**Silicon Photodetectors Based on Internal Photoemission Effect: The Challenge of Detecting Near-Infrared Light**

Maurizio Casalino, Luigi Sirleto, Mario Iodice and Giuseppe Coppola *Istituto per la Microelettronica e Microsistemi, Consiglio Nazionale delle Ricerche, Naples Italy* 

#### **1. Introduction**

50 Photodetectors

Yu, Z., M. Aceves-Mijares, K. Monfil, R. Kiebach, R. Lopez-Estopier, J. Carrillo, Room

embedded in oxide films, J. Appl. Phys., 2008, 103, 063706.

Science Publishers, Inc., 2008, 233.

Focus on Nanomaterials Research, Editor: B. M. Carota, ISBN 1-59454-897-8 Nova

temperature current oscillations in naturally grown silicon nanocrystallites

Silicon Photonics has emerged as an interesting field due to its potential for low-cost optical components integrated with electronic functionality. In the past two decades, there has been growing interest in photonic devices based on Si-compatible materials (Kimerling et al., 2004; Jalali & Fathpour, 2006) in the field both of the optical telecommunications and of the optical interconnects. In this contest, tremendous progresses in the technological processes based on the use silicon-on insulator (SOI) substrates have allowed to obtain reliable and effectiveness full complementary metal-oxide semiconductor (CMOS) compatible optical components such as, low loss waveguides, high-Q resonators, high speed modulators, couplers, and optically pumped lasers (Rowe et al., 2007 ; Vivien et al., 2006 ; Xu et al., 2007 ; Michael et al., 2007; Liu et al., 2007 ; Liu et al., 2006). All these devices have been developed to operate in the wavelength range from C optical band (1528–1561 nm) to L optical band (1561–1620 nm). However one of the crucial steps toward the integration of photonics with electronics resides in the development of efficient chip-scale photodetectors (PD) integrated on Si. Bulk photodetectors are perhaps the oldest and best understood silicon optoelectronic devices. Commercial products in Si operate at wavelengths below 1100 nm, where band-toband absorption occurs. For the realization of photodiodes integrated in photonics circuits operating at wavelengths beyond 1100 nm silicon is not the right material because its transparency. In the last years, in order to take advantage of low-cost standard Si-CMOS processing technology, a number of photodetectors have been proposed based on different physical effects, such as: defect-state absorption (Bradley et al., 2005), two photon absorption (TPA) (Liang et al., 2002) and internal photoemission absorption (Zhu et al., 2008a). Physical effects, working principles, main structures reported in literature and the most significant results obtained in recent years were reviewed and discussed in our previous paper (Casalino et al., 2010a). In this paragraph, we go into more depth on photodetectors based on the internal photoemission effect (IPE). Silicon infrared photodiodes based on IPE are not novel, in fact PtSi/p-Si, Ir/p-Si and Pd2Si/p-Si junctions are usually used in the infrared imaging systems (Kosonocky et al., 1985). The main advantages of these devices resides in their extremely high switching speed and in their simple fabrication process, but, due to high background current density these devices can only work at cryogenic temperature.

Silicon Photodetectors Based on Internal

**2.1.1 Standard fowler's theory of IPE** 

the photocurrent, is given by:

*F*

(Fig. 1).

strictly linked to a device′s quantum efficiency by the formula:

the design and realization of the amplifier circuitry that follows.

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 53

indicating the current produced (Iph) by a certain optical power (POPT). Responsivity is

*I nm*

Reasonable responsivities are necessary for an acceptable signal-to-noise ratio and to ease

IPE is the optical excitation of electrons into the metal to an energy above the Schottky barrier and then transport of these electrons to the conduction band of the semiconductor

Fig. 1. Energy band diagram for a metal/n-semiconductor junction. "Reprinted with permission from M. Casalino *et al.*, "A silicon compatible resonant cavity enhanced photodetector working at 1.55 μm," Semicond. Sci. Technol., 23, 075001, 2008 (doi:

The standard theory of photoemission from a metal into the vacuum is due to Fowler (Fowler, 1931). In a gas of electrons obeying the Fermi-Dirac statistic, if photon energy is close to potential barrier (hν≈ФB), the fraction (Fe) of the absorbed photons, which produce photoelectrons with the appropriate energy and momenta before scattering to contribute to

<sup>0</sup>

( ) 2( ) <sup>3</sup>

*kT <sup>h</sup> kT e*

*kTE e*

8 log 1

*F*

2 2

where hν is photons energy, ФB0 is the potential barrier at zero bias, ΔФB is the lowering due to image force effect (as we will see later), EF is the metal Fermi level, k is the Boltzmann's

0

*kT*

( )

*B B*

 

2 ( )

*h*

*B B*

 

(3)

*kT*

10.1088/0268-1242/23/7/075001). IOP Publishing is acknowledged."

