2009. 05. Hanjo Lim School of Electrical &amp; Computer Engineering hanjolim @ajou.ac.kr

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

2009. 05. Hanjo Lim School of Electrical &amp; 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.

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
• 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.

consider

• 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.
• 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

likewise

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.
• 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.