Post on 02-Feb-2016
description
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
7
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
8
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
9
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
11
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
12
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 /
13
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
15
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
16
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
17
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
18
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