# S V Gryshchenko - Quantum efficiency of the ingaasgaas resonant photodetector for the ultrashort optical connections - страница 1

Telecommunications and Radio Engineering, 67(19):1749-1762 (2008)

SOLID-STATE ELECTRONICS

Quantum Efficiency of the InGaAS/GaAs Resonant Photodetector for the Ultrashort Optical Connections*

S.V. Gryshchenko, A.A. Dyomin, V.V. Lysak, and S.I. Petrov

Kharkiv National University of Radio Engineering and Electronics, 14, Lenin Ave, Kharkiv, 61166, Ukraine

ABSTRACT: A theoretical analysis has been conducted for the quantum efficiency of an InGaAs/GaAs resonant cavity p-i-n photodetector for the ultrashort optical connections. The quantum efficiency at resonance has been estimated. The mathematical model allows for the physical parameters of the photodetector, radiation wavelength, mirror reflectivity and optical absorption in all detector layers. The relations between the quantum efficiency and the reflection coefficient of the upper mirror of the Ala65Gaa35As/GaAs resonator in combination with other variable parameters of the structure have been determined. The magnitudes of the upper mirror reflection coefficient depending on physical parameters of the structure have been found.

The rapid growth of computer technologies makes new demands to the highspeed communication systems, including the ultrashort connections. The state-of-the-art computers utilize optical connections not only instead of stubs but also instead of electron logical elements of the motherboard. Thus, the research on the optical connection elements (receiver, transmitter, waveguide) is still on the agenda. Among the transmitters, the vertical resonator lasers (VRL) are the most attractive choice. The receivers of the laser radiation are p-i-n photodiodes, distinguished by a high response speed and the frequency characteristic being shifted to the higher frequency range in relation to other types of photodiodes [1-4]. Good promises show also the resonant cavity photodetectors (RCPD), whose p-i-n structure is embedded into an optical resonator with the mirrors, as a

Originally published in Radiophysics and Electronics, Vol. 12, No 2, 2007, pp. 401^107.

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ISSN 0040-2508 © 2008 Begell House, Inc.

rule, being the Bragg reflectors. The advantage of this engineering solution is that the optical field is amplified at the resonance frequency inside RCPD. By a small i-region, the achievable quantum efficiency is very high. For example, at the order of dozens of nanometer it is almost 100% [1,2,5,6]. Hence, the response speed of the diode that depends on the i-region width remains high. Having a high response speed and perceptibility toward the required narrow oscillation spectrum, RCPDs are the most suitable devices for data transmission systems. The QE parameter ( r ) actually indicates the efficiency of transformation of the optical power into the electric one and is the key factor for all types of photodetectors. The QE magnitude depends on the physical parameters of the detector and the resonator mirrors. The investigation of these dependences determines the aim and the problems stated in this work.

The aim of presented paper is to explore the influence of the physical parameters of a resonance cavity detector on its quantum efficiency. The necessary values of the reflection coefficient to provide the high QE should be achieved allowing for the given detector material and the wavelength to be detected.

The problem to be solved is building, by means of the analytical model, the dependence of QE on the reflection coefficient of the resonator mirrors, the efficient layer width and the absorption coefficients. The results should be analyzed to reveal the mechanism of controlling the QE by varying the physical parameters. The obtained dependences will be used to determine the necessary number of layers in the upper Bragg reflector that will ensure a high QE.

mathematical model

The quantum efficiency of the detector is defined as a probability for a photon inside the detector to excite an electron that would produce the photocurrent. The total QE includes the optical QE, taking into account the photon absorption in the substance, the internal QE, taking into account the inelastic electron scattering in the quantum well, and the barrier QE, taking into account the electron scattering at the contact barrier.

In view of the above, we can write down the total QE

r = ra%rc, (1)

where ra is the optical QE, rb is the barrier QE, rc is the internal QE.

The diagram of RCPD is presented in Fig. 1. The efficient absorbing layer is situated between two distributed Bragg reflectors (DBR) and is determined by the thickness d and the absorption coefficient a. The distances between the efficient layer and the upper and the bottom mirrors are denoted with L1 and L2,

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respectively, that means the length of the separating layer. The case in question concerns a direct incidence of the radiation on the photodetector, so we confine the consideration to the one-dimensional variant. Assume that the tangential component of the electric vector incident on the photodetector is Ei. By using the plane wave approximation, decompose the electromagnetic field in the RCPD structure into the forward wave Ef and the back wave Eb. The resonator mirrors,

the upper and the bottom ones, are described by the amplitude reflection coefficients

"M and re

- M

respectively, where q\ and q>2 are the phase shifts conditioned

by the optical field penetration inside the mirror; r1 and r2 are the absolute values of the amplitude reflection coefficients. The energy reflection coefficients of the upper

and the bottom mirrors are expressed as R1 r2 and R2 ■ r22, respectively.

z = 0

z=L

L1 ax

d a

r1e

DBR

1

2

DBR

E

L2 a

3

2 ™>ex

E

FIGURE 1. Shematic view of the resonance cavity detector: 1 and 3 are the separating layers, 2 is the efficient layer.