0

*B B e h*

  *ph OPT*

*P*

[ ] 1242

(2)

However, in recent years IPE has emerged as a new option for detecting also near infrared (NIR) wavelengths at room temperature. Unfortunately, the photoemission quantum efficiency is low compared to that of detectors based on inter-band absorption and this limits the application both in power monitoring and in the telecommunication field. Low quantum efficiency is a direct result of conservation of momentum during carrier emission over the potential barrier, in fact, the majority of excited carriers which do not have enough momentum normal to the barrier are reflected and not emitted. Moreover as incoming photons can excite carriers lying in states far below the Fermi energy, which can not overcome over the metal-semiconductor potential barrier, the quantum efficiency of these devices is further decreased. In recent years new approaches and structures are proposed in order to circumvent these limitations.

In this paragraph an overview of the state of the art of NIR all-Si photodetectors based on IPE is presented. First, the physical effects of IPE and the main figures of merit of devices based on IPE are elucidated. Then, the main structures reported in literature, starting from historical devices for imaging application at infrared wavelengths, up to new NIR devices which could be adapted for telecommunications and power monitoring, are described in detail. Finally the most significant results obtained in the last years are reviewed and discussed comparing the performances of devices based on different approaches.

#### **2. IPE-based device performances**

We hereafter present a theoretical background useful to clarify the physics behind the working principle of devices and we analyze crucial points affecting device performances.

#### **2.1 Device efficiency and IPE theory**

IPE is the optical excitation of electrons in the metal to energy above the Schottky barrier and then transport of these electrons to the conduction band of the semiconductor. The standard IPE theory is due to Fowler (Fowler, 1931). However, the Fowler's theory was obtained without taking into account the thickness of the Schottky metal layer. In 1973, Archer and Cohen (Archer & Cohen, 1973) proposed thinning the electrode to increase the emission probability. The enhancement of IPE in thin metal film was theoretically investigated by Vicker who introduce a moltiplicative factor to the Fowler's formula (Vickers, 1971). However the resulting electrode was so thin that it was semitransparent producing low metal absorbance. More recently Casalino et al. proposed to improve the absorbance in NIR photodetector based on IPE by using a microcavity Fabry-Perot (Casalino et al., 2008a, 2010b). Finally a further enhancement of IPE can be obtained due to image force effect which modify the Schottky barrier under a reverse bias favouring IPE. This effect can be taken into account by adding a barrier collection efficiency term. The resulting quantum efficiency of a photodetector based on IPE, can be written by the formula (Casalino et al., 2008b):

$$
\eta = A\_T F\_e P\_E \eta\_c \tag{1}
$$

where AT is the total optical absorbance of the metal, Fe is the Fowler' factor, PE is the Vicker's factor and ηc is the bias dependent barrier collection efficiency. Very often, device efficiency is described in term of responsivity, a very important property of a detector indicating the current produced (Iph) by a certain optical power (POPT). Responsivity is strictly linked to a device′s quantum efficiency by the formula:

$$\Re = \frac{I\_{ph}}{P\_{OPT}} = \frac{\mathcal{A}[nm]}{1242}\eta \tag{2}$$

Reasonable responsivities are necessary for an acceptable signal-to-noise ratio and to ease the design and realization of the amplifier circuitry that follows.

#### **2.1.1 Standard fowler's theory of IPE**

52 Photodetectors

However, in recent years IPE has emerged as a new option for detecting also near infrared (NIR) wavelengths at room temperature. Unfortunately, the photoemission quantum efficiency is low compared to that of detectors based on inter-band absorption and this limits the application both in power monitoring and in the telecommunication field. Low quantum efficiency is a direct result of conservation of momentum during carrier emission over the potential barrier, in fact, the majority of excited carriers which do not have enough momentum normal to the barrier are reflected and not emitted. Moreover as incoming photons can excite carriers lying in states far below the Fermi energy, which can not overcome over the metal-semiconductor potential barrier, the quantum efficiency of these devices is further decreased. In recent years new approaches and structures are proposed in

In this paragraph an overview of the state of the art of NIR all-Si photodetectors based on IPE is presented. First, the physical effects of IPE and the main figures of merit of devices based on IPE are elucidated. Then, the main structures reported in literature, starting from historical devices for imaging application at infrared wavelengths, up to new NIR devices which could be adapted for telecommunications and power monitoring, are described in detail. Finally the most significant results obtained in the last years are reviewed and

We hereafter present a theoretical background useful to clarify the physics behind the working principle of devices and we analyze crucial points affecting device performances.

IPE is the optical excitation of electrons in the metal to energy above the Schottky barrier and then transport of these electrons to the conduction band of the semiconductor. The standard IPE theory is due to Fowler (Fowler, 1931). However, the Fowler's theory was obtained without taking into account the thickness of the Schottky metal layer. In 1973, Archer and Cohen (Archer & Cohen, 1973) proposed thinning the electrode to increase the emission probability. The enhancement of IPE in thin metal film was theoretically investigated by Vicker who introduce a moltiplicative factor to the Fowler's formula (Vickers, 1971). However the resulting electrode was so thin that it was semitransparent producing low metal absorbance. More recently Casalino et al. proposed to improve the absorbance in NIR photodetector based on IPE by using a microcavity Fabry-Perot (Casalino et al., 2008a, 2010b). Finally a further enhancement of IPE can be obtained due to image force effect which modify the Schottky barrier under a reverse bias favouring IPE. This effect can be taken into account by adding a barrier collection efficiency term. The resulting quantum efficiency of a photodetector based on IPE, can be written by the

discussed comparing the performances of devices based on different approaches.

