Design of resonant cavity structure for efficient high-temperature operation of single-photon avalanche photodiodes Mahdi Zavvari,1,* Kambiz Abedi,2 and Mohammad Karimi3 1
Department of Electrical Engineering, Urmia Branch, Islamic Azad University, Urmia, Iran 2
3
Department of Electrical Engineering, Shahid Beheshti University, Tehran, Iran
Department of Electrical Engineering, Mahabad Branch, Islamic Azad University, Mahabad, Iran *Corresponding author: [email protected] Received 10 February 2014; revised 14 April 2014; accepted 14 April 2014; posted 15 April 2014 (Doc. ID 206174); published 20 May 2014
A novel design of a single-photon avalanche photodiode (SPAD) is proposed based on resonant cavity (RC) structure, and its performance is studied. In the proposed structure, InAlAs/InGaAs distributed Bragg reflectors (DBRs) are employed as top and bottom mirrors and the quantum efficiency (QE) of the absorption region is calculated considering the effect of the RC. Results show that using 12 periods of DBRs as a bottom reflector without incorporation of a top mirror can enhance the QE to about 90% at room temperature. For this RC-enhanced SPAD, a single-photon quantum efficiency (SPQE) is obtained of about 0.35 at T 300 K. For temperatures lower than T 260 K, SPQE is about 1. Results show that although the RC doesn’t affect the dark current, for a given SPQE the dark count rate is lower for the RC-SPAD. © 2014 Optical Society of America OCIS codes: (040.1345) Avalanche photodiodes (APDs); (040.5160) Photodetectors; (230.1480) Bragg reflectors; (040.3060) Infrared. http://dx.doi.org/10.1364/AO.53.003311
1. Introduction
Single-photon avalanche photodiodes (SPADs) are of much interest because of their significant role in photon counting applications including satellite laser ranging [1], deep-space laser communication [2], time-resolved photon counting [3], quantum key distribution [4], quantum imaging [5], and quantum cryptography [6]. In order to achieve single-photon detection in an avalanche photodiode, the electric field within the multiplication region should be increased to above breakdown. There are several operation modes of SPADs depending on the application of above-breakdown voltage and quenching of output [7,8]. In gated-mode operation, the photodiode is
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biased below breakdown and the excess voltage is applied as short-period pulses [9]. In the duration of the gate pulse any photogenerated carrier can result in high detectable output. Once the pulse goes down, the avalanche gain is reduced and the output is quenched [10]. In free-running mode the detector is biased above breakdown and quenching can be done either by active or passive circuits [11]. A self-quenched structure has also been proposed as another quenching mechanism in which the avalanche-generated carriers accumulate at the interface of the multiplication region and impose an additional charge that reduces the electric field and results in reduction of avalanche gain [12]. InGaAs SPADs are an interesting option for some applications such as chemical detection and laser ranging, because the 1.55 μm wavelength is used in optical fiber communication and is in the eye-safe range 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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[13,14]. Single-photon quantum efficiency (SPQE) of this type of SPADs is reported to be about 3%–5% at room temperature [15]. This means that to obtain higher efficiencies, it is necessary to lower the operation temperature, commonly by thermoelectric Peltier cooling and with increased cost. Although higher efficiencies can be achieved by increased absorption via employing a thick active region, this results in speed limitations of the detector and increases the complexity and cost of fabrication. In order to achieve room temperature operation and eliminate the need for a thick region, an alternative is to confine the optical field within the active region by using a resonant cavity (RC) structure. The proposed RC consists of top and bottom reflectors to enhance the optical field in the active region, leading to the creation of a large number of carriers as a result of enhanced absorption in the active region, and hence it is expected to have higher quantum efficiency (QE). Previously some groups have reported improvement in characteristics of photodetectors with distributed Bragg reflectors (DBRs) in their structures [16]. Lai et al. proposed a RC structure consisting of a DBR as the bottom mirror and a top mirror of subwavelength grating to enhance the performance of an InGaAs multiple-quantum-well-based photodetector [17]. Balram et al. enhanced the absorption of a germanium photodetector at wavelengths beyond the material’s direct absorption edge by utilizing a subwavelength planar metal–dielectric RC [18]. Detectivity of quantum dot IR photodetectors (QDIPs) with 8–12 times enhancement using a RC structure has been reported by Asano et al. [19]. In addition, Wang and Lin used GaAs∕Al2 O3 and Ge∕SiO2 as the bottom and top reflectors, respectively, of a RC to enhance the performance of a QDIP at a wavelength of 8 μm [20]. However, there is only one experimental report regarding the application of a RC to a SPAD for enhancement of its performance [21]. To our knowledge, there is no study about the design of a resonant-cavity-enhanced SPAD and its room temperature achievement. In this paper we study the application of such a RC structure to a SPAD and investigate its improved efficiency. The advantage of this RC-enhanced SPAD comes from its design with no increase in the length and thickness of the active region, which satisfies two contradictory properties: (1) high operation speed, which requires a short optical path (thin active region) and (2) high probability of photon absorption, which requires a long optical path. The paper is organized as follows. In Section 2, the modeling of the RC is presented and a RC structure is proposed for room temperature enhancement in the QE of the absorption layer. Geiger mode operation of the device is modeled in Section 3, and the results and discussion are in Section 4. The paper is concluded in Section 5. 2. Resonant Cavity Model
