Analytical solution for photonic band-gap crystals using Drude conductivityBrian Schulkin,a) Laszlo Sztancsik, and John F. FedericiDepartment of"Physics. New Jersey Institute of Technology, University Heights. Nev.,;ark, New Jersey 07l02

(Received 18 July 2002; accepted 9 April 2004)

An analytical solution for the photonic band-gap of semiconductor structures in the terahertz (THz)frequency range is discussed. In analogy with the Kronig- Penney model for electronic band-gaps inperiodic potentials, Maxwell's equations for the propagation of light in the photonic crystal aretransformed into an equivalent form of Schrodinger's equation. In the THz frequency range, thereti'active index of the semiconductor is well represented by a frequency-dependent Drude model.We thus find a Kronig-Penney type solution for the photonic band-gap crystal. ([) 2004AmericanAssocialion of"Physics Teachers.

[DOT: 10.1119/1.1758223]

I. INTRODUCTION

Photonic crystals were first discussed by Yablonovitch Iand John" and have been the focus of growing interest asnew applications and technologies emerge,3,4 Textbook treat-ments of photonic crystals5,6 typically draw an analogy be-tween the solution to Schrodinger's equation for a periodicpotential, which leads to solid-state band-gap structures, andthe solution of Maxwell's equations with a periodic dielectricconstant, which leads to photonic band-gap structures.

The analysis of periodic potentials and their impact onelectronic band structures and lattice dynamics are the cen-tral theme of the study of solid-state physics. A mechanicalanalog of periodic lattice dynamics has been developed forlecture demonstrations.? However, the mathematical analogybetween electron band-gaps and photonic band-gaps is lim-ited. Analytical solutions~ for photonic band-gap crystals arerare, and we must generally resort to numerical solutions, inparticular, for a full three-dimensional treatment.6 One rea-son is that Maxwell's equations for photonic band-gap crys-tals have the added constraint that the electromagnetic wavesare transverse, whereas the wave function of Schrodinger'sequation does not have this limitation.

For numerical solutions, it is beneficial to cast Maxwell'sequations in terms of the magnetic field intensity, H:

1

(w

)"

VX E(X) vXH= 7;, H,

whcre E(X) is the dielectric constant of the material as afunction of position, w is the frequency of the electromag-netic wave, and c is the speed of light. In this formulation,the vector operator VX l/Evx simplifies the numericalanalysis.

Our goal is to present a simple, one-dimensional analyticalmethod that utilizcs undergraduate levcl solid-state physicsto introduce photonic band-gap crystals. We solve for theelectric field E instead of H for simplicity, and consider aclass of materials that exhibit Drude-like frequency-dependent conductivity, which leads to an equation that canbe solved analytically in analogy to the Kronig-Penneymodel of solid-state band-gap structures, Our model utilizesalternating layers of doped and un-doped silicon sheets as thephotonic band-gap material. The altemating layers form aone-dimensional lattice that fOlIDs a periodic potential en-ergy in the propagation direction, normal to the silicon lay-ers.

Am. J. Phys. 72 (7), July 2004 http://aapt.org/ajp

n"""", ~",nv nA7U'7A ".,

Schrodinger's equation for a one-dimensional periodicsquare-well potential energy in a crystal, theKronig - Pennel-l! model, can be solved analytically. Bytransforming Maxwell's equations into a form equivalent toSchrodinger's equation, a Kronig-Penney-type solution canbe found to calculate the photonic band-gap structurc. In thisfommlation, the solution is given in terms of the potentialenergy of thc pcriodic laycrs. Thc layer's free carricr conccn-tration takes on thc role of the pcriodic potential cncrgy. Byusing this formalism, simple one-dimensional ,photonic crys-tals can bc modeled in the terahertz (THz) (far-infrared) ti'c-quency range.

II. KRONIG-PENNEY MODEL

Thc time-independent Schrodinger equation may be writ-ten as

rJ" (I"\(I- - - + V(z)\(I = E\(I

2m (Jz2 '

where \(I is an eigenfunction and E is the energy eigenvalueof the particle with mass m moving in a square-well pcriodicpotential energy V(z), shown in Fig, 1. Ifwe divide throughby TJ2/2m, we obtain

(2)

(1)(/"\1.1 2mV(z) 2mE.

- -+ \1.1=-\(1Jz2 TJ"' TJ2 .

The potential of the un-doped silicon, Va, is assumed to benegligible compared to the potential of the doped silicon,Vb, The condition for traveling wave solutions, \(I = A exp(- ikz) , through a square-well periodic potential of finitewidth b and lattice spacing a + b is 10

Q2- K2

cos(k(a+b))= 2QK sinh(Qb)sin(Ka)

+ cosh( Qb )cos(Ka),

(3)

(4)

~m

..

