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2009. 05. Hanjo Lim School of Electrical & Computer Engineering hanjolim 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

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.

At BZ boundary with the crystal potential satisfying

    • with an integer
    • Let (Note Fourier theorem)
  • 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

  • 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

=> Show the existence of bandgap and the dependence of on

  • 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

    • 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.


  • 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?

Decaying field, i.e., evanescent wave from the surface should have a complex wave vector as giving the skin depth as

  • If normal incidence,
  • Consider near the band minimum at
  • for band minimum
  • with


Reflecting metal

Non ideal conductor

1D photonic crystal


For (i.e. in the 2nd 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

    • 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.

  • 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.