Ž .Infrared Physics & Technology 40 1999 25–32

Enhancement of radiative lifetime in semiconductors usingphotonic crystals

Zoran Djuric a,), Zoran Jaksic a, Danijela Randjelovic a, Tatjana Dankovic a,´ ˇ ´ ´ ´Wolfgang Ehrfeld b, Andreas Schmidt b

a Belgrade UniÕersity, IHTM, Institute of Microelectronic Technologies and Single Crystals, NjegoseÕa 12, Belgrade, YugoslaÕiaˇb Institute of Microtechnology Mainz, Karl-Zeiss Straße 18-20, Mainz, Germany

Received 10 November 1997

Abstract

In this work we present a structure consisting of a semiconductor sample immersed between two one-dimensionalphotonic crystals, designed with the aim to investigate the possibility to increase radiative lifetime in infrared photodetectors,i.e., to decrease the background radiation limit. Since we used a HgCdTe detector for the detection of CO laser radiation,2

we formed a defect in the front-side photonic crystal with the transmission peak on 10.6 mm. Our numerical calculationsshow that the 1-D photonic crystal structures at the front and at the back side of the semiconductor sample significantlyincrease the radiative lifetime in the photodetector. q 1999 Elsevier Science B.V. All rights reserved.

PACS: 72.20.Jv; 72.80.Ey; 78.20.Ci; 78.66.ywKeywords: Photonic crystals; PBG; Radiative lifetime; Reabsorption; Photodetectors; Mercury cadmium telluride

1. Introduction

During the last several years a number of papersappeared with the results of investigation of periodicdielectric structures which may be one-, two- orthree-dimensional. These artificial structures are

w xcalled photonic crystals 1 . The propagation of elec-tromagnetic waves in photonic crystals is very simi-lar to the well-known case of propagation of electronwaves in single crystals. Consequently many terms

Žcommonly used in solid state physics such as for-bidden band gap for electrons, reciprocal lattice,Brillouin zones, dispersion relations, Bloch wave

) Corresponding author. E-mail: zdjuric@helix.chem.bg.ac.yu.

.functions, etc. can be applied to these artificialcrystals, where they can describe the propagation ofphotons.

There are a number of phenomena peculiar tophotonic crystals. Under certain conditions a rangeof photon frequencies exists for which a photonic

Ž .band gap PBG appears and photons with thesefrequencies cannot propagate through such struc-tures, regardless of their direction. Within a photonicband gap optical modes, spontaneous emission and

Ž . w xzero vacuum fluctuations are absent 1 .To date, many investigations have been directed

towards the study of spontaneous emission in elec-tro-optic devices such as semiconductor lasers, withthe aim of improving characteristics using a PBG.

1350-4495r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S1350-4495 98 00039-5

( )Z. Djuric et al.r Infrared Physics & Technology 40 1999 25–32´26

To the authors’ knowledge, this work is the firstto utilise the properties of a photonic crystal toincrease the radiative lifetime of charge carriers in aninfrared photodetector, thus improving its perfor-mance limits.

The first part of this work presents the theory ofthe radiative lifetime within a PBG, making theassumption that photonic crystals do not emit ther-

w xmal radiation within the forbidden energy band 1 ,which agrees with the fundamental properties ofthese crystals. The theory assumes that the photonlifetime in the active region of semiconductor isshort compared to the macroscopic radiative lifetime.

As an example we designed a structure consistingof a photodetector confined within a 1-D photoniccrystal and performed numerical calculation of itsparameters for the case when the detector tempera-ture was in the range 77–300 K, while the back-ground was at 300 K.

2. Theory

Let us consider the structure presented in Fig. 1.Ž .The active area semiconductor is immersed be-

tween two 1-D photonic crystals which can be de-scribed by their frequency-dependent reflection coef-

Ž . Ž .ficients R n and R n . The choice of these coef-1 2

ficients depends on the desired spectral range forimprovement of the limiting parameters of the detec-tor.

We are interested in the component of the radia-tive lifetime of carriers influencing the noise inphotodetectors. For these calculations it is conve-

w xnient to use the theory of Humphreys 2,3 , whichtakes into account the effects of reabsorption. Thetheory is valid only for a detector with an infiniteactive area. However, since the absorption coeffi-cient of detector active material must be large inorder to obtain high quantum efficiency, we assumedthat a detector with finite dimensions could be re-garded as a good approximation of the ideal case andthat the edge effects could be neglected. Of course, amethod to approximate most closely the ideal casedescribed by Humphreys would be to confine thedetector fully within a 3-D photonic crystal.