In order to increase the optical field within the active region of the photodetector, we propose a RC 3312
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structure consisting of DBRs. For the first step, we assume that the structure consists of DBRs in both the bottom and top, as shown in Fig. 1. In the figure the absorption region is based on an InGaAs material with an InP large bandgap multiplication region. We use periodic repetitions of InGaAs/InAlAs latticematched layers to serve as DBR mirrors. The DBR layers reflect the transmitted light back into the active region, which leads to an increase in absorption, and hence higher QE is achieved. To reach the maximum possible value of QE, higher reflection of the DBR is required. We utilize a transmission matrix method to calculate the DBR reflection considering the refractive index, wavelength, and number of DBR layers. Assuming the incident and reflected beams as the electric and magnetic field, respectively, the amplitudes of these fields can be related to each other by a 2 × 2 transfer matrix at the boundary of the two mirrors by the following equation [22]:
Ea Bb
i sin δ cos δ γ1 iγ 1 sin δ cos δ
Eb ; Bb
(1)
where δ is the phase difference and γ is the inductance. For the Nth layer of the DBR, transfer matrix method (TMM) can be written as
Ea Bb
MT
EN ; BN
M T M 1 M 2 M 3 …M N :
(2)
(3)
The transmission coefficient, r, is defined as r
γ 0 m11 γ 0 γ s m12 − m21 − γ s m22 ; γ 0 m11 γ 0 γ s m12 m21 γ s m22
(4)
where
Fig. 1. Schematic of RC-SPAD (left) with separated absorption, grading, charge, and multiplication structure of active region (right).
p γ j nj ε0 μ0 cos θj ;
δj
2π n d cos θj : λ0 j j
(5)
The calculated reflectivity for the DBR consisting of InGaAs/AlGaAs alloys as a function of the number of layers at a wavelength of 1550 nm is shown in Fig. 2. As can be seen, the reflection can reach near unity for 30 periods of InGaAs/InAlAs materials, but such a large period will inevitably increase the burden and reliability during the fabrication. Therefore, we choose 12 pairs for the DBR in our simulation, which is sufficient to reflect the predominant part of the optical field about 90%. Figure 3 shows the reflection spectrum as a function of wavelength for 12 pairs of DBR materials. It is evident that 90% of the optical field is reflected into the absorption region at 1550 nm, while the RC structure is not sensitive to other ranges of radiation. This means that the absorption probability of photons with a wavelength of 1550 nm can be considerably increased, and hence it is expected to have a higher QE. The QE of the photodetector considering the effect of the RC structure is defined as
Fig. 3. Reflectivity of the DBR as a function of wavelength for 12 pairs of InGaAs/InAlAs layers.
where Ega is the bandgap energy, which can be calculated at different temperatures by the Varshni method using the following equation [23]: Eg E0 −
aT 2 : Tb
(8)
(7)
Figure 4 shows the calculated QE with and without the RC structure versus temperature for different reflectivities of the top and bottom mirrors. It is evident that the QE is lower for the structure without a RC. Figure 4 reveals that a higher reflectivity of the top mirror is desired for low temperature operation. However at higher temperatures, a lower reflectivity of top mirror can enhance the reflectivity to near unity. This is illustrated in Fig. 5, where the QE is depicted as a function of R1 and for different temperatures. From this figure the required R1 for the maximum QE varies for different temperatures. For room temperature operation, R1 should be set at around 0.1. Since the air–semiconductor interface creates a reflectivity of about 0.3, it’s not practically achievable to have R1 0.1 by means of DBRs [24]. Therefore we design the RC top mirror with the
Fig. 2. Reflectivity of the DBR as a function of its period at 1550 nm wavelength.
Fig. 4. QE versus temperature for different values of top and bottom mirror reflections.
η
1R2 e−KT a
p 1−2 R1 R2 e−KT a cos2βLφ1 φ2 R1 R2 e−2KT a
×1−R1 1e−KT a ;
(6)
where K is the absorption coefficient, T a is the effective absorption region width, R1 and R2 are the reflectivities of the top and bottom reflectors, respectively, φ1 and φ2 are the phase, L is the cavity length, and β is the corresponding propagation constant. Obviously, the QE is maximized when 2βL φ1 φ2 2mπm 1; 2; 3…. K can be obtained from an empirical expression as [23] q K 22900 0.79987 − Ega cm−1 ;
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Fig. 5. QE of the detector as a function of reflectivity of the top mirror at different temperatures. (Inset shows the QE versus R2 at various temperatures when R1 0.3.)