E

K= " 'TJ

(5)

and

Q=2m Vb 2mE---

r/ rJ" .(6)

iE;2004 American AssocIation of Physics Teachers

1i

~ Vb~..

~

:;c~ Va""

z

Fig. 1. Plot of the dopant density of the silicon layers as a function of depththrough the crystal. The crystal is assumed to be semi-infinite.

In the limit of delta function periodic potential baniers,where b-'>O and Ub-,>oo, the condition for traveling wavesolutions is 10

p

cos(ka)= Ka sin(Ka) + cos(Ka),

where

p= Q2ba2 .

III. PERIODIC PLASMA FREQUENCY

In this section our goal is to transfonn Maxwell's equa-tions into an equivalent formulation, in analogy to Eq. (3).From this formulation, the photonic band gaps can be calcu-lated analytically from Eqs. (4), (5), and (6). The periodicityof the potential energy in Schrodinger's equation will cone-spond to a periodic electron density in Maxwell's equations.Maxwell's equations in CGS units are

YXE= - ~aBc at'

, I aE 4v XB= -c - 'iTJ(z)c 00at

+c (10)

where Coois the contribution to the dielectric constant due to

bound charges, which does not depend on the dopant density.J(z) is the periodic CUlTentdensity induced by free charges.

Most approaches that relate the Kronig-Penney modelwith photonic band-gap materials are directed toward solvingEq. (3) for which the dielectric constant varies periodicallyin space and is purely real (that is, there are only travelingwave solutions) and J(z)=0.12,13,16 Moreover, these ap-proaches assume that the dielectric constant, or equivalentlythe index of refraction, is independent of frequency. Clearly,this assumption is not valid when the frequency of the elec-tromagnetic wave is comparablc to the plasma frequency ofthe doped regions.

In the present case, it is the free charges that are periodic.If we substitute Eq. (10) into the curl of Eq. (9), we obtain

0 1 a2E 4'iT aJ(z)- Y~E= - J c",,-- -:;--.c at2 c at

(11)

2 Am. J. Phys., Vol. 72, No.7, July 2004

"""""" ~ ,..", "'A"7"""7A "..,

3 4 52

Frequency [THz)

Fig. 2. Plot of the index of refraction as a function of frequency for both thedoped and un-doped layers. For frequencies below 2.8 THz, the doped lay-ers act as potential energy barriers in analogy with electronic band-gapstructures.

(7)Ifwe apply Ohm's law, J(z)= CT(z)E, and assume a solutionto Eq. (11) of the form of E = Eo exp( - ikz+ iwt), Eq. (11)takes the form

(8)

W21- y2E= -

(

4'iTCT(z)i\

2 Coo- )Ec \ w .I

(12)

For the semi-infinite crystal, O<z<oo, the dielectric constantIS

C(z)=Coo- 4'iTCT(Z)iw

(13)

(9)

In the THz frequency range, the propagation of electro-magnetic waves through doped semiconductors such as sili-con is well characterized by the Drude model,11,14 whichassumes that the collision frequency, 1/7, of the free elec-trons in the doped silicon is negligible compared to THzfrequencies, that is, WP> I. From experimental data for thedirect current (de) conductivity versus dopant density, thescattering time 7 can be estimated from

ne27

CTde= -;;- . (14)

For doped silicon with Ne~ 1O18cm-3, 11r~6.46 MHz. IS Ifwe neglect collisions, the dielectric constant can be ex-pressed as

C(z) = C - wp(z)200 -w2 '

(15)

where w~= 4'iTN e2hn is the plasma frequency. The index ofrefraction can be expressed as 11= Fe:

l1(z)= Coc- wp(Z )2? .ur

(16)

By using Eqs. (13) and (15), we can express Eq. (12) as

2 (,)p(Z)2. (,)2- y E+ :;- E= ---:;-cocE.

c~ c-(17)

Brian Schulkin, Laszlo Sztancsik, and John F. Federici 2

.... a+b---tIb .... a""

-

2,5

2=

1.5

1

0.5

0

0

,.

J.

2

1.8

1.6

1::-if0.8....~

0.6

0.4

0.2

0

0 1.56

k(a+b)

2.34 3.120.78

Fig. 3. The dispersion relation for the crystal with a = 50 ,urn, b = 5 ,urn,

and N= I X 1018 cm-3 There is a photonic band gap centered at 1.1 THz.