A result of the theory of Humphreys is that thecharge carriers and photons within the sample reachequilibrium very quickly, so that radiative recom-bination combined with reabsorption does not con-tribute to the noise within the usual photodetectorbandwidth. The photon lifetime is approximatelymrac, where a is the absorption coefficient, c isthe velocity of light and m is the refractive index of

Fig. 1. Idealised structure of an infrared detector with photonic crystals above and below its active area. The photonic crystal on the backsurface consists of a PBG material with three Ge layers with thickness 0.57 mm and PbF 1.24 mm thick. The structure of the upper layer is2

identical, but a defect is introduced instead of the middle Ge layer, i.e., Ge is deposited with thickness of 1.503 mm, which creates a defectmode for the 10.6 mm wavelength. The right side of the figure shows the corresponding spectral dependencies of the reflection coefficients.

( )Z. Djuric et al.r Infrared Physics & Technology 40 1999 25–32´ 27

the semiconductor under consideration, and in practi-cal situations the order of magnitude of this lifetimeis 10y13 to 10y12 s.

According to the above we conclude that in calcu-lation of carrier lifetime and noise we can neglect thecomponent of radiative recombination which doesnot produce photons leaving the sample. We can alsoneglect all the photons incident to the sample whichdo not increase the number of electron-hole pairs.

To determine the radiative lifetime we start fromthe fact that the photon density per unit frequencywithin the sample in thermal equilibrium is

8pn 2m3

q n s f n r s , 1Ž . Ž . Ž .0 n 3 hn r kTc e y1Ž .

Ž .where f n is the Bose–Einstein distribution, r isn

the density of states, T is the absolute temperature,and h is the Planck constant.

It is necessary to stress here that we assume thedimensions of the semiconductor sample to be largeenough to neglect edge effects and that r sn

8pn 2m3rc3. In the general case, if the sample di-mensions were small compared to the light wave-length, it would be necessary to modify the expres-

w xsion for the density of states 1 .It is known that the radiative recombination rate

per unit frequency in nondegenerate semiconductoris proportional to the concentration of electrons and

w xholes 4 :

npr sr P , 2Ž .rad r E2ni

where besides the usual coefficient r we introducer

P , the probability for a photon to exit the sampleE

after being generated in recombination of an elec-tron-hole pair.

The parameter r can be calculated according tor

the principle of detailed balance if we know thedependence of the absorption coefficient a on fre-quency n .

In the thermal equilibrium the number of the actsof recombination with photon emission in the fre-quency range between n and nqdn is equal to thenumber of electron-hole pairs generated by thermalequilibrium radiation at the same frequency. Thus,

r sW n f n r , 3Ž . Ž . Ž .r n

Ž .where W n is the probability of photon absorptionper unit time,

cW n san sa . 4Ž . Ž .g

m

We assumed that there is no dispersion within thesample medium, so that the group velocity is Õ sg

crm. In this way the radiative recombination rate isdetermined by

ac npr s f n r P . 5Ž . Ž .rad n E 2m ni

Radiative recombination rate per unit of semicon-ductor volume is determined by the absorbed photonflux arriving from the environment

cq A0g s P , 6Ž .rad A34m Ad

where A is the area of the semiconductor sample, dits thickness and VsAd is its volume. We dividethe expression by m3 because we assume that theflux arrives from the free space, ms1. P is theA

probability of the absorption of the arriving photons.Now we apply again the principle of the detailed

balance, but this time it regards the balance of thephotonic fluxes inside and outside the sample whenthe whole system is in thermodynamic equilibrium.This enables us to determine the equilibrium proba-bility of the photon emission P by taking that rE0 rad