minimum achievable reflectivity (R1 0.3) by using no DBRs in top layer. The inset of Fig. 5 shows that as R2 increases, the QE improves for all ranges of temperature. 3. Geiger Mode Operation
In order to evaluate the performance of the proposed SPAD, we study the dark count rate (DCR) and the SPQE as two important figures of merit of a SPAD. The dark count is defined as the generation of an output pulse when no photon is incident and is related to the dark current of the detector. We assume Poisson statistics for the detection of dark counts, and therefore the probability of the dark count is described as [25] Pd 1 − exp−Pa N d ;
N d N DM1 τ N DM2 τtr M 0 Pd
(10)
where τ is the gate pulse width. N DM1 corresponds to the average number of dark carriers generated in SPAD, which is given by [27] N DM1 N D;ABS N D;MUL ;
(11)
where N D;ABS and N D;MUL are the number of dark carriers generated in the absorption and multiplication regions, respectively, and correspond to the dark current of each region. The second term of Eq. (10) corresponds to dark carriers generated before the gate pulse. These 3314
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N th
ni τSRH
;
(12)
where ni is the intrinsic carrier density and τSRH is the lifetime of Shockley–Read–Hall process. The tunneling term of the dark current can originate from the band-to-band term (BBT) or trap-assisted term (TAT). The number of dark carriers generated by each of the BBT and TAT processes can be expressed as [23,29] N BBT
(9)
where Pa is the avalanche probability calculated by the dead space multiplication theory [26]. N d is the total number of dark counts originating from different mechanisms and can be expressed by
c expτ∕τd − 1 M 1 − c g expΔT∕τd − 1 c expτtr ∕τd − 1 ; M Pd 1 − c g expΔT∕τd − 1
carriers may remain in the multiplication region when the pulse arrives and hence are treated as seed carriers. N DM2 is calculated similarly to N DM1 for the DC bias voltage of the detector, which is a few volts (∼1 V) less than V br . M 0 is the avalanche gain at the DC bias voltage, and τtr is the transit time. The last two terms of Eq. (10) correspond to the number of detrapped carriers during the gate pulse and before it, respectively. c and M g are the ratio of the number of trapped carriers to the total number of carriers and the number of carriers created per avalanche, respectively, τd is the detrapping time, and ΔT is the period of the gate pulses. The dark carriers in SPAD are generated from different mechanisms including thermally generated and tunneling-generated bulk carriers. The thermal generation rate per unit volume is obtained from [28]
N TAT
2mr Eg
1∕2
−π2mr E3g 1∕2 q2 F 2 p exp ; (13) 4π 3 ℏ2 2 2ℏqF
3∕2 3∕2 −B1 EB1 B2 EB2 N T exp E (14) 3∕2 3∕2 ; −B1 EB1 −B2 EB2 N N v exp exp c E E 2mr 1∕2 q2 F 2 Eg 4π 3 ℏ2
respectively, where mr is the reduced effective mass, Eg is the bandgap energy of the avalanche region, F is the electric field, W is the avalanche region thickness, Rs is the junction parasitic leakage resistance, N T is the density of the traps, and N c and N v are the effective densities of the states in the conduction and valance bands, respectively. B1 πmlh ∕21∕2 ∕2qℏ and B2 πmc ∕21∕2 ∕2qℏ, with mc and mlh corresponding to the effective masses of the electron and the light hole, respectively, EB1 is the barrier height from the valance band to the trap, and EB2 is the barrier height from the trap to the conduction band. Here we consider EB1 aEg and EB2 1 − aEg , with a < 1. Based on the above-mentioned equations, the DCR can be described as [30] DCR Pa N d ∕τ:
(15)
On the other hand, the SPQE is another performance characteristic of a SPAD and can be expressed as [28]
Table 1.
Parameters Used in our Calculations
Parameters
Values
Gate pulse width Pulse repetition rate Geiger mode avalanche gain Av. no. of incident photons Refractive index of AlGaAs Refractive index of InAlAs AlGaAs thickness in DBR InGaAs thickness in DBR Detector area Density of traps Trap energy Detrapping time
SPQE
2 ns 500 kHz 108 0.3 3.6 3.23 107 nm 119 nm 2000 μm2 1015 cm−3 0.75Eg 200 ns
Pp − Pd ; P0
(16)
where Pp is the probability that a current pulse is triggered by either a photogenerated carrier or a dark carrier when the detector is under illumination and is calculated from [28] Pp 1 − exp−Pa N p :
(17)
N p is the total number of generated carriers given by N p N DM1 τ N DM2 τtr M 0 c expτ∕τd − 1 M 1 − c g expΔT∕τd − 1 c expτtr ∕τd − 1 ηN 0 ; M Pp 1 − c g expΔT∕τd − 1 Pp
(18)
where N 0 is the number of photons and η is the QE of the QD absorption region; the details of its calculation can be found in [31]. The term ηN 0 is added to the number of dark carriers to describe the total number of generated carriers. In Eq. (16) P0 1 − exp−N 0 is the probability that the incident beam contains at least one photon during the gate pulse. The parameters and their quantities used in our calculations are listed in Table 1.