The forward wave Ef at the point z = 0 can be obtained from the self-

consistence condition, i.e., Ef is the total of the wave transmitted through the

upper mirror and the wave transmitted through the whole resonator forth and back. Thus, the expression for E f can be presented as a self-consistent

equation [1,5,6]

Ef = tE + r1r2e-ad-aex(Ll+L2) x e-1 {2PL+V1+M)Ef, (2)

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where t1 is the transmission factor of the upper DBR; a is the absorption coefficient of the efficient layer; aex is the absorption coefficient of the separating layer; в is the propagation constant.

According to (2), the forward wave Ef is written as

E = t1

f 1 - rre~ad-ae, (L +L2)e-i(2pL+p +рг)

The back wave Eb has the form

Eb = yie~ad ,2e~(ae*/2)(L1+L2) e~1 (eL+M) Ef (3) The optical power inside the resonator is [1]

P =1T~\ES I2, (where s = f or b), (4)

where n0 is the vacuum impedance; n is the refraction index of the detector material.

Here, the optical power absorbed in the efficient layer P can be derived from the input power Pi in the following form [1,7,8]:

P = (pfe-aexL + Pbe-ae'L2) (1 - e-ad ) =

(5)

(1 -1 )(e-aexL + r22 e^2 e-a'L ) (1 - e-ad )

1 - 2rr2e aL cos (ipL + q\ + q>2 ) + (rr2) e

2 -2a„L

where ac is the normalized absorption coefficient equal to a a = (aeXL + aexL2 +ad) / L .

Bearing in mind that the optical QE is

Па = % , (6)

obtain

( e-aexL + e~a"L2 R2eacL ) Па =(1 - R )(1 - e-ad )T----'--=•. (7)

Г1 - 2yJRR2 e~acL cos (2eL + p+ p2) + RR2e~

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As can be easily seen from (7), QE has a periodical spectrum. The peak QE will be observed at the resonance wavelength. At the resonance, (+ p+p2 = 2mn (m = 1, 2, 3...)) , expression (7) can be reduced to

( I .. r .. r ..гЛ

R2eacL )

[1 -y[RRR2e-acL

2

-I-2 J J

(1 - r )(1 - e-ad ). (8)

It should be noted that in eq. (8) the reflections at the boundary surfaces between the efficient layer and the separating layers are neglected. This approximation is true when the considered object is a heterostructure with small contrasts of the dielectric constant [1,3,7].

The influence of the standing wave (SW), i.e., the spatial distribution of the optical field in the resonator, is also neglected in this equation. While dealing with detectors having thick efficient layers covering several SW periods, one can neglect the SW influence. However, the SW contribution is considerable for very thin efficient layers [8].

The SW contribution can be taken into account by using the concept of an efficient absorption coefficient [1,8]

aeff=SWC- a, (9)

where SWC is the corrected SW coefficient; aeff is the efficient absorption

coefficient in the efficient region, taking into account the SW influence. Substituting a by aeff in (8), we obtain QE that allows for SW.

For a case when p = 0 , p2 = 0, i.e., the radiation wavelength corresponds to the Bragg mirror wavelength A = AB, and Lj = L2, i.e., the efficient layer is situated directly in the center of the resonator, one can apply the following

formula for SWC [8]:

2r2sin ( /3d )

SWC = 1+ 2 ,K I, (10) 3d (1 + r22)

r, 2nneff Ln + L2 n2 + dna ...... . ,

where в =-; nef =—L-1-—-- is the efficient refraction index, n,

n2 and na are the refraction indices of the separating and the efficient layer.

The internal QE parameter should be considered only when the photodetector contains quantum-dimensional layers. The optically excited electrons in the efficient layer will experience an inelastic scattering, as a result

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lose their energy, thus the probability of escaping the potential well will decrease. The parameter can be presented in the form

% = exp(-d / 4), (11)

where Lz is the free path length of the electron.

The barrier QE is defined as a probability for the electron to run a distance of x near the barrier without scattering

% = exp(-x / Ls), (12)

where Ls is the free path length in the potential well created by force of "electric image"; x is the distance between the barrier peak and the boundary. The force of "electric image" characterizes the interaction between the electric charges and the boundary of two semiconductors [9].

The location and the height of the barrier peak depends on the applied bias voltage, as the external field changes the force of "electric image" [10]. As a result, the dependence of the photocurrent on the bias voltage is conditioned by the electron scattering in the potential well at the boundary and by the variation of the maximum barrier energy. Distance x can be presented as [10]

x =

V 16n£0£aF J

(13)

where q is the electron charge, є0 is the dielectric constant, sa is the dielectric permittivity of the efficient layer, F = (Vb -V0)/d is the electric field in the efficient region, Vb is the bias voltage, V0 is the plane region energy.

the structure under investigation

The detecting structure is a three-layered In02Ga08As/GaAs compound, whose optical length is a half-wavelength of the incident radiation (X = 0.98 um). The structure and the materials were chosen on the basis of experimental data obtained by various research teams, with the regard of the information on actual devices [3-5,11-13]. The diagrammatic view of the structure in question is shown in Fig. 2.