Ž . Ž .and g from Eqs. 5 and 6 are equal for n p srad 0 0

n2. We assume further that for small perturbations ofi

the equilibrium P fP . Thus, the following ex-E E0

pression is obtained for the resultant recombina-tion–generation rate per unit frequency,

np cUs y1 q n P n . 7Ž . Ž . Ž .0 A2 3ž /n 4m di

After integration over frequency we obtain thefollowing expression for the net recombination

npUs y1 G2ž /ni

np cs y12ž / dni

=q`

2 31 8pn mP n dn , 8Ž . Ž .H A3 3 hn r kT4m c e y1Ž .E rhg

( )Z. Djuric et al.r Infrared Physics & Technology 40 1999 25–32´28

where the lower limit is determined by the frequencyn sE rh, where E is the band gap of the semicon-g g g

ductor. We determine the absorption probability forthe photons generated from the interval around P inA

the following manner:

1 a 1yRŽ .d 1P n sŽ . HA ya d½q 1yR R e00 1 2

= ya y y2 a dqa yq e qR e d y0 2

a 1yRŽ .d 2qH ya d1yR R e0 1 2

= ya y y2 a dqa yq e qR e d y , 9Ž .0 1 5which gives

1yeya d

P n sŽ .A ya d1yR R e1 2

= ya d2y R qR 1yeŽ . Ž .1 2

ya dy2 R R e . 10Ž .1 2

where we used for flux the expression for the genera-w xtion function from 5 which neglects the interference

effects, and we neglected the effects of the semicon-ductor side edges:

1yR FŽ .1g z sŽ .

1yR R exp y2a dŽ .1 2

= exp ya z qR exp y2a dqa z ,Ž . Ž .2

11Ž .where F is the incident photon flux at the front side.

Radiative lifetime is defined by the relation

1 U n qp0 0s sG 12Ž .2t Dn nR i

while the radiative lifetime in intrinsic semiconduc-Ž .tor n sp sn is given as0 0 i

1 2Gs

t nRi i

q`22 2pn

s H 2 hn r kTdn C e y1Ž .E rhi g

=1yeya Žn .d

ya Žn .d1yR n R n eŽ . Ž .1 2

= Ž .ya n d2y R n qR n 1yeŽ . Ž . Ž .Ž .1 2

ya Žn .dy2 R n R n e dn , 13Ž . Ž . Ž .1 2

where we stressed explicitly which variables aredependent on frequency n .

3. Design of photonic crystal

Our aim was to enhance the radiative lifetime inour HgCdTe photodetector whose peak sensitivity isin the wavelength range 6–12 mm. Thus our goalwas to calculate parameters of a 1-D photonic crystalstructure with the PBG in the mentioned wavelengthrange which simultaneously transmitted radiation atls10.6 mm.

We consider a one dimensional PBG structurewhich can be fabricated by the sputtering technique.This consists of alternating layers of germanium andlead fluoride. We calculated the photonic band gapsof our structure by applying the formalism presented

w xin 6 to solve the wave equation obtained from theMaxwell’s equations for a traditional dielectric mul-tilayer film analogously to the Kronig–Penney modelfor 1-D case. For the refractive index of lead fluoride

w xwe used the data from Ref. 7 , which we fitted in

Fig. 2. The band gaps for a 1-D lattice of GerPbF slabs in2

dependence on d rd ratio.Ge

( )Z. Djuric et al.r Infrared Physics & Technology 40 1999 25–32´ 29

the range 1.5 mm to 11.862 mm as ns1.738yy4 2 Ž .0.0018ly8.4636P10 l l in mm . For germa-

nium we used the value ´s16.1 in the 6–12 mmw xrange 8 . Thus, we obtained a transcendental equa-

tion which determined the photonic band gap mapw xfor 1-D lattice of dielectric slabs 9 . Fig. 2 shows

our calculated bandgap map for 1-D Ge and PbF2

PBG combination.We performed our calculation for two structures.

First we consider the structure denoted by A, consist-ing of three periods of alternating layers of Ge andPbF . By the above mentioned method we calculated2

that the photonic band gap covered the largest por-tion of the desired wavelength range if the thicknessratio of the layers was d rds0.31, where dsGe

d qd s1.81 mm, i.e., d s0.57 mm andPbF Ge Ge2

d s1.24 mm.PbF2

The reflection coefficient of the structure obtainedin such a manner was calculated using a matrix

Fig. 3. Spectral dependence of the 1-D photonic crystal reflectionŽ . Ž . Ž .coefficient: a three GerPbF layers case A ; a seven GerPbF2 2

Ž .layers case B . Solid lines denote the structures without defect,and dotted lines are valid for the structures with built-in defects at10.6 mm.

method based on the multilayer thin films theoryw x10 . To make the photonic crystal transmit the radia-tion of CO laser we introduced a defect in it by2

using the trial method for various values of thicknessof one of the germanium layers.