Fig. 6. Calculated SPQE as a function of overbias for different temperatures.
about 0.35, which is comparable with the SPQE of previously reported conventional SPADs at lower temperatures, below T 220 K [23,28,32]. The reason is addressed in Fig. 7, which depicts the behavior of the two terms of probability used in the calculation of SPQE as a function of the normalized overbias and for two temperatures. 1 − Pd is the probability that a dark carrier does not trigger an avalanche breakdown, while Pp is the probability that an avalanche occurs with either a photogenerated or dark carrier. As can be seen in the figure, at T 260 K, the dark count probability is near zero. Hence the output is only generated by photon absorption, and consequently the SPQE reaches 1. For T 300 K and at lower voltages, the dark count probability is still lower, and it is expected to have optically generated output pulses. Although for higher biases the singlephoton detection probability increases, the dark count probability is also enhanced and therefore the SPQE decreases.
4. Results and Discussion
According to the results in Section 2, we calculate the performance characteristics of the proposed SPAD with DBR reflectors in only its bottom. Figure 6 shows the calculated SPQE of the RC-SPAD as a function of normalized applied overbias and for different temperatures. The normalized overbias is defined as ΔV∕V br V − V br ∕V br, where V br is the breakdown voltage, i.e., the applied voltage of the device needed to reach breakdown in the multiplication region, and V is the voltage across the device. According to the figure, SPQE reaches a value of about 1 for temperatures up to 260 K. This means that for such temperatures all of the incident photons can trigger an avalanche and generate a detectable output. Also in Fig. 6, for T 300 K, the SPQE takes a value of
Fig. 7. Calculated probabilities for proposed RC-SPAD at T 260 K and 300 K (solid lines show Pp and dashed lines show the probability of not having a dark count). 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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Fig. 8. SPQE versus normalized overbias for the SPAD with and without a RC at different temperatures.
For comparison, Fig. 8 shows the calculated SPQE for SPAD with and without DBR reflectors. As can be seen in the figure, a considerable improvement in SPQE can be achieved by adding a bottom reflector to the structure. Actually, the QE of the absorption layer increases for RC-SPAD, and hence the probability of triggering an avalanche with a photogenerated carrier is higher. As expected, the SPQE is lower for higher temperatures, because of the increased dark count probability. Figure 9 shows the calculated DCR of the detector as a function of normalized overbias and for different temperatures. It is evident that both the temperature and bias result in higher orders of DCR because of the strong dependence of the dark current generation rate on these parameters. For our structure we design the charge layer such that the electric field of the absorption region is low enough to have a lower population of dark carriers from the absorption region. Since the RC structure just confines the optical field, it doesn’t influence the DCR, and this quantity is identical for SPADs with and without RCs. However, based on Fig. 10, which depicts the DCR versus
Fig. 9. DCR versus normalized overbias for different temperatures. The results are the same both with and without a RC. 3316
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Fig. 10. DCR versus SPQE for the SPAD with and without a RC at different temperatures.
SPQE with and without a RC and for two temperatures, it is found that for a given SPQE, the DCR is lower for a RC-based SPAD. 5. Conclusions
We proposed the application of DBR layers as a bottom reflector of a SPAD structure for more confinement of the optical field within its active region and enhancement of the QE of the absorption layer. Simulation results show that considerable improvement in the SPQE is achieved, about 0.35 at T 300 K, which is comparable to the SPQE of conventional SPADs at temperatures below 220 K. Although the DBR layers don’t directly influence the DCR, for a given SPQE the DCR is lower for a RC-SPAD. References 1. M. Oh, H. Kong, T. Kim, S. Jo, B. Kim, and D. Park, “Development and analysis of a photon-counting three-dimensional imaging laser detection and ranging (LADAR) system,” J. Opt. Soc. Am. A 28, 759–765 (2011). 2. G. Smith, K. McIntosh, J. Donnelly, J. Funk, L. Mahoney, and S. Verghese, “Reliable InP-based Geiger-mode avalanche photodiode arrays,” Proc. SPIE 7320, 73200R (2009). 3. B. E. Kardynał, Z. L. Yuan, and A. J. Shields, “An avalanchephotodiode-based photon-number-resolving detector,” Nat. Photonics 2, 425–428 (2008). 4. D. Stucki, G. Ribordy, H. Z. A. Stefanov, and J. G. Rarity, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APD’s,” J. Mod. Opt. 48, 1967–1981 (2001). 5. M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Dispersion-cancelled and dispersion-sensitive quantum optical coherence tomography,” Opt. Express 12, 1353–1362 (2004). 6. H. Kosaka, “Single-photon interference experiment over 100 km for quantum cryptography system using balanced gated-mode photon detector,” Electron. Lett. 39, 1199–1201 (2003). 7. S. Cova, M. Ghioni, A. Lacaita, C. Samori, and F. Zappa, “Avalanche photodiodes and quenching circuits for singlephoton detection,” Appl. Opt. 35, 1956–1976 (1996). 8. M. Zavvari and V. Ahmadi, “Self quenched quantum dot avalanche photodetector for mid-infrared single photon detection,” Infrared Phys. Technol. 62, 7–12 (2014). 9. K. Sugihara, E. Yagyu, T. Nishioka, T. Kurata, M. Matsui, Y. Tokuda, and K. Itoh, “Analysis of single-photon-detection
