PHYSICAL REVIE%' B VOLUME 48, NUMBER 15 15 OCTOBER 1993-I

Surface eFects and band measurements in photonic crystals

Kenneth W. -K. Shung and Y. C. Tsai

Physics Department, National Tsing Hua University, Hsinchu, Taiwan 30043, The Republic of China
(Received 30 November 1992; revised manuscript received 30 April 1993)
An analytic study of Bloch waves near photonic crystal surfaces has been carried out. This is a gen-
eralization of Heine's theory of metal/semiconductor interfaces. The main concern of this study is how
the surfaces of a finite system affect band determination— which usually involves transmission measure-
ments on small photonic crystals. An analytic expression for transmission is obtained. The results show
that the usual band measurements could lead to considerable errors, especially when they involve narrow
band gaps. A possible method to improve band measurements has been suggested. The analytic surface
analysis has been demonstrated to be fairly accurate, even for a small system. The theory is expected to
be applicable to other surface-related problems of photonic systems.

Recently, Yablonovitch and Gmitter (YG) (Ref. 1) by a distance —1/q from the interface. Heine's 1D
demonstrated the existence of photonic band structure in theory has been the basis for understanding
periodic dielectric materials. This system has generated metal/semiconductor interfaces. It is generalized here
many research efforts. ' Fascinating phenomena such for finite photonic crystals. Analytic results for Bloch
as anomalous Lamb shifts and possible applications in states at surfaces and for transmission have been derived.
semiconductor lasers ' have been suggested and studied. The transmission spectra of YG's crystal have been
Electromagnetic (EM) waves in a periodic dielectric, like evaluated in the I ~X and the I ~L directions. The re-
electrons in a solid, are described by the Bloch theory. In sults are then compared with YG's measurements. Ex-
solids, however, the many-body effect due to the perimentally, band gaps are put at where the transmis-
Coulomb interaction causes significant modifications to sion rate drops substantially. This study shows that band
+he one-body Bloch states, as is exemplified by the Na gaps can, at best, be qualitatively determined this way.
conduction band. " The interaction among photons is The error may be considerable (e.g. , -60%%uo) and it could
negligible, suggesting that photonic crystals can be an get even worse if the gap size (Es ) becomes smaller. This
ideal test ground for the decades-old Bloch theory. Ac- is mainly because the imaginary part of the crystal
curate band determination is important in regard to pho- momentum (q) is proportional to E; therefore, a large
tonic crystal applications. system of sizes ~1/q cc 1/E would be necessary for
Photonic bands are normally determined' with transmission to diminish inside a small gap. Based on the
transmission measurements on a finite, usually small, analytic results, we suggest an alternative but accurate
photonic structure. For example, the photonic crystal way to employ finite-crystal transmission to determine
employed in the YG measurement is made of an A1203 the band dispersions — not just the gap sizes. This will be
bulk with 8000 air cavities, which serve as "atoms" and discussed in connection with Fig. 1.
are arranged into the fcc structure. About one-quarter of The surface effect revealed from this 1D theory is ex-
these atoms are on the surfaces. An interesting question pected to hold, probably qualitatively, in 3D photonic
here is how the band structure, which is a property of an crystals. The results could even by quantitatively correct,
infinite system, should be determined from such a small for example, near the X point where a single lattice
sample. Some insights are necessary before a detailed scattering component dominates. Experiences with sim-
comparison of measured photonic bands with various ple metals have illustrated that" 1D models are valuable
theoretic band calculations ' ' *' is possible. in analyzing important effects in 3D systems. A
Briefly stated, the purpose of this work is to investigate thorough 3D study of the photonic surface effect should
the surfaces of photonic crystals in general, and their be very interesting, but that would require extensive nu-
effect on band measurements in particular. A photonic merical calculations. An analytic 1D study could be very
crystal surface is very similar to a metal/semiconductor helpful in our understanding of such numerical results.
interface. Near the Fermi level of the latter, there is a The electric field, E(r), of an EM wave in a periodic
band gap on the semiconductor side and there are propa- medium obeys the Maxwell equation
—
gating states on the metallic side just like a photonic
crystal surface in the gap region. Heine' investigated
the metal/semiconductor interfaces with a one-
dimensional (1D) model. Mainly, he established that
states inside a band gap have complex crystal momen- where the dielectric function has been separated into two
tum, i.e., k =G/2+iq, where G is a reciprocal wave vec- parts: e(r)=@0+@'(r). eo is the averaged dielectric con-
tor. The imaginary part implies that electrons with ener- stant and E'(r) the periodic part. This equation is very
gy in the gap can actually tunnel into the semiconductor similar to the Schrodinger equation for electrons in a

0163-1829/93/48(15)/11265(5)/$06. 00 48 11 265 1993 The American Physical Society

