Free charge carrier induced refractive indexmodulation of crystalline Silicon

Aashish Singh

Department of Electrical and Computer Systems Engineering,Monash University, Clayton,

Victoria 3800, AustraliaEmail: aashish physics@rediffmail.com

Abstract—New relationships for the free carrier induced re-fractive index modulation of crystalline silicon at 1.3 and 1.55µm are derived. Free electrons are more effective in perturbingthe refractive index compared to free holes.

I. INTRODUCTION

Silicon (Si) based active photonic devices have shown stu-pendous progress in the past couple of years at the importanttelecommunication wavelength of 1.3 µm and 1.55 µm [1],[2]. For the purpose of modeling and analysis of the activedevices the important relationships relating the influence ofthe free carrier density to the dielectric constant is essential.For example, modeling of the devices based on silicon’snonlinear optical properties require the estimation of the freecarrier absorption (FCA) loss [induced by the two photonabsorption (TPA) phenomenon] to the optical beam [1]. Onthe other hand the devices based on plasma dispersion effectlike modulators and switches requires the calculation of thechange in the material refractive index by the free carrierdensity [2], [3]. This is essential for evaluating the phase shiftof the propagating optical beam [2], [3].

The inter-relationship between the doping density and theSi dielectric constant was empirically derived by Soref et al.[4]. They used the experimentally measured doping dependentabsorption spectra for both n-type and p-type Si at 300 K (seeFig. (5) and Fig. (7) in [4]) for their analysis. A straight linefit to the change in absorption coefficient vs doping densityspectra (see Fig. (12-15) in [4]) was used to evaluate theempirical relations. These relations are widely used in themodeling of silicon nonlinear optical properties based devicesto account for FCA loss [1]. It is to be noted though, thatthe simple Drude model they used to verify the empiricalfit missed the experimental points by over half an orderof magnitude. For deducing the carrier refraction relations,they first subtracted the un-doped spectra from the dopedspectra (for both n-type and p-type absorption spectra) andsubsequently performed the Kramers-Kronig (KK) transformof the resultant spectra to obtain the change in the refractiveindex with the doping concentration (see Fig. (10-11) in [4]). Aleast square fit to these data points then resulted in the carrierrefraction relations (also see Eqs. (3-6) in [2]) which are al-most universally used for modeling silicon optical modulators

[2], [3]. However, recent reports in silicon optical modulatorsrevealed marked discrepancy between the experimental andtheoretical predictions [3], [5]. This resulted in the motivationfor the present study.

II. SCATTERING DEPENDENT DRUDE MODEL

The dielectric constant of a homogenous and isotropicsemiconductor medium can be defined as [6], [7]:

ε =εLε0− σ0q

ε0µ0m∗

⟨τ2

1 + ω2τ2

⟩+ i

σ0q

ωε0µ0m∗

⟨τ

1 + ω2τ2

⟩(1)

where εL is the lattice dielectric constant [8], ε0 is the permit-tivity of free space, τ is the energy dependent relaxation time,〈..〉 represents the energy average of the enclosed quantity, m∗

is the conductive effective mass of the free carriers, µ0 and σ0

represents the d.c mobility and d.c conductivity respectively,ω is the optical frequency, and q is unit electronic charge.

For any semiconductor, the relationship between the freecarrier relaxation time (τ ) and the energy (E) can be expressedas [7]

〈τ〉 = A×∫∞0E

32+se−E/kT dE∫∞

0E

32 e−E/kT dE

(2)

where A is a proportionality factor and s is a constantwhose value depends on the dominant form of the scatteringmechanism. For ionized impurity scattering s = 3/2 and whenthe scattering due to lattice vibration (acoustic phonon) isdominant, s is found to have a value −1/2 [7]. Let the Gammafunction, denoted as Γ(.), be defined as

Γ(x) =∫ ∞

0

tx−1exp(−t)dt (3)

then using Eq. (3), Eq. (2) can be written in terms of theGamma function as

〈τ〉 = A(kT )s Γ( 52 + s)

Γ( 52 )

(4)

similarly⟨τ2⟩

can be expressed as⟨τ2⟩

= A2(kT )2s Γ( 52 + 2s)Γ( 5

2 )(5)