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Department of Electrical Engineering, Urmia Branch, Islamic Azad University, Urmia, Iran 2
3
Department of Electrical Engineering, Shahid Beheshti University, Tehran, Iran
Department of Electrical Engineering, Mahabad Branch, Islamic Azad University, Mahabad, Iran *Corresponding author: [email protected] Received 10 February 2014; revised 14 April 2014; accepted 14 April 2014; posted 15 April 2014 (Doc. ID 206174); published 20 May 2014
A novel design of a single-photon avalanche photodiode (SPAD) is proposed based on resonant cavity (RC) structure, and its performance is studied. In the proposed structure, InAlAs/InGaAs distributed Bragg reflectors (DBRs) are employed as top and bottom mirrors and the quantum efficiency (QE) of the absorption region is calculated considering the effect of the RC. Results show that using 12 periods of DBRs as a bottom reflector without incorporation of a top mirror can enhance the QE to about 90% at room temperature. For this RC-enhanced SPAD, a single-photon quantum efficiency (SPQE) is obtained of about 0.35 at T 300 K. For temperatures lower than T 260 K, SPQE is about 1. Results show that although the RC doesn’t affect the dark current, for a given SPQE the dark count rate is lower for the RC-SPAD. © 2014 Optical Society of America OCIS codes: (040.1345) Avalanche photodiodes (APDs); (040.5160) Photodetectors; (230.1480) Bragg reflectors; (040.3060) Infrared. http://dx.doi.org/10.1364/AO.53.003311
1. Introduction
Single-photon avalanche photodiodes (SPADs) are of much interest because of their significant role in photon counting applications including satellite laser ranging [1], deep-space laser communication [2], time-resolved photon counting [3], quantum key distribution [4], quantum imaging [5], and quantum cryptography [6]. In order to achieve single-photon detection in an avalanche photodiode, the electric field within the multiplication region should be increased to above breakdown. There are several operation modes of SPADs depending on the application of above-breakdown voltage and quenching of output [7,8]. In gated-mode operation, the photodiode is
1559-128X/14/153311-07$15.00/0 © 2014 Optical Society of America
biased below breakdown and the excess voltage is applied as short-period pulses [9]. In the duration of the gate pulse any photogenerated carrier can result in high detectable output. Once the pulse goes down, the avalanche gain is reduced and the output is quenched [10]. In free-running mode the detector is biased above breakdown and quenching can be done either by active or passive circuits [11]. A self-quenched structure has also been proposed as another quenching mechanism in which the avalanche-generated carriers accumulate at the interface of the multiplication region and impose an additional charge that reduces the electric field and results in reduction of avalanche gain [12]. InGaAs SPADs are an interesting option for some applications such as chemical detection and laser ranging, because the 1.55 μm wavelength is used in optical fiber communication and is in the eye-safe range 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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[13,14]. Single-photon quantum efficiency (SPQE) of this type of SPADs is reported to be about 3%–5% at room temperature [15]. This means that to obtain higher efficiencies, it is necessary to lower the operation temperature, commonly by thermoelectric Peltier cooling and with increased cost. Although higher efficiencies can be achieved by increased absorption via employing a thick active region, this results in speed limitations of the detector and increases the complexity and cost of fabrication. In order to achieve room temperature operation and eliminate the need for a thick region, an alternative is to confine the optical field within the active region by using a resonant cavity (RC) structure. The proposed RC consists of top and bottom reflectors to enhance the optical field in the active region, leading to the creation of a large number of carriers as a result of enhanced absorption in the active region, and hence it is expected to have higher quantum efficiency (QE). Previously some groups have reported improvement in characteristics of photodetectors with distributed Bragg reflectors (DBRs) in their structures [16]. Lai et al. proposed a RC structure consisting of a DBR as the bottom mirror and a top mirror of subwavelength grating to enhance the performance of an InGaAs multiple-quantum-well-based photodetector [17]. Balram et al. enhanced the absorption of a germanium photodetector at wavelengths beyond the material’s direct absorption edge by utilizing a subwavelength planar metal–dielectric RC [18]. Detectivity of quantum dot IR photodetectors (QDIPs) with 8–12 times enhancement using a RC structure has been reported by Asano et al. [19]. In addition, Wang and Lin used GaAs∕Al2 O3 and Ge∕SiO2 as the bottom and top reflectors, respectively, of a RC to enhance the performance of a QDIP at a wavelength of 8 μm [20]. However, there is only one experimental report regarding the application of a RC to a SPAD for enhancement of its performance [21]. To our knowledge, there is no study about the design of a resonant-cavity-enhanced SPAD and its room temperature achievement. In this paper we study the application of such a RC structure to a SPAD and investigate its improved efficiency. The advantage of this RC-enhanced SPAD comes from its design with no increase in the length and thickness of the active region, which satisfies two contradictory properties: (1) high operation speed, which requires a short optical path (thin active region) and (2) high probability of photon absorption, which requires a long optical path. The paper is organized as follows. In Section 2, the modeling of the RC is presented and a RC structure is proposed for room temperature enhancement in the QE of the absorption layer. Geiger mode operation of the device is modeled in Section 3, and the results and discussion are in Section 4. The paper is concluded in Section 5. 