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2009. 05. Hanjo Lim School of Electrical & Computer Engineering hanjolim @ajou.ac.kr - PowerPoint PPT Presentation

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2009. 05. Hanjo Lim School of Electrical & Computer Engineering hanjolim @ajou.ac.kr. Lecture 4. Electrons and lights in 1D periodic structures. Electrons in a 1D potential : Nearly free and Kronig-Penny model.

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2009. 05.

Hanjo Lim

School of Electrical & Computer Engineering

[email protected]

Lecture 4. Electrons and lights in 1D periodic structures

  • Electrons in a 1D potential : Nearly free and Kronig-Penny model.

    • Free electrons ; electrons at BZ boundary meets total reflection

    • corresponding to interference effect

    • represented by standing waves of

    • the form

    • and

  • Charge density peaks at

  • Charge density is zero there and peaks between the atoms.

  • Electrons in state see more of the attractive potential than those in free

  • electron state which have Note that

  • Electrons in see less of the attractive potential than free electrons.

    • ∴ Electrons in state lie lower (higher) in energy than the free

    • electron value at the BZ boundary.

  • and normalization of the wave length of the box.

  • multiply

  • Then, kinetic term, let

  • equals not zero only when

  • Since the potential is real and symmetric the integral becomes

    • Likewise, we can prove that normalized wave function crystal potential satisfying

    • satisfying the Schroedinger eq.

    • gives the eigenenergy which is higher than

    • We could also extend our calculation to evaluate and at the 2nd, 3rd,

    • etc, BZ boundaries, i.e. at with the bandgaps given

    • by the Fourier components of the crystal potential.

    • Kronig-Penny Model

    • with

    • => Sol.: with and

  • When electron energy the bandgap appears at the BZ boundaries.

  • As electrons are confined more around the atoms, becomes larger.

  • Note the correlation between Fourier components of the and value.

  • What if the electrons are far from the BZ boundary? Free electrons.

    • Multilayer Film: Physical origins of the PBGs on

      • 1D PhC ; alternating layers of materials with and and a period commonlyusedfor dielectric mirror, optical filters, and resonators

      • 1) Traditional approach; propagation of plane

      • wave and multiple reflections at the interfaces.

      • 2) PhC approach ; symmetry approach with index

      • of the modes; and band number

      • Let the modes have as a Bloch form with the

      • translational invariance for

      • The CTS in the xy-plane can take any value.

    • The DTS in the z-direction representation of in the 1st BZ

    • => photonic band(PB) diagram.

    • consider on

    • i) If uniform dielectric medium,

    • with an assigned artificial period

    • bands are continuous.

    • ii) If nearly uniform dielectric medium light line and a small PBG between the upper and lower branches of the PB structure.

    • PBG : frequency range in which no mode can exist regardless of value.

    • iii) If periodic medium with high dielectric constant a PB diagram showing a large PBG.

    • Note) Most of the promising applications of 2D or 3D PhCs rely on the location and width of PBG.

    a wave propagating in the z-direction

    for the 3 cases of periodic dielectric films in the z-direction.

    • Physical origin of the PBG formation ; understandable considering the field mode profiles for the states immediately above and below the gap

    • Occurrence of the gap between bands at the BZ edge means that the PBG appears at

    • Note) Standing wave formation at is the origin of the band gap (nearly free electron model in solid state theory).

    • ∴ PBG is formed by the multiple reflections forming the standing waves.

    • The way of standing wave formation; from the EM variational theorem.

    • i) If nearly uniform dielectric medium

    • standing waves at

    • Note) Any other distribution with same frequency violates the symmetry.

    • Origin of frequency difference ; due to field concentration to a high-

    • and low- dielectrics (not fully sinusoidal). => dielectirc band, air band.

    • ii) If periodic medium with a higher dielectric

    • contrast the field energy for both

    • band is primarily concentrated in the high-

    • layers but the 1st being more concentrated in

    • the high- material.

    high- material: lowest energy distribution,

    low- material: field distribution normal to ground state.

    • Note that, in 1D PhCs, i) PBG always appears for any dielectirc contrast

    • the smaller the contrast, the smaller the gaps. ii) Occurs between

    • every set of bands at BZ’s edge or its center. Why?

    • Evanescent modes in PBGs: defect or surface state.

    • EM wave propagating in the 1D PhC; Bloch wave

    • Meaning of no states in PBG; no extended states given by Bloch form.

    • What happens if an EM wave whose frequency falls in the PBGs is sent

    • to the surface? No EM modes are allowed in the PhC: No purely real

    • exists for any mode at that frequency. Then is it reflected just

    • from the surface or exists in the PhC as an evanescent modes localized

    • at the surface? What determinates the field distribution in the reflection phenomena? If evanescent modes from the surface, how behaves?


    Reflecting metal

    Non ideal conductor

    1D photonic crystal

    • For (i.e. in the 2 should have a complex wave vector as giving the skin depth asnd band), real Bloch states.