11 266 KENNETH W. -K. SHUNG AND Y. C. TSAI

Imaginary part of k (G)

0.00 0.0'I 0.02 0, 03 0.04 0.05 ~0 ck UG~k —G
45
~ I I I
I
I L 1 I
I
I I 1 I
I
I I I I c2 c2
(5)
(k —G) —eo CO
CI, G= 67
2 UGCI, ,
c2 c
where G =Gz for bands near X. This two-band mixing
3.5
formalism should work equally well for bands near L,
with 6=61. Effectively, the problem has been reduced
in Eq. (5) to a 1D one. This approximation enables us to
study the problem analytically.
Nontrivial solutions of Eq. (5) have the general form
('-')
2. 5
Q3 k=
6 +V F—(co),
2
U
Q3
where
I
F(co)=(G /4)+eo(co /c )
0.25 0.30 0.35 0.40 0.45 0.50
Rea I part of k (G) —QG eo(co /c )+ UG(co /c ) .
FIG. 1. The calculated band structure near the X point of a F (co) ) 0 correspondsto band modes, for which k is real.
fcc photonic crystal. The crystal parameters are taken from F(co) &0 corresponds to gap modes, which have complex
YG's measurement, with the atomic size R, =0. 372bo. Inside k =G/2+iq, where q =V~F(co)I. Figure 1 is the calcu-
the band gap, the wave vector is complex: k =G/2+iq. The lated band structure near the X point of YG's photonic
nonvanishing q implies a nonvanishing transmission rate in the crystal. Both the real and the imaginary parts of k have
gap region. The crosses represent bands determined from the been shown in units of G (Gx here). The result illustrates
calculated transmission spectra of Fig. 2 (see text below). the general band feature in a gap region: q increases from
zero at gap edge to a finite value near midgap. These gap
modes exist only at surfaces or interfaces since, other-
solid. Following the standard theory of band electrons,
wise, the amplitude of the field would grow exponentiaHy
we treat E(r) as scalar waves and expand it into Bloch
and that is unphysical. The presence of these gap modes
states is important for transmission in the gap region—
especially for a small crystal.
E(r)= g g Cq Ge' (2) Consider a semi-infinite photonic crystal first which, by
6 k assumption, occupies the x )
—a/2 half-space, where a
is the layer separation. The allowed modes inside the
where gz is summed over the first Brillouin zone. The crystal thus have the form
periodic e'(r) can be expanded as
e' "+ye'k '~ for band modes,
e'(r)= gG UGe' r

—
e q cos
2
x+6 for gap modes .
For a fcc crystal, like the one in YG's measurement,

— The factors g and 5 are determined from Eq. (5):

UG =— (e, —e& ) sin( GR, ) cos( GR, ) g=[k /—
c )] [/U (Gco /c )] and sin25=qG/
b06 6 eo(co
[ UG(co /c )]. In deriving these results we have assumed
(4) a small gap size (i.e., UG «
eo), which is normally
satisfied [see Eq. (4)]. We have also assumed that the first
b0 is the lattice constant and R, the radius of the atom. layer atoms are located at x =0; that is one-half the layer
e, and eb, respectively, are the dielectric constants of the separation from the crystal surface. Take the [100] sur-
atom and of the background. b0=1. 27 cm, e, =1, and face in the fcc structure, for example, which corresponds
eb =12.25 in YG's measurement. They have employed to transmission in the I — +X direction; the crystal surface
different R, 's (i.e. , for diff'erent UG's) and studied the is put at x = — a/2= —b0/4. Such a surface condition,
band structures. on which the following calculations are based, is how a
Of particular interest here are bands near the X surface is normally defined in a solid. ' We do not have
(k = Gz/2) and L (k = Gr /2) points, where detailed to put the surface this way, but different surface condi-
band measurements exist. ' Near X, the Bloch scattering tions would give different factors g and 5, which, in turn,
due to UG dominates, since k =+G~/2 are degenerate would result in different transmission spectra (see below).
X
states. With only 'this dominating potential component We remark that the gap states at surfaces we study
retained, the scalar-wave approximation employed in Eq. here correspond to the ED surface states of Meade
(2) is reasonable. Equation (1) can thus be rewritten as et al. , and the band states to their EE states. The first
48 SURFACE EFFECTS AND BAND MEASUREMENTS IN. . . 11 267