2. Resonant Cavity Model
In order to increase the optical field within the active region of the photodetector, we propose a RC 3312
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structure consisting of DBRs. For the first step, we assume that the structure consists of DBRs in both the bottom and top, as shown in Fig. 1. In the figure the absorption region is based on an InGaAs material with an InP large bandgap multiplication region. We use periodic repetitions of InGaAs/InAlAs latticematched layers to serve as DBR mirrors. The DBR layers reflect the transmitted light back into the active region, which leads to an increase in absorption, and hence higher QE is achieved. To reach the maximum possible value of QE, higher reflection of the DBR is required. We utilize a transmission matrix method to calculate the DBR reflection considering the refractive index, wavelength, and number of DBR layers. Assuming the incident and reflected beams as the electric and magnetic field, respectively, the amplitudes of these fields can be related to each other by a 2 × 2 transfer matrix at the boundary of the two mirrors by the following equation [22]:
Ea Bb
i sin δ cos δ γ1 iγ 1 sin δ cos δ
Eb ; Bb
(1)
where δ is the phase difference and γ is the inductance. For the Nth layer of the DBR, transfer matrix method (TMM) can be written as
Ea Bb
MT
EN ; BN
M T M 1 M 2 M 3 …M N :
(2)
(3)
The transmission coefficient, r, is defined as r
γ 0 m11 γ 0 γ s m12 − m21 − γ s m22 ; γ 0 m11 γ 0 γ s m12 m21 γ s m22
(4)
where
Fig. 1. Schematic of RC-SPAD (left) with separated absorption, grading, charge, and multiplication structure of active region (right).
p γ j nj ε0 μ0 cos θj ;
δj
2π n d cos θj : λ0 j j
(5)
The calculated reflectivity for the DBR consisting of InGaAs/AlGaAs alloys as a function of the number of layers at a wavelength of 1550 nm is shown in Fig. 2. As can be seen, the reflection can reach near unity for 30 periods of InGaAs/InAlAs materials, but such a large period will inevitably increase the burden and reliability during the fabrication. Therefore, we choose 12 pairs for the DBR in our simulation, which is sufficient to reflect the predominant part of the optical field about 90%. Figure 3 shows the reflection spectrum as a function of wavelength for 12 pairs of DBR materials. It is evident that 90% of the optical field is reflected into the absorption region at 1550 nm, while the RC structure is not sensitive to other ranges of radiation. This means that the absorption probability of photons with a wavelength of 1550 nm can be considerably increased, and hence it is expected to have a higher QE. The QE of the photodetector considering the effect of the RC structure is defined as
Fig. 3. Reflectivity of the DBR as a function of wavelength for 12 pairs of InGaAs/InAlAs layers.
where Ega is the bandgap energy, which can be calculated at different temperatures by the Varshni method using the following equation [23]: Eg E0 −
aT 2 : Tb
(8)
(7)
Figure 4 shows the calculated QE with and without the RC structure versus temperature for different reflectivities of the top and bottom mirrors. It is evident that the QE is lower for the structure without a RC. Figure 4 reveals that a higher reflectivity of the top mirror is desired for low temperature operation. However at higher temperatures, a lower reflectivity of top mirror can enhance the reflectivity to near unity. This is illustrated in Fig. 5, where the QE is depicted as a function of R1 and for different temperatures. From this figure the required R1 for the maximum QE varies for different temperatures. For room temperature operation, R1 should be set at around 0.1. Since the air–semiconductor interface creates a reflectivity of about 0.3, it’s not practically achievable to have R1 0.1 by means of DBRs [24]. Therefore we design the RC top mirror with the
Fig. 2. Reflectivity of the DBR as a function of its period at 1550 nm wavelength.
Fig. 4. QE versus temperature for different values of top and bottom mirror reflections.
η
1R2 e−KT a
p 1−2 R1 R2 e−KT a cos2βLφ1 φ2 R1 R2 e−2KT a
×1−R1 1e−KT a ;
(6)
where K is the absorption coefficient, T a is the effective absorption region width, R1 and R2 are the reflectivities of the top and bottom reflectors, respectively, φ1 and φ2 are the phase, L is the cavity length, and β is the corresponding propagation constant. Obviously, the QE is maximized when 2βL φ1 φ2 2mπm 1; 2; 3…. K can be obtained from an empirical expression as [23] q K 22900 0.79987 − Ega cm−1 ;
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Fig. 5. QE of the detector as a function of reflectivity of the top mirror at different temperatures. (Inset shows the QE versus R2 at various temperatures when R1 0.3.)