    • For (i.e. within the gap), purely imaginary decay of the wave with attenuation coefficient).

    • As ∴ band gap must be wide enough for a good reflection.

    • Note ; Evanescent modes

    • There is no way to excite them in a perfect crystal of infinite extent. But

    • a defect or edge in the PhCs might sustain such a mode. => defect states,

    • defect modes, surface states, surface modes.

    • One or more evanescent modes localized at the defect (defect states) may

    • be compatible depending on the symmetry of a given defect.

    • The states near the middle of the gap are localized much more tightly

    • than the states near the gap’s edge.

    are solutions of the eigenvalue problem,

    do not satisfy the translational symmetry.

    • Localized states near the surface: surface states should have a complex wave vector as giving the skin depth as

    • Similarity of localized states between the PhCs and semiconductors;

    • shallow donors and acceptors, extrinsic or intrinsic defects.

    • Off-axis propagation in the 1D PhCs (ex: let )

    • 1) Because of non-existence of periodic dielectric arrangements in the

    • off-axis direction, there are no band gaps for off-axis propagation when

    • all possible are considered.

    • 2) For on-axis propagation (normal incidence), field in the x-y plane;

    • degenerate, i.e., x- or y-polarization differ only bya rotationalsymmetry.

    • ∴ We may take field (polarization) as x- or y-direction as convenient.

    • * For a mode propagating insome off-axis -direction, broken symmetry

    • → lifted degeneracy

    • must be wide enough for a good reflection.

    • exist a perfect mode

    • Off-axis propagation

    • ex)

    • 1) Nonexistence of band gaps for off-sxis propagation when all possible

  • are considered. Because of no periodic dielectric arrangement.

    • ex) A wave propagating in y-direction (reflect. invariance on yz-plane)

    • Possible polarizations; x-direction or in the yz-plane.

    • Absence of rotational symmetry between the x-polarized wave and yz-

    • polarized wave → different relations for x- and yz-polarized waves.

    • ∴ Degenerate bands for the waves propagating in the z-direction split into two distinct polarizations.

    • ① Different slopes for different polarization

    • means different velocity, i.e., from

    • with the band and polarization index smaller

    • slope of the photonic band => smaller velocity

    • due to different field confinement.

    • ②Approximately linear relations for any band in the long-wavelength

    • limit => homogeneous dielectric medium.

    • The variation of in the photonic crystal is smoothed out in the scale of

    • the long wavelength EM wave: homogenization phenomena. => effective

    • dielectric constants depending on and polarization direction.

    • ex) x-polarized modes have a lower frequency than the modes polarized in the yz-plane for the wave with on 1D PhC of

    • The field distributions at a long-wavelength limit show the reason.

    • The field lies in the high- regions for the x-polarized wave and crosses

    • the low- & high- regions for the wave polarized in the yz-plane.

    • Asymptotic behavior of the modes for large

    • (short ) region: Bandwidth

    • for large value, especially below the line

    • because of the exponential decay of the modes.

    • Defect modes: modes localized at a defect. in the scale of

    • Defects: a structure that destroys a perfectly periodic lattice (ex: a layer

    • having different width or than the rest in 1D PhCs).

    • Consider the on-axis propagation of a mode with the frequency in the

    • PBG via a defect in 1D PhCs.

    • Introducing a defect will not change the fact that there are no extended

    • modes with freq. inside the periodic lattice, since the destruction of

    • periodicity prevents describing the modes

    • of the system with wave vector

    • Then a resonant mode of the defect ↔

    • extended states inside the rest of PhC? (Yes)

    • Defect state: can be interpreted as localized at defect and exponentially

    • decay inside the rest, i.e. a wave surrounded by two dielectric mirrors.

    • If the thickness of a defect becomes of the order of quantized modes

    • → Fabry-Perot resonator/filter (band pass filter)

    • If a defect is the high- material, as increases (why?) with the

    • increase of decay rate as

    • Density of states : # of allowed states per unit increase in frequency

    • Interaction (or interference) between two different localized states.

    • Interaction of modes at the interface between two different PhCs:

    • possible if the two PBG overlap. Existence of a mode having

    • Surface states: localized modes at the surface of a PhC.

    • Surface: there is a PBG only in the PhC, and no PBG in the air.

    • Therefore, we should consider four possibilities depending on whether

    • the EM wave is decaying or extended in the air or PhC for all possible

    • If an EM mode is decaying in the PhC

    • (a mode whose lies in the PBG) and

    • also in the air ( below the light line)

    • → EM mode is localized at the surface

    • → Surface states.

    • Note: All four cases are possible in the case of

    • the structure described at the legend of left

    • figure.

    • It can be shown that every layered material (1D PhC) has surface modes

    • for some termination.

    Band structure of 1D PhC with =13( =0.2a)

    and =1( =0.8a) with the termination of high dielectic layer with 0.1a thickness.