letter E stands for extended waves in vacuum and the determining the gap size from transmission measure-
second letter D (E) for decaying (extended) waves in the ments.
crystal. These are the only surface modes that are Figure 2 illustrates the calculated T(co) (solid curves)
relevant to transmission measurements. Other surface for four different finite crystals. N =4, 8, 16, and 32 indi-
modes, denoted as DD and DE states, also exist on sur- cate the numbers of lattice layers of the crystals. The pa-
faces. They are not discussed here; their analytic proper- rameters chosen here simulate YG's measurements near
ties can be studied in a similar fashion. the X point. The atomic size R, =0. 372bo gives a gap
The EM wave transmission through a finite crystal is size of E =1.312 GHz, the location of which is shown
analogous to quantum tunneling across a potential by the two vertical curves in the figure. There is a steady
barrier. Consider a 2N-layer crystal located at decrease in T as m enters the gap from the band region.
—a/2 &x & (2N ——,' )a and an incident wave traveling in This result suggests no obvious criterion as to how the
the +x direction, which, for example, could represent a band edge should be determined from the spectra. The
finite crystal with-N layers of fcc cells. The wave is de- rate does not drop abruptly at the edge as might have
scribed by been expected, ' at least not for X (30. If we had em-
ployed a smaller E,
the outcome would have been even
e
lkpx
+re, Ekpx
x ( ——,
a worse. To see this, we note that at the gap center [i.e. ,
where eo(co /c )=6 /4] Eq. (6) can be expanded for a
Ck0k(x)+ C —k Pk( small E and gives q = GUG/4en o- E in the lowest-order
E(x) = approximation. Small q, then, would need a large N for T
——&x &(2N ——
')a, 2 (8) to drop down appreciably. There is thus a systematic er-
2
te ', x)(2N — ')a
—, .
ror involving the gap measurement and it gets worse rap-
idly as E reduces in size.
In deriving Eq. (9) we have made several assumptions.
ko=Qeok is the wave vector in vacuum. The com- A very important one is that the waves in a finite crystal
ponent —x ) comes from the surface reffection at
pk ( are describable by Bloch waves [Eq. (8)]. Its validity
x =(2N — ')a.
—, It is clear from Eqs. (7) and (8) that the needs to be checked, especially for small 1V's. Recall from
transmission in the band (gap) region is like quantum tun- Eq. (5) that the problem is essentially one dimensional.
neling across a potential well (barrier). The coefficients r, The band near k =G&/2 can thus be studied in an alter-
t, Ck, and C k are determined by the continuity of E(x) native way by, for example, employing a periodic mul-
and dE(x)ldx at the surfaces. The transition rate so ob- tislab structure in which the band parameters a, UG, and
tained can be expressed analytically: eo are kept unchanged. This can easily be achieved with
a construct which contains two uniform slabs in the unit
cell, each with a different dielectric constant. The
in band regions, transmission rate of such a multislab system can easily be
g2 —1
1+ sin [2kNa] calculated by means of the so-called transfer-matrix
A
(9)
in gap regions, „Gap
g&2 TTVVT
1+ A'
+1 sinh [2qNa]
L

N=32
A =2ko[g 6 —k(1+q )],
B =(kc+k )(I+q )+g G 2' Gk, —
A
' = 2ko [q cos25 —( 6/2 ) sin25 ], —0
C)

K
and
B'=[ko+(6 l4) —q ]cos25+qG sin25 .
Within a band, T exhibits an oscillatory pattern as a func-
tion of co, and is enveloped between 1 and A 2 /B 2 &1.
N=4
— Bloch
---
Matrix
There is a finite transmission rate inside a gap. The rate 0 I I I I I I I I

could be considerably large near band edges where q goes 2

to 0 (see Fig. 1). For a sufficiently large crystal, however, Frequency (4. 775 GHz)
T-e '~, which vanishes as 1V~~. Figure 1 shows FIG. 2. The calculated transmission spectra in the [100]
that q =0.02G at the gap center where q reaches its max- direction for finite photonic crystals with 2N layers; N=4, 8,
imum. Thus, it takes N &&10 for T to diminish even at 16, and 32. The same crystal parameters as those in Fig. 1 have
the midgap; X is only about 10 in YG's measurement. It been employed. The two vertical curves indicate where the
is interesting to examine how, in a finite crystal, the ideal band gap is located (i.e., when N — + ~ ). The finite
transmission rate drops as co crosses from a band region transmission rate inside the gap is important for small systems
into a gap. Such studies are obviously important in and can cause errors in the band-gap measurements.
11 268 KENNETH W. -K. SHUNG AND Y. C. TSAI 48

method, ' in which EM waves in neighboring slabs are point, for example. If T, =0.01 is employed, the mea-
related by a 2X2 matrix. The method is equivalent to f —
sured gap would be within 5% of error at =0.20, but
solving the Maxwell equation numerically for the finite
multislab system. Dashed curves in Fig. 2 are such exact
the error would be as high as 60% at f =0. 55 even
though the actual gap sizes at these two f's are about the
results, in which the same band parameters as those used same. It is clear that such band measurements are, at
for the solid curves have been used. Very close agree- best, qualitatively accurate. The measured results (the
ment between the two calculations is obtained. It sug- circles) atf) 0. 2 generally lie within the shaded region.
gests the validity of the analytic expressions, Eqs. (7) and That they are narrower than the actual gap sizes is prob-
(9), for photonic crystals with N &4. The close agree- ably due to the finite tunneling inside the gaps. The mea-
ment at such small K's is surprising —the band nature is sured E at f (0. 2 is larger than all theoretical values.
already eminent in an N =4 crystal. The close agreement This result cannot be understood based on our present
found in Fig. 2 also indicates that similar Bloch state
analysis may be applied to other surface-related prob-
analysis. We note that the gap must vanish as f +0 (i.—
when the air cavities shrink to zero sizes); measured re-
e.,

