1

2009. 05.

Hanjo Lim

School of Electrical & Computer Engineering

hanjolim@ajou.ac.kr

Lecture 4. Electrons and lights in 1D periodic structures

2

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.

.,2,,0 etcaax

)exp(ikx

,)/(exp)/(exp axiaxi

)/cos(.,. axei )./sin( ax

*2ee

2

e

.

2const

)(

)2/()/( 22 ma

e

.1ikxe

CV

a aa

3

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

EV

dxd

m c2

22

2

)()()( laxVaxVxV ccc

K

Kl

lc iKxValxiVxV )exp()/2exp()(

;,1)()(0

* LdxxxL

0)/cos(2)/2exp(

2 2

22

axL

EalxiVdxd

m ll

:)/(2

)/cos()/cos(22

22

0 2

22

am

dxaxdxdax

LmL

)/exp()/exp()/2exp()/exp()/exp(

21

0axiaxialxiVaxiaxidx

L ll

L

.0E

.)exp()exp(11 ,0

kk

Ldxxkiikx

Ll

,11 VV

.);()2/()( 011011 EthanlowerisEnegativeVVEEVLOOLLV

.l

)/cos(2 axL

4

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

)./sin(2)/sin(22 2

22

axL

EaxL

Vdxd

m C

)(10 VEE

)()()(2 2

22

rErcVdxd

m

bndxaforV

andxforxV

0

00)(

kaKaKaKaP coscossin)/(

.0E

E E

,,/3,/2 etcaak

lV

)/sin()/2( 2/1 axL

2/2abQP .2/22 mQEV

5

=> 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.

gE .PgE

,cVE

gE

KaKaKaP cossin)/(

Ka

15

10

5

2 3 4

cVlV Q

6

Multilayer Film: Physical origins of the PBGs

1D PhC ; alternating layers of materials with and and a period commonly used for 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.

//k

)( zz k

).//( aka z

1

)()( zuRzu

,...).2,1( llaR

)()(//,

//

//, ,,zueerH

kkn

zikki

kkn z

z

z

2

11d 22d

y

z

x

zkk ,//

.n

:a

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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

),12/13/( 21

a wave propagating in the z-direction

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

),0( // kkk z

k

),1/13/( 21

,// nnckvp

.:/)( linelightcalledckk

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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.

ak /2&1 nn

aaka 2/)/2(2 )/(2 aka

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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.

),12/13/( 21

high- material: lowest energy distribution,

low- material: field distribution normal to ground state.

),1/13/( 21

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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?

),( 21

)()(//

//

// ,,,,zueerH

kkn

zikki

kkn z

z

z

),( // zkkk

k

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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

1 2 21

◉

0

~E

H)(k

likewise

zReflecting metal

)(k

x

ze

zx eJzJ 0)(

Non ideal conductor 1D photonic crystal

z

21 zzz ikkk 2/1 zk

)(2 k ak / ).exp()()exp()()()(exp)( 2121 zkzuzikrHzuzikkirH zzzz

2222 )()/()/()( kakak

.ftenvelope

zsx eEzE )(

)/(21

22

22 k

kakletakk

akk

kak

ak

/)/(!2

1)/()( 222

2

/

2/22

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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.

0

0

;k

;k

:)(2 k

;(ik

.,

are solutions of the eigenvalue problem,

do not satisfy the translational symmetry.

iakkk ZB /

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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 by a rotational symmetry.

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

* For a mode propagating in some 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.

ykk y ˆ//

yk

E~

E~

k

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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.

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

limit => homogeneous dielectric medium.

)(k

)(k

kkck ii )()(

)(k

,i

)0,0( k

1p

2p

k

x

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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.

D

k

yk

0)( 0/ zz kak

k

ykk y ˆ//

).(z

,yck

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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)

.k

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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.

,

deft

.2/G

.

.0, // kikz

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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.

.//k

ht

lt