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

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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

◉

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.

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.

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