We performed the calculation of the structure Afor a photonic crystal with three periods ds1.81mm, where we assumed that the medium on the front

Ž .side was air ms1 , and on the back side mercuryŽ .cadmium telluride Hg Cd Te with ms3.51. The1yx x

thickness of the second Ge layer was increased tod s1.503 mm, thus producing a defect at 10.6Ge

mm. Fig. 3a shows the spectral dependence of thereflection coefficient of the structure A for the case

Ž .without an introduced defect solid line and with theŽ .defect at 10.6 mm dotted line .

We also performed the calculation for a photonicŽ .crystal with seven GerPbF pairs structure B . The2

crystal parameters were again d s1.24 mm andPbF2

d s0.57 mm. The 10.6 mm defect within theGe

structure B was fabricated by increasing the thick-Ž .ness of the middle fourth Ge layer to 1.545 mm.

Spectral dependencies of the front- and back-sidereflection coefficient for this structure are shown in

Ž .Fig. 3b for the case without defect solid line andŽ .with the defect dotted .

4. Radiative lifetime calculation

We performed our calculation for bulkHg Cd Te samples with different compositions x1yx x

and at different temperatures ranging between Ts77and 300 K, confined between two 1-D GerPbF2

photonic crystals designed in the above describedmanner.

The band gap of HgCdTe was determined accord-w xing to the relation of Hansen et al. 11 , while the

Fermi level was calculated using the approximationw xof Altschul and Finkman 12 . To calculate the

HgCdTe refractive index we used a modified versionw xof the approximation from Ref. 13 .

To calculate the interband absorption coefficientw xin MCT we used the Kane’s kP model for InSb 14 ,

w xapplied to Hg Cd Te by Blue 15 and Anderson1yx xw x w x16 , and modified in Refs. 17,18 . As is well known,the absorption coefficient in the Kane region can be

( )Z. Djuric et al.r Infrared Physics & Technology 40 1999 25–32´30

written as a sum of coefficients for transitions fromthe light-hole and heavy-hole bands to the conduc-

w xtion band. We used ´ s19–11 x 13 for the high-`

frequency dielectric permittivity in nondegenerateŽ .semiconductor x is the material composition , Ds

0.9 eV for the spin split-off energy value, and Ps8P10y10 eV m for the Kane’s matrix element.

w xIn our calculations we used our modification 18of the expression for the absorption coefficient dueto the exponential band-tailing below the cut-off

Ž . w xenergy Urbach region 19 . We used the value50 000 my1 for the crossover point between the

w xKane region and the exponential region 19 . Theradiation frequency corresponding to this point wasdetermined numerically.

ŽThe integral of intrinsic radiative lifetime Eq.Ž ..12 was numerically calculated by the Gauss–

w xKronrod procedure 20 . The allowed relative calcu-lation error was 0.01. The intrinsic concentrationwithin Hg Cd Te was determined according to1yx x

w xSchmit 21 .Fig. 4 shows the calculated dependence of radia-

tive lifetime on the front-side reflection coefficientfor an idealised Hg Cd Te structure mounted on a1yx x

Žhighly polished reflective surface R f1 for all2.wavelengths . The operating temperatures were 77 K

Žas required for BLIP photoconductive detectors for. Ždetection of 10.6 mm radiation , 220 K achievable

.with thermoelectric coolers and, for comparison,300 K. The cadmium molar fractions x were 0.20,

Ž0.185 and 0.165, respectively each of them corre-.sponding to a peak sensitivity at 10.6 mm .

Fig. 4. Radiative lifetime versus front-side reflection coefficientfor Hg Cd Te detector thickness 5 mm and for temperatures 771y x x

Ž . Ž . Ž .K xs0.20 , 220 K xs0.185 and 300 K xs0.165 .

Fig. 5. Radiative lifetime versus front-side reflection coefficientfor Hg Cd Te detector thickness 3, 5, 10, and 20 mm at 220 K1y x x

and xs0.185.