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Further, the carrier relaxation time 〈τ〉 can also be representedas [7]

〈τ〉 =µ0m

∗

q(6)

Thus, using Eq. (4) and Eq. (6), the parameter A can beevaluated and is defined as

A =µ0m

∗

q× 1

(kT )s× Γ(5/2)

Γ(5/2 + s)(7)

Now writing the averaging terms appearing in Eq. (1) in termsof Eq. (4) and Eq. (5), (with x = E/kT ) we get⟨

τ

1 + ω2τ2

⟩=

qλ2

4π2c2µ0m∗L(D) (8)

and ⟨τ2

1 + ω2τ2

⟩=

λ2

4π2c2J(D) (9)

which transforms Eq. (1) into

ε =εLε0− σ0qλ

2

4π2c2ε0m∗µ0J(D) + i

σ0q2λ3

8π3c3ε0m∗2µ20

L(D)

(10)

where

L(D) =Γ(5/2 + s)[Γ(5/2)]2

∫ ∞0

xs+3/2e−x

D + x2sdx

J(D) =1

Γ(5/2)

∫ ∞0

x2s+3/2e−x

D + x2sdx

D =q2λ2

4π2c2m∗2µ20(N,T )

[Γ(5/2 + s)]2

[Γ(5/2)]2

Note that, neglect of the scattering effects in Eq. (10) reducesit to simple Drude model [4].

III. RESULTS AND DISCUSSION

Using the electron and hole conductivity effective massvalue of 0.26 m0 and 0.39 m0 [4] respectively, s= 3/2 fordominant ionized impurity scattering and mobility model ofArora et al. [9], Eq. (10) was solved numerically to obtain thechange in the absorption coefficient with the doping concentra-tion at λ= 1.3 µm and at λ= 1.55 µm. The results are plottedin Fig. (1) without any mathematical fitting. For comparisonpurposes, the experimental data and the theoretical curve takenfrom corresponding Figs. (12-15) in Ref. [4] are also plotted.It should be noted that the theoretical curve shown by brokenline corresponds to the simple Drude model used in Ref. [4].It is evident from these plots that λ2- dependence of FCA aspredicted by simple Drude model is not valid for near infraredFCA, which is consistent with the measurement data of Spitzerand Fan [10]. Moreover, compared with the simple Drudemodel which deviates from the measurements by a factor of7-8 approximately over the entire doping concentration, thepresent model agrees very well especially at higher dopingconcentration.

Next, change in the refractive index with change in the

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ΔN (cm−3)

Δα (

cm−

1 )

This work for elec.Schmid et al.Spitzer et al.Soref et al.

λ = 1.3 μm

(a)

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−1

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ΔN (cm−3)

Δα (

cm−

1 )

This work for holesSchmid et al.Soref et al.

λ = 1.3 μm

(b)

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−1

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ΔN (cm−3)

Δα (

cm−

1 )

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λ = 1.55 μm

(c)

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101

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103

ΔN (cm−3)

Δα (

cm−

1 )

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λ = 1.55 μm

(d)

Fig. 1. Change in absorption coefficient vs change in (a) n-type dopantconcentration at λ = 1.3 µm (b) p-type dopant concentration at λ = 1.3 µm (c)n-type dopant concentration at λ = 1.55 µm (d) p-type dopant concentrationat λ = 1.55 µm. Curves with solid line obtained using Eq. (10). Experimentaldata and the broken curves (simple Drude model) digitized from correspondingFigs. (12-15) in Ref. [4].

doping concentration at λ= 1.3 µm and λ= 1.55 µm wereevaluated. To this end, the n-type absorption spectra (Fig. (5)in [4]) and p-type absorption spectra (Fig. (7) in [4]) weredigitized. The spectra of undoped silicon were subtracted fromthe doping dependent spectra, the resultant ∆α curves wereused in KK relations (Eq. (2) in Ref. [4]) and numericallyintegrated [11]. The results are plotted as data markers in Fig.(2) along with the theoretical curve obtained using Eq. (10) atT = 300 K.

