside walls of a box

The ball does not leave the confines of the box, because it does not have enough energy

More general analogy: a charged particle confined within a potential well which has infinite potential at its walls

The charged particle can never escape from the confines of the potential well –it has zero probability of being found outside the well

Its wavefunction in the regions of space x<0, x>L is ZERO

For 0<x<L, what do we expect classically?

U(x)

U=0 x=Lx=0

Finding Ψ(x) Inside the Box

1-D time independent problem where U(x)=0inside the box and U(x)=∞at x=0 and x=L

Consider the region inside the box. SE for this system is given by:

This is a simple 2nd order differential equation whose most general solutions are given by a combination of sin and cos functions:

At x = 0, [ψ(0) = 0] & [ψ(0) = A sin 0 + B cos 0 = 0 + B] ⇒B = 0

Eψx () dx 2

U(x)

U=0 x=Lx=0

Finding Ψ(x) Inside the Box

This trigonometric Stationary State solution indicates the solution is an oscillating wavefunction of wavenumber k

kcan be found from the boundary and continuity conditions atx=Lto yield the wavefunctions and associated energy EIGENSTATES for 0<x<L:

The constant A can be found by using the normalisation condition, to give:

ψn =Asin n πx n=1,2,3..

xn sin

Finding Ψ(x) Inside the Box

For each value of n(the principal quantum number) there is a specific wavefunction describing the state of the particle in the box

Each wavefunction has an associated energy

The energies are “quantised”-is this predicted classically?

What do the wavefunctions look like inside the box?

Finding Ψ(x) Inside the Box

Fit an integer number of half wavelengths in the box. It’s a standing wave problem, just like the circular orbits of the electron

Quantum Dots

Nanocrystals (~ 5 nm diameter), typically of semiconductor e.g. CdSe, PbS

Show effect of confinementi.e. energy level structure coming from size of dot - like a quantum well

Quantum Dots

UQ - Jamie Warn er

Energy

1-D:

Quantum Dots

Light sources -pick wavelength by size of dot

Biological marker -attach dot to cell etc. and observe fluorescence

Quantum technologies -quantum computer gates; nanoelectronics http:// w. sciencenews. org /a rticles /200 30215/bob 10.a sp

Particle in a Box (Finite Square Well)

The finite square well is a more difficult problem to solve, buta similar approach can be taken

We find that the energyeigenstatesinside the potential well are still quantised, but that there is a finite probability of finding theparticle outside the confines of the potential, i.e. for 0>x>L, the wavefunction is non zero –it is a decreasing exponential, and not quantised

The wavefunction “penetrates”into the space outside the well

The Quantum Oscillator

The classical oscillator is any particle which is subject to restoring force The particle undergoes small displacements around some equilibrium position

Classically, it is modelled by a particle of mass m, attached to a spring with a force constant K. The particle oscillates with an angular frequency ω, and the potential energy is given by:

1 Kx biong

We can treat this system with quantum mechanics –useful model for molecular vibrations, photons, and nanomechanical oscillators

The Quantum Oscillator SE (time independent) can be written for this system as:

The simple trig functions do not represent adequate solutions tothis equation, and it is satisfied by Gaussian-type solutions for the ground state (maths is beyond the scope of the course):

() xExm h h xexpCx

Energy levels for the harmonic oscillator

Nanomechanical oscillator

Cantilever (or beam) is like a pendulum – but restoring force comes from stiffness, not gravity

Size of oscillation quantized (i.e. displacement)

ω0 is the resonant frequency, and √<x2 > is the root-mean-square displacement

Quantization gets “washed out”at higher temperatures: kT >> hω0 /2 π

0 time time n n+1

∆x = 114 fm LaHaye et al, Science 304, 74 (2004)

Position measurement

Capacitive readout -like a microphone!

Change in position changes electrical properties of a coupled circuit

Tiny changes in capacitance measured!

When the nanoscopic quantum mechanical oscillator is coupled to an electrical circuit, it is called a nanoelectromechanical system (NEMS) http://hyp er physics.ph y-astr.gs u.ed u/h bas e/au di o/mic.h t ml capacitor

Using the NEMS to measure http:// w.its .calt ech. edu/~h amm el/mrfmpc h.html

Spin of a nucleus or electron is like a small magnet

A force F exists between a permanent magnet on a cantilever (or other resonator), and the magnetization of the spin.

Modulating the direction of the spin periodically causes a (very!) small oscillation in the cantilever -measure it to detect the spin

The direction of the spin can be modulated by using a radio-frequency magnetic field generated by a coil

Summary & conclusions

1.Quantum problems can be solved with the Schrödinger equation 2.Tight confinement causes quantum effects -discretization 3.The energy level spacing depends on the degree of confinement 4.Each discrete level has an associated wavefunction

5.Quantum problems with a restoring force lead to the quantum harmonic oscillator

6.The harmonic oscillator has energy levels spaced by the oscillation anglular frequency (times h/2π)

7.Nanomechanical resonators are quantum harmonic oscillators at very low temperatures