foundation of nanophotonics

7/27/2019 Foundation of Nanophotonics

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NANOPHOTONICS

Three main categories:

Nanoscale confinement of matter Nanoscale confinement of light Nanoscale photonic/electronic processes

Photons and Electrons in nanoscale

Although they look very different (one is particle and the other is a wave) classically,they are very similar in nano(quantum) scale.Both can be described by waves (optical and electron microscope). This is becauseboth Maxwell and Schrdinger equations can be put into a very similar form (eigen

value equation) resulting in very similar result.


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Foundation of Nanophotonics

Photon-Electron Interaction and Similarity

Basic equations describing propagation of photons in dielectrics have some

similarities to propagation of electrons in crystals

Similarities between Photons and Electrons

Wavelength of Light,

Wavelength of Electrons,


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Light Interaction with Matter

Maxwells Equations

D =

D = Electric flux density B =Magnetic flux density

E =Electric field vector H =Magnetic field vector

= charge density J= current density


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Constitutive Relations

Constitutive relations relate flux density to polarization of a medium

Electric ED = 0E + P(E) = E

When P is proportional to E

0 = Dielectric constant of vacuum = 8.85 10-12C2N-1m-2 [F/m]

= Material dependent dielectric constant

Total electric flux density = Flux from external E-field + flux due to material polarization

MagneticB = 0H + M(H)

0= permeability of free space = 4x10-7H/m

Magnetic polarization vector

we will focus on materials for which M = 0 B = 0H


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The Wave Equation

Curl equations: Changing E field results in changing H field results in changing E field.

Goal: Derive a wave equation:

Solution: Waves propagating witha (phase) velocity v


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Step 1: Obtain a partial differential equation that depends only on E

Apply curl on both side of a)

Step 2: Substitute b) into a)

D = 0E + P wave equation


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Comparing

Vector identity:

E= 0when1)= 02)(r) does not vary significantly within a distance


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Linear, Homogeneous, and Isotropic Media

P linearly proportional to E: P = 0E

is a scalar constant called the electric susceptibility

Define relative dielectric constant as: r= 1 +

In anisotropic media P and E are not necessarily parallel:

In non-linear media:


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Dispersion Relation

Dispersion relation: = (k)

Derived from wave equation

Substitute:

Result:

Group velocity:

Phase velocity:


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Maxwells Equations for Light

Eigen value Wave Equation:

For plane wave

Describes the allowed frequencies of light

Schrodingers Eigen value Equation for Electrons

Describes allowed energies of electrons


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Free Space Solutions

Photon Plane Wave:

Electron Plane Wave:

Interaction Potential in a Medium:

Propagation of Light affected by the Dielectric Medium (refractive index)

Propagation of Electrons affected by Coulomb Potential


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Free space propagation of both electrons and photons can be described by planewaves. Momentum for both electrons and photons, p = (h/2)k For Photons, k = (2/) while for electrons, k = (2/h)mv For Photons, Energy E = pc =(h/2)kc while for electrons,

Wave vectors and Dispersion


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Their dif ferences are:

Electron has nonzero rest mass; photon does not.

Electrons generate a scalar field while the photons are vector fields (light is polarized). Electrons possess spin, and thus their distribution is described by FermiDiracstatistics. For this reason, they are also called fermions.(Pauliexclusion principle)Photons have no spin, and their distribution is described by BoseEinstein statistics. Forthis reason, photons are called bosons. (They like to stay at the same energy level)

Electrons bear a charge while the charge of photons is zeroElectron can be localized indefinitely while photons can not


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Electron Quantum Confinement

Quantum-confined materials are structures which are constrained to nanoscalelengths in 1, 2, or all 3 dimensions. The length along which there is quantum

confinement must be smaller than de Broglie wavelength of electrons for thermalenergies in the medium.

Thermal Energy, E =

De Broglie Wavelength,

For T = 10 K, the calculated deB in GaAs is 162 nm for electrons and 62 nm for holes

For effective quantum-confinement, one or more dimensions must be less than 10 nm.

Artificially created structures with quantum confinement on 1, 2, or 3 dimensions arecalled,- Quantum Wells,- Quantum Wires and- Quantum Dots respectively.

Ph t C fi t


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Confinement of Light results in field variations similar to the confinement of electron ina potential Well.

Photon Confinement

For light, the analogue of a potential well is a region of high refractive index bounded

by a region of lower refractive-index.

Electrons confine to much smaller area then photons


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(A) Electric field distribution for TE modes n = 0, 1, 2 in a planar waveguidewith one-dimensional confinement of photons.

(B) Wavefunction for quantum levels n = 1, 2, 3 for an electron in a one-dimensional box.

P ti l i i fi it ll


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Particle in an infinite wellParticle trapped in an infinitely deep one-dimensional potential well with a specificdimensionObservations

Energy is quantized, even the lowestenergy level has a positive value and notzero The probability of finding the particle isrestricted to the respective energy levelsonly and not in-between Classical E-p curve is continuous. Inquantum mechanics, p = k with k = n/lwhere n = 1, 2, 3 etc.

En = 2k2/2m = n222/2ml2

In fact the negative values are notcounted since the probability of findingthe electrons in n=1 and n=-1 is thesame and also E is the same at thesevaluesWhen l is large, energies at En and En+1move closer to each other => energy is

continuous in classical systems.


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In the case of a fini te well, possible solutions also outside the well

If (E-V) positive, allowed solutions with real kif (E-V) negative (outside the well), k is immaginary

evanescent solut ions : finite probability (decreasing) of finding electrons justoutside the well |x|>a/2 (tunneling).

Increasing probability with E (and n)

Particle in a finite well


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a) Shallow well with single allowed

levelb) Increase of allowed levelsc) Comparison of the finite-well(solid line) and infinite well(dashed line) energies

Observations The wave functions are not zeroat the boundaries as in the infinitepotential well Allowed particle energiesdepend on the well depth

Finite well energy levels V0

Particle in a finite well


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nl (l=0) = n

For l=0 the result converges to the case of a one-dimensional box

Particle in a spherically symmetric potential

nlroots of the spherical Bessel functions


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1. particle in infinite 1D square wellparticle confined, different solutions possible (sine/cosine)- wavefunction zero at well boundary

- particles had zero point energy: lowest possible state has finite energy- spacing between energy levels increases with increasing n- infinite number of allowed modes

2. particle in finite depth 1D square well- wavefunction nonzero at well edge

- finite number of allowed modes- energy levels slightly modified

3. particle in a 3D infinite spherical square potential well- three quantum numbers required to describe wavefunction (n,l,m)- infinite number of modes

4. particle in 3D Coulomb binding potential- discrete bound levels- energy spacing decreases as principal quantum number increases

foundation of nanophotonics

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