Several features are noticeable in Fig. (2). First, the agree-ment between the theory and the experimental data is excellentfor the free electrons over the entire doping concentrationrange both at λ = 1.3 and 1.55 µm, however for the free holestheoretical curve completely misses the measured values andthe disagreement diverges towards the low doping concentra-tion range. For example at ∆N = 1017cm−3 the variation inthe modeled ∆n due to the free holes is approximately a factorof 5 smaller than the straight line fit to the experimental data.Second, free electrons comparatively induce larger changes inthe refractive index than the free holes throughout the dopingconcentration range at both the wavelengths. Third, theoreticalmodel predicts parallel running lines for electrons and holes.The second and third point listed above were also observedby Soref et al. using simple Drude model (see Fig. 2 in [12]).

The mismatch between the theory and the experimentalvalues in the case of the free holes and the nature of themismatch (divergence at the low doping concentration) is notclearly evident at this point. However following aspects wereinvestigated to reason out the discrepancy:

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1017

1018

1019

102010

−4

10−3

10−2

10−1

ΔN (cm−3)

− Δ

n

This work for Elec.This work for HolesFree ElectronsFree Holes

λ = 1.3 μm

(a)

1017

1018

1019

102010

−4

10−3

10−1

10−2

ΔN (cm−3)

− Δ

n

This work for Elec.This work for HolesFree ElectronsFree Holes

λ = 1.55 μm

(b)

Fig. 2. Change in the refractive index with the change in dopant density atT = 300 K for (a) λ= 1.3 µm (b) λ= 1.55 µm. Solid lines for electrons andbroken lines for holes, obtained using Eq. (10). Data markers for electronsand holes are obtained using KK transform (see text for details).

1) The mismatch of the modeled ∆α plots with the ex-perimental data at lower doping concentration range:It should be noted that within the entire doping con-centration range of 1017-1020 (in cm−3), at λ= 1.3 andλ= 1.55 µm, the theoretical model used (Eq. (10)) forestimating the ∆α, matches the experimental data well(see Fig. 1) both for n-type and p-type dopants. Thus, ifthe theoretical model gives even a reasonable fit for the∆α vs doping concentration plots (with the experimentaldata), it is expected to show a similar trend for the ∆nvs doping concentration plots, which indeed is the casefor the free electrons (see Fig.(2)).

2) The effect of the Burstein-Moss bandfilling, Coulombinteraction of the carriers with the impurities, and theabsorption due to the band to band (αbb) transitionbetween the energy bands and the impurity band tails:Near the indirect band gap of heavily doped Si, thesethree effects act simultaneously. The former two areopposite in nature and partially cancel each other. How-ever, these effects are much smaller, compared with αbb

as demonstrated experimentally [13]. Out of the twowavelength (λ = 1.3 and 1.55 µm) investigated, αbb willbe more pronounced at λ = 1.3 µm. However recentexperimental measurements [14] of αbb in moderatelydoped n-type and p-type Si at room temperature revealsthat at 1.3 µm, αbb is 3-4 orders of magnitude smallerthan FCA. For higher concentration of dopants eventhough αbb increases significantly, it will be completelymasked off by rapidly increasing FCA.

3) The effect of neglecting acoustic phonon scattering andthe use of non-degenerate statistics in deriving Eq. (10):For heavily doped semiconductors classical statistics isnot the appropriate choice. However for near-infraredwavelengths and in the doping concentration rangeconsidered in this paper the fit of the model with themeasurement is good as verified by various authors (see[15]–[17] and references in these publications).

For easy use in the modeling, the theoretical plots of Fig. (2a)and of Fig. (2b) are mathematically represented as:

For λ= 1.3 µm

∆n = ∆ne + ∆nh

= −[8.4× 10−22∆Ne + 5.6× 10−22∆Nh

](11)

For λ= 1.55 µm

∆n = ∆ne + ∆nh

= −[1.2× 10−21∆Ne + 8× 10−22∆Nh

](12)

IV. CONCLUSION

The scattering dependent Drude model was reviewed in thispaper. The model is used to determine the carrier refractionrelations for silicon at 1.3 and 1.55 µm. Analysis shows thatthe free electrons are more effective in perturbing the refractiveindex compared to free holes.

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