The choice of In0.2Ga0.8As as the active material was conditioned by the presence of the absorption peak in the region of 1 um , since much of the

radiators used in the data transmission systems, operate in this very spectral range. The separating layer material was chosen to be GaAs, in the first place,

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due to the minor absorption at the operating wavelength [2,3]. For DBR we selected a multilayered mirror Al0.65Ga0.35As/GaAs. This structure is characterized by low barrier voltages, high heat conductance, high electric conductance, the required contrast of the refraction indices, low losses of the radiation absorption at the free carriers and the possibility to apply the methods of the current and optical field transverse limiting. The thickness of the Bragg mirror layers was selected such that the Bragg mirror wavelength XB 0.98 um.

I I I I AlGaAs/GaAs

was

n

GaAs

GaAs

-InGaAs

■ AlGaAs/GaAs

ччч^ЛчУ-ч,ч:чЧ-''олл-чЧ'чЧ''>1|—h contact

FIGURE 2. The structure under investigation.

The Table 1 lists the parameters used for the numerical modeling of QE and the reflectance of DBR.

Table 1: Structure parameters

Parameter

Value

Efficient layer thickness, d (In02Ga08As)

from 4 to 15 nm

Separating layer thickness, h (GaAs)

from 61 to 67 nm (depending on d value)

Mirror layer thickness (Al0 65Ga0 35As)

77.44 nm

Mirror layer thickness (GaAs)

69.5 nm

Refraction index Al0 65Ga0 35As

3.1637

Refraction index GaAs, n1, n2

3.5256

Refraction index In02Ga08As, na

3.5691

Absorption coefficient of the efficient region, a (In0.2Ga0.8As)

from 1.5-104 to 24-104 cm-1

The absorption coefficient of the separation layer, aex (GaAs)

1.5-102 cm-1

Plane wave energy, V0

2.1 V

Carrier free path length, Lz

250 A

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result analysis

Figures 3 and 4 demonstrate the computation results for QE as a function of the upper mirror reflection coefficient R1 for various values of the bottom mirror

reflection coefficient R2. The graphs are built for two values of the efficient layer absorption coefficient at the resonance. According to the computations, the only dependence having a peak is QE v.v. R1, hence below we will consider this dependence for various values of other structure parameters. The value of R1 at the peak of QE depends on R2 . To reach the maximum QE, R1 should be decreased simultaneously with R2 . The maximum amplification in the resonator could be reached provided that the radiation is retained in the central part, i.e., the condition R2> R1 should be satisfied.

FIGURE 3. Dependence of QE on the reflection coefficient of the upper and the bottom mirrors for a = 1.5104 cm-1. The number in brackets denotes the DBR layers for the bottom mirror. The solid curve corresponds to R2 = 1 at N = <». The rest of curves top-down R2 = 0.98(N = 50); R2 = 0.95(N = 40); R2 = 0.86(N = 30); R2 = 0.63(N = 20).

As seen from the graphs, n decreases as R2 goes down at any value of R1. The reason is that while decreasing R2 , the losses increase considerably, as a part of radiation after every passage inside the resonator leaves it through the bottom mirror. Due to the quantum-dimensional nature of the efficient layer, the absorption per one passage is very small. Hence, the reflection coefficient of the bottom mirror should be as high as possible. Interesting moment is that the stronger the influence of the R2 magnitude on QE, the lower the absorption coefficient in the efficient layer (compare the curve bias in Figs. 3 and 4), as

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FIGURE 4. Dependence of QE on the reflection coefficient of the upper and the bottom mirrors for a = 24-104 cm-1. The number in brackets denotes the DBR layers for the bottom mirror. The solid curve corresponds to R2=1 at N = да. The rest of curves top-down R2 = 0.98(N= 50); R2 = 0.95(N = 40); R2 = 0.86(N = 30); R2 = 0.63(N = 20).

As known, the reflection coefficient grows as the number of the DBR layers increases [2,12], that is shown in Fig. 5, illustrating the dependence between the reflection coefficient of the bottom mirror and the layer number. Obviously, by a small number of layers, the dependence is almost linearly ascending. It enables one to obtain a relatively high reflection coefficient by simple adding layers to the bottom DBR. However, when the layers are more than 15, an inflection of the curve occurs and the growth of the reflection coefficient slows down. The reflection of every individual layer diminishes due to their bigger total number, which causes the saturation. Thus, the 100% reflection can hardly be achieved just by adding new layers. The unlimited piling of the DBR layers will lead to the increase in the production cost and growth of the optical losses in DBR. So the reflection coefficient of R2 = 0.98 for the exit mirror seems to be sufficient [3,4,13]. This value can be reached by the layer number N = 50. Even more efficient method of obtaining the maximum-close-to-unity reflection coefficient implies usage of materials with a high refraction index contrast [14].

The reflection coefficient in the multiple DBR layers, i.e., the external layers, the substrate and the separating layers, were taken into account as well. As distinct from the bottom mirror, air is also numbered among the external layers of the upper one, so that the contrast of the air-AlGaAs refraction index is quite high. It causes an essential increase in the mirror reflection coefficient by the same number of layers (Fig. 6), enabling one to reduce their number.