In one dimension, - V2E= - a2E/ az2, and Eq. (17) may be

written as

a2E w (z)2 w2--+~E=-E E

clz2 c2 c2 00 .(18)

Ifwe compare Eq. (18) toEq. (3), we see that the eigen-

function 'It and electric field E are analogous. The quantity

w2/c2 takes on the role of the eigenvalue 2m E/ 7/ of Eq. (3).

The condition for travelingwave solutions for the photonicband-gap is given by Eq. (4) with

K=~£,c (I 9)

and

Q- I (2---2--\jwn-w EC P "" (20)

where wp is the plasma frequency of the doped layers. Wenote that the similar forms ofEqs. (18) and (3) are a result ofthe particular form of the frequency dependent dielectric

constant, that is, the w~/w2 term in Eq. (15). If Q is imagi-

0.7

0.6

0.5

g 0.4...::: 0.3~c 0.2..~ 0.]

0

-0.\

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Frequency [THz]

Fig. 4. The effective index ofrefraction is used to plot the overall transmis-sion of the photonic crystal vs frequency for normal incidence.

3 Am. 1. Phys., Vol. 72, No.7, July 2004

n,"""",, ~ n" n.""""7A on

1.5

" 0.5",E~ 0E« -0.5

-1

1 I.1THZ-O.8THzl-1.50 55 ]\0 ]65 220

Depth (microns]

Fig. 5. Plot of two waves propagating through the crystal; the solid line isfor a frequency of 0.8 THz (below the photonic band gap) and the dottedline is for 1.1 THz (middle of photonic band gap).

nmy (that is, if there are traveling wave solutions in thedoped layers rather than exponentially damped solutions),Eq. (4) is still the solution of the photonic crystal16 withcosh Qb->cosIQlb and

Q2-K2 . IQI2+K2 .2QK smh Qb-> - 21QIK sml Qlb.

For typical applications of the Kronig-Penney model tocrystals, the value of Q in Eq. (6) is real because the potential

barrier is higher than the energy of the moving particles. Tn

order to make Q real in Eq. (20), w;> w2 E",. In this regime,the electromagnetic frequency is below the plasma frequencycutoff for free electrons in the doped layers. Consequently,the wave function is exponentially damped in these regions.This condition is analogous to that of a solid state system forwhich the energy of the particle is smaller than the barrierpotential ( E- U < 0). A quick approximation for the requiredcarrier concentration for the plasma frequency cutoff at 1THz is N= 1.1178X 1017cm-3 with Eoo=10. At 0.3 THz, thisvalue decreases to 1.006X 1016cm-3.

(21)

IV. CRYSTAL DESIGN

In this section, we apply realistic parameters toEq. (18)and produce a dispersion relation. The photonic structure, asstated before, is comprised of a stack of alternating dopedand un-doped silicon layers. The resistivity of the doped sili-con is 0.023 fl cm, which cOlTesponds to a carrier concen-tration of N = 1018cm - 3. The parameters a and bare 50 and5 ,urn, respectively. The refractive index, n, is plotted againstfrequency in Fig. 2 using Eq. (16) to illustrate which fre-quencies are solely absorbed by the doped layers, that is, theoperating range. The un-doped layers have negligible carrierconcentration so that the index of refraction is dominated bythe bound charges. In order for the index of the doped layers

to be imaginary, w;> w2E"" the fi:equency of the incidentlight must be less than ~ 2.8 THz. A plot of the arc-cosine ofthe left-hand side of Eq. (4) versus frequency yields the dis-persion relation of the crystal, as shown in Fig. 3.

An effective index of refraction for the crystal is uscd todescribe the overall reflectivity from the photonic structure.The relation

dw

n=c dk (22)

Brian Schulkin, Laszlo Sztancsik, and John F. Federici 3

...

is used to plot the crystal transmittance as a function of fre-quency, as shown in Fig. 4. Figure 5 is a plot of the wavepropagation through the crystal for two frequencies: one inthe photonic band gap and one outside of the gap.

V. CONCLUSION

By assuming a Drude-like frequency dependent conduc-tivity for free carriers in alternating layers of a semiconduc-tor photonic band-gap material, an analytical solution in theTHz range can be formulated. The solution is analogous tothe Kronig-Penney model of electronic band-gaps. This for-malism serves as a tool for introducing undergraduate stu-dents to photonic band-gap materials using the familiar for-malism of the Kronig-Penney model and Drude-likeconductivity.

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Brian Schulkin, Laszlo Sztancsik, and John F. Fcdcrici 4