Fig. 5 shows the dependence of radiative lifetimeon the front side reflection coefficient for xs0.185and for a temperature Ts220 K, for sample thick-ness 3, 5, 10, and 20 mm.

To determine the influence of the PBG materialaround the detector first we performed calculationsfor an idealised case when the detector ds5 mm

Ž .was mounted on a highly reflective substrate R f12Ž .and coated with an antireflection coating R f0 .1

The next calculation was performed for our structureA, i.e., for the R corresponding to a photonic1

crystal with three paired layers GerPbF and with a2

defect at 10.6 mm; R corresponds to the same case2

without defect. The last calculation was performedfor the structure B, i.e., the photonic crystal withseven paired layers GerPbF and with a defect at2

10.6 mm; R corresponds to the same crystal with-2

out defect. The above calculations were performedfor those structures at three different temperaturesŽ .77, 220 and 300 K and for compositions x tailored

Žto give the optimum response at 10.6 mm 0.20,.0.185 and 0.165, respectively . The results of the

calculations are summarised in Table 1.Although the utilised PBG structures are simple,

1-D and with a small number of layers, it can beseen that they enhance radiative lifetime in all of thecalculated cases. At 300 K the improvement is al-most an order of magnitude. It is also almost anorder of magnitude at 220 K, and about 15 times atthe liquid nitrogen temperature. Even in the worst

Ž .case at 300 K and for three layers the PBG struc-

( )Z. Djuric et al.r Infrared Physics & Technology 40 1999 25–32´ 31

Table 1Carrier radiative lifetime for bulk mercury cadmium telluride photodetectors at 77, 220 and 300 K, calculated with and without 1-DGerPbF photonic crystal enhancement2

w x w x w xt s , Ts77 K, xs0.21 t s , Ts220 K, xs0.185 t s , Ts300 K, xs0.165Ri Ri Ri

y3 y5 y5Ž .Idealised detector R s0, R s1 1.406P10 4.632P10 3.52P101 2y2 y4 y5Ž .Structure A three layers GerPbF 2.023P10 2.099P10 8.541P102y2 y4 y4Ž .Structure B seven layers GerPbF 3.362P10 5.720P10 1.197P102

ture improves the radiative lifetime more than twice.In all the cases the useful signal at 10.6 mm is fullyretained.

5. Conclusion

In this work we present a structure consisting of amercury cadmium telluride infrared photodetectorimmersed between two 1-D GerPbF photonic crys-2

tals designed to enhance the radiative lifetime in thedetector. A defect is introduced into the 1-D crystalat the front side with the aim to retain the usefulsignal at 10.6 mm while suppressing the radiativerecombination in the range 6–12 mm. Our calcula-tions show that the proposed structure should im-prove the radiative lifetime for at least an order ofmagnitude. With a convenient choice of PBG mate-rial this could be increased significantly.

An increase in the number of GerPbF layers in2

the photonic crystal structure results in a more dis-tinct and narrower peak at 10.6 mm and the radiativelifetime is longer, but at the same time the amplitudeof the peak decreases. This means that it is necessaryto reach a compromise between these opposed ef-fects in order to achieve an optimum in increasingthe radiative lifetime while simultaneously retainingthe useful signal.

Regarding the influence of other recombinationmechanisms to the overall performance of detectorswith PBG improvement of radiative lifetime, it mustbe said that the high temperature operation regionŽ .intrinsic material, 300 K in mercury cadmium tel-luride is dominated either by Auger processes orShockley–Read recombination, depending on the

w xquality of single crystals used 13 . At lower temper-atures radiative lifetime becomes more important,

Ž .and in the BLIP operation 77 K it may becomew xpredominant 13,22 . Even at temperatures achiev-

Ž .able by thermoelectric coolers typically 220 Kradiative lifetime may dominate if some of non-equi-librium methods is used for suppression of Auger

Žgeneration–recombination either a combination ofw xexclusion–extraction 23–25 or magnetoconcentra-

w x.tion 26 . Thus it appears that the method proposedherein could be used in such situations to improvefurther the performance of the photodetectors.

The theory presented herein is general and can beŽapplied in the case of wider bandgap materials for

example, in detectors for optical communications.and in 3–5 mm devices , where radiative lifetime is

important or even prevailing.

Acknowledgements

This work was partially funded by the Volkswa-gen Foundation within the framework of the projectIr72 957.

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