minimum achievable reflectivity (R1 0.3) by using no DBRs in top layer. The inset of Fig. 5 shows that as R2 increases, the QE improves for all ranges of temperature. 3. Geiger Mode Operation
In order to evaluate the performance of the proposed SPAD, we study the dark count rate (DCR) and the SPQE as two important figures of merit of a SPAD. The dark count is defined as the generation of an output pulse when no photon is incident and is related to the dark current of the detector. We assume Poisson statistics for the detection of dark counts, and therefore the probability of the dark count is described as [25] Pd 1 − exp−Pa N d ;
N d N DM1 τ N DM2 τtr M 0 Pd
(10)
where τ is the gate pulse width. N DM1 corresponds to the average number of dark carriers generated in SPAD, which is given by [27] N DM1 N D;ABS N D;MUL ;
(11)
where N D;ABS and N D;MUL are the number of dark carriers generated in the absorption and multiplication regions, respectively, and correspond to the dark current of each region. The second term of Eq. (10) corresponds to dark carriers generated before the gate pulse. These 3314
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N th
ni τSRH
;
(12)
where ni is the intrinsic carrier density and τSRH is the lifetime of Shockley–Read–Hall process. The tunneling term of the dark current can originate from the band-to-band term (BBT) or trap-assisted term (TAT). The number of dark carriers generated by each of the BBT and TAT processes can be expressed as [23,29] N BBT
(9)
where Pa is the avalanche probability calculated by the dead space multiplication theory [26]. N d is the total number of dark counts originating from different mechanisms and can be expressed by
c expτ∕τd − 1 M 1 − c g expΔT∕τd − 1 c expτtr ∕τd − 1 ; M Pd 1 − c g expΔT∕τd − 1
carriers may remain in the multiplication region when the pulse arrives and hence are treated as seed carriers. N DM2 is calculated similarly to N DM1 for the DC bias voltage of the detector, which is a few volts (∼1 V) less than V br . M 0 is the avalanche gain at the DC bias voltage, and τtr is the transit time. The last two terms of Eq. (10) correspond to the number of detrapped carriers during the gate pulse and before it, respectively. c and M g are the ratio of the number of trapped carriers to the total number of carriers and the number of carriers created per avalanche, respectively, τd is the detrapping time, and ΔT is the period of the gate pulses. The dark carriers in SPAD are generated from different mechanisms including thermally generated and tunneling-generated bulk carriers. The thermal generation rate per unit volume is obtained from [28]
N TAT
2mr Eg
1∕2
−π2mr E3g 1∕2 q2 F 2 p exp ; (13) 4π 3 ℏ2 2 2ℏqF
3∕2 3∕2 −B1 EB1 B2 EB2 N T exp E (14) 3∕2 3∕2 ; −B1 EB1 −B2 EB2 N N v exp exp c E E 2mr 1∕2 q2 F 2 Eg 4π 3 ℏ2
respectively, where mr is the reduced effective mass, Eg is the bandgap energy of the avalanche region, F is the electric field, W is the avalanche region thickness, Rs is the junction parasitic leakage resistance, N T is the density of the traps, and N c and N v are the effective densities of the states in the conduction and valance bands, respectively. B1 πmlh ∕21∕2 ∕2qℏ and B2 πmc ∕21∕2 ∕2qℏ, with mc and mlh corresponding to the effective masses of the electron and the light hole, respectively, EB1 is the barrier height from the valance band to the trap, and EB2 is the barrier height from the trap to the conduction band. Here we consider EB1 aEg and EB2 1 − aEg , with a < 1. Based on the above-mentioned equations, the DCR can be described as [30] DCR Pa N d ∕τ:
(15)
On the other hand, the SPQE is another performance characteristic of a SPAD and can be expressed as [28]
Table 1.
Parameters Used in our Calculations
Parameters
Values
Gate pulse width Pulse repetition rate Geiger mode avalanche gain Av. no. of incident photons Refractive index of AlGaAs Refractive index of InAlAs AlGaAs thickness in DBR InGaAs thickness in DBR Detector area Density of traps Trap energy Detrapping time
SPQE
2 ns 500 kHz 108 0.3 3.6 3.23 107 nm 119 nm 2000 μm2 1015 cm−3 0.75Eg 200 ns
Pp − Pd ; P0
(16)
where Pp is the probability that a current pulse is triggered by either a photogenerated carrier or a dark carrier when the detector is under illumination and is calculated from [28] Pp 1 − exp−Pa N p :
(17)
N p is the total number of generated carriers given by N p N DM1 τ N DM2 τtr M 0 c expτ∕τd − 1 M 1 − c g expΔT∕τd − 1 c expτtr ∕τd − 1 ηN 0 ; M Pp 1 − c g expΔT∕τd − 1 Pp
(18)
where N 0 is the number of photons and η is the QE of the QD absorption region; the details of its calculation can be found in [31]. The term ηN 0 is added to the number of dark carriers to describe the total number of generated carriers. In Eq. (16) P0 1 − exp−N 0 is the probability that the incident beam contains at least one photon during the gate pulse. The parameters and their quantities used in our calculations are listed in Table 1.