lems; for example, the localized, interfacial states between sults do not reflect this trend. A smaller Eg, as we argued
two photonic crystals. Localized impurity states have earlier, implies a smaller q, which in turn means higher
been suggested ' to have important applications in semi- transmission rates inside a gap. Therefore, a larger sam-
conductor lasers. ple may be necessary to determine the band in the small-f
Attempts have been made to follow YG's procedure in regime.
—
determining the band gaps at X and L with our calcu- This study suggests no particular T, to use for an accu-
lated transmission spectra. The results are then com- rate band measurement. However, an analytic relation
pared with YG's measurements (Fig. 3). As we have ex- ri. e., Eq. (9)] between the transmission spectra and the
plained, there is no obvious criterion in locating the gaps, band parameters (e.g. , UG, eo, . . . ) has been established.
and it is not clearly described how gaps are determined This relation has been checked to be valid even for a
experimentally. ' We therefore choose an arbitrary cutoff small system. With the interaction among photons negli-
(
T„and put gaps at where T T, . Results shown are for gible, this result is essentially exact for a photonic system.
T, =0. 1 (long-dashed curves) and for T, =0.01 (short- It is thus possible to gain useful band information by
dashed curves). The region between these two sets of analyzing T(co) in accordance with Eq. (9). One possible
data has been shaded. The horizontal axis here method is described in the following. It is reasonable to
represents the packing fraction f, which is a measure of expect that large transmission is associated with Bloch
R, and is defined as the volume fraction that is occupied states of the crystal. Equation (9) shows that the maxima
by the air cavities. The actual gap sizes (the solid curves), of T(co) are at the crystal momentum k =me/2Na, with
in general, lie inside the shaded area. However, a serious m an integer. It is convenient to express k within the
problem here is that a reasonable T, cannot be deter- reduced-zone scheme. Then the frequencies at which the
mined prior to the band analysis. Take the gap at the X transmission maxima are located can be associated with
k =( —,' — m'l2N)G, where 1~m'&N. In regard to the
& 0.3 spectra of Fig. 2, m'=1 should be associated with the
———Tc=0. 1 first peaks on either side of the gap, m'=2 with the
second peaks, etc. The band dispersion determined from
the calculated N = 32 spectrum is shown in Fig. 1 by the
crosses — which actually trace out the bands faithfully.
The same band structure would have been obtained if the
N=8 or 16, or even the 4 spectra were employed for
analysis. That this is true can easily be seen by matching
the peak positions of the various spectra of Fig. 2. This
method appears to be useful for determining the whole
band structure, not just the gap sizes. It should be very

'
interesting to compare bands measured in this fashion
with those of the various band calculations. ' '
In summary, we have included surfaces in a modified
Bloch theory for photonic crystals, and the results are ex-
pressed analytically. Comparison with exact, numerical
calculations indicates that the analytic expressions
remain valid for systems as small as only four layers.
0.00 0.25 0.50 0.75 1.00 Transmission rates of finite photonic crystals have been
packing fraction calculated and employed in band studies. It was pointed
FICz. 3. The gap widths at different packing fractions {i.e., out that the existing band analysis could lead to large er-
different R, ). The gap sizes have been normalized by the band rors, especially when E is small. The analytic results of
energy at the center of the gap at X. The long- (short-) dashed this study have been shown to be helpful for an improved
curves are from this calculation with the cutoff T, =0. 1 (0.01). band measurement. Our main results, Eqs. (7) and (9),
The solid curves represent the ideal gap sizes for infinite systems may also be useful in studies of other surface-related
and the circles {0
and o ) are taken from YG's measurements. problems too. They have been employed to examine the
48 SURFACE EFFECTS AND BAND MEASUREMENTS IN. . . 11 269

localized EM mode at interfaces between photonic crys- K.S. thanks G. D. Mahan and others at the Oak Ridge
tals. ' The "one-body" Bloch theory for photonic sys- National Laboratory for useful discussions. This work
tems is essentially exact within the 1D model. The result was supported in part by the National Science Council of
—
is expected to hold in 3D possibly quantitatively, in gap Taiwan, the Republic of China under Grant Nos. NSC
regions around certain high-symmetric points in k space. 81-0208-M-007-09 and NSC 82-0208-M-007-008.

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