Fig. 6. Calculated SPQE as a function of overbias for different temperatures.
about 0.35, which is comparable with the SPQE of previously reported conventional SPADs at lower temperatures, below T 220 K [23,28,32]. The reason is addressed in Fig. 7, which depicts the behavior of the two terms of probability used in the calculation of SPQE as a function of the normalized overbias and for two temperatures. 1 − Pd is the probability that a dark carrier does not trigger an avalanche breakdown, while Pp is the probability that an avalanche occurs with either a photogenerated or dark carrier. As can be seen in the figure, at T 260 K, the dark count probability is near zero. Hence the output is only generated by photon absorption, and consequently the SPQE reaches 1. For T 300 K and at lower voltages, the dark count probability is still lower, and it is expected to have optically generated output pulses. Although for higher biases the singlephoton detection probability increases, the dark count probability is also enhanced and therefore the SPQE decreases.
4. Results and Discussion
According to the results in Section 2, we calculate the performance characteristics of the proposed SPAD with DBR reflectors in only its bottom. Figure 6 shows the calculated SPQE of the RC-SPAD as a function of normalized applied overbias and for different temperatures. The normalized overbias is defined as ΔV∕V br V − V br ∕V br, where V br is the breakdown voltage, i.e., the applied voltage of the device needed to reach breakdown in the multiplication region, and V is the voltage across the device. According to the figure, SPQE reaches a value of about 1 for temperatures up to 260 K. This means that for such temperatures all of the incident photons can trigger an avalanche and generate a detectable output. Also in Fig. 6, for T 300 K, the SPQE takes a value of
Fig. 7. Calculated probabilities for proposed RC-SPAD at T 260 K and 300 K (solid lines show Pp and dashed lines show the probability of not having a dark count). 20 May 2014 / Vol. 53, No. 15 / APPLIED OPTICS
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Fig. 8. SPQE versus normalized overbias for the SPAD with and without a RC at different temperatures.
For comparison, Fig. 8 shows the calculated SPQE for SPAD with and without DBR reflectors. As can be seen in the figure, a considerable improvement in SPQE can be achieved by adding a bottom reflector to the structure. Actually, the QE of the absorption layer increases for RC-SPAD, and hence the probability of triggering an avalanche with a photogenerated carrier is higher. As expected, the SPQE is lower for higher temperatures, because of the increased dark count probability. Figure 9 shows the calculated DCR of the detector as a function of normalized overbias and for different temperatures. It is evident that both the temperature and bias result in higher orders of DCR because of the strong dependence of the dark current generation rate on these parameters. For our structure we design the charge layer such that the electric field of the absorption region is low enough to have a lower population of dark carriers from the absorption region. Since the RC structure just confines the optical field, it doesn’t influence the DCR, and this quantity is identical for SPADs with and without RCs. However, based on Fig. 10, which depicts the DCR versus
Fig. 9. DCR versus normalized overbias for different temperatures. The results are the same both with and without a RC. 3316
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Fig. 10. DCR versus SPQE for the SPAD with and without a RC at different temperatures.
SPQE with and without a RC and for two temperatures, it is found that for a given SPQE, the DCR is lower for a RC-based SPAD. 5. Conclusions
We proposed the application of DBR layers as a bottom reflector of a SPAD structure for more confinement of the optical field within its active region and enhancement of the QE of the absorption layer. Simulation results show that considerable improvement in the SPQE is achieved, about 0.35 at T 300 K, which is comparable to the SPQE of conventional SPADs at temperatures below 220 K. Although the DBR layers don’t directly influence the DCR, for a given SPQE the DCR is lower for a RC-SPAD. References 1. M. Oh, H. Kong, T. Kim, S. Jo, B. Kim, and D. Park, “Development and analysis of a photon-counting three-dimensional imaging laser detection and ranging (LADAR) system,” J. Opt. Soc. Am. A 28, 759–765 (2011). 2. G. Smith, K. McIntosh, J. Donnelly, J. Funk, L. Mahoney, and S. Verghese, “Reliable InP-based Geiger-mode avalanche photodiode arrays,” Proc. SPIE 7320, 73200R (2009). 3. B. E. Kardynał, Z. L. Yuan, and A. J. Shields, “An avalanchephotodiode-based photon-number-resolving detector,” Nat. Photonics 2, 425–428 (2008). 4. D. Stucki, G. Ribordy, H. Z. A. Stefanov, and J. G. Rarity, “Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APD’s,” J. Mod. Opt. 48, 1967–1981 (2001). 5. M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Dispersion-cancelled and dispersion-sensitive quantum optical coherence tomography,” Opt. Express 12, 1353–1362 (2004). 6. H. Kosaka, “Single-photon interference experiment over 100 km for quantum cryptography system using balanced gated-mode photon detector,” Electron. Lett. 39, 1199–1201 (2003). 7. S. Cova, M. Ghioni, A. Lacaita, C. Samori, and F. Zappa, “Avalanche photodiodes and quenching circuits for singlephoton detection,” Appl. Opt. 35, 1956–1976 (1996). 8. M. Zavvari and V. Ahmadi, “Self quenched quantum dot avalanche photodetector for mid-infrared single photon detection,” Infrared Phys. Technol. 62, 7–12 (2014). 9. K. Sugihara, E. Yagyu, T. Nishioka, T. Kurata, M. Matsui, Y. Tokuda, and K. Itoh, “Analysis of single-photon-detection
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