*ATeEc F P*

where AT is the total optical absorbance of the metal, Fe is the Fowler' factor, PE is the Vicker's factor and ηc is the bias dependent barrier collection efficiency. Very often, device efficiency is described in term of responsivity, a very important property of a detector

(1)

order to circumvent these limitations.

**2. IPE-based device performances** 

**2.1 Device efficiency and IPE theory** 

formula (Casalino et al., 2008b):

IPE is the optical excitation of electrons into the metal to an energy above the Schottky barrier and then transport of these electrons to the conduction band of the semiconductor (Fig. 1).

Fig. 1. Energy band diagram for a metal/n-semiconductor junction. "Reprinted with permission from M. Casalino *et al.*, "A silicon compatible resonant cavity enhanced photodetector working at 1.55 μm," Semicond. Sci. Technol., 23, 075001, 2008 (doi: 10.1088/0268-1242/23/7/075001). IOP Publishing is acknowledged."

The standard theory of photoemission from a metal into the vacuum is due to Fowler (Fowler, 1931). In a gas of electrons obeying the Fermi-Dirac statistic, if photon energy is close to potential barrier (hν≈ФB), the fraction (Fe) of the absorbed photons, which produce photoelectrons with the appropriate energy and momenta before scattering to contribute to the photocurrent, is given by:

$$F\_e = \frac{\left[\left(h\nu - \left(\phi\_{\rm B0} - \Delta\phi\_{\rm B}\right)\right)^2 + \frac{\left(kT\pi\right)^2}{3} - 2\left(kT\right)^2 e^{-\frac{\ln\nu - \left(\phi\_{\rm B0} - \Delta\phi\_{\rm B}\right)}{kT}}\right]}{8kTE\_F \log\left[1 + e^{\frac{\ln\nu - \left(\phi\_{\rm B0} - \Delta\phi\_{\rm B}\right)}{kT}}\right]}\tag{3}$$

where hν is photons energy, ФB0 is the potential barrier at zero bias, ΔФB is the lowering due to image force effect (as we will see later), EF is the metal Fermi level, k is the Boltzmann's

Silicon Photodetectors Based on Internal

considered for thin metal protection purpose.

Fig. 3. Sketch of IPE photodetector based on optical cavity.

Carballar, 1997).

**Layer 1**

**Einc**

**ER**

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 55

thin metal film, the use of a microcavity Fabry-Perot has been proposed. The sketch of proposed photodetector is shown in Fig. 3. The resonant cavity is a Fabry–Perot vertical tothe-surface structure. It is formed by a distributed Bragg reflector (DBR), a metallic top mirror and in between a Si cavity. A dielectric layer on top of the metal is generally

The proposed structure of Fig. 4 can be modelled by the multilayer shown in Fig. 3 and absorbance calculation can be carried out by the Transfer Matrix Method (TMM) (Muriel &

**n1=1 n2 n3 n4 n5 n6=3.45**

Normal incidence condition and the unidimensional variations of refractive index (n) along the propagation direction z, are taken into account. Let Einc and ER be the electric field complex amplitude of the incident and reflected waves at the interface between the air and the dielectric

Fig. 4. Multilayer schematizing device in fig. 3 for absorbance calculation by TMM.

layer, and ET the waves transmitted by final plane of structure, by TMM we obtain:

**Layer 2 Layer 3 Layer 4 Layer 5 Layer 6**

**E2,B E4,B**

**Dielectric layer Metal Silicon cavity**

**<sup>E</sup> ET 2,F E4,F**

**D B R** **z**

constant and T is the absolute temperature. As it is possible to see in Eq. 3, Fe is strongly depending from the potential barrier height of the metal-semiconductor interface.

#### **2.1.2 IPE enhancement in thin metal films**

In order to study the quantum efficiency for thin metal films, the theory must be further extended, taking into account multiple reflections of the excited electrons from the surfaces of the metals film, in addition to collisions with phonons, imperfections and cold electrons. Assuming a thin metal film, a phenomenological, semiclassical, ballistic transport model for the effects of the scattering mechanisms resulting in a multiplicative factor for quantum efficiency was developed by Vickers (Vickers, 1971) and recently reviewed by Scales *et al.* (Scales & Berini, 2010). According to this model the accumulated probability PE that the electrons will have sufficient normal kinetic energy to overcome potential barrier is given by:

$$P\_E \cong \frac{L\_e}{d} \left[ 1 - e^{-\frac{d}{L\_e}} \right]^{\frac{1}{2}} \tag{4}$$

where d is the metal thickness and Le the metal mean free path. For example, as it can be seen by plotting Eq. 4 in Fig. 2 (assuming a copper mean free path of 45 nm (Chan et al., 1980a), the lower metal thickness, the higher PE.

Fig. 2. Vicker's factor PE versus metal thickness for copper metal.

It is worth noting that in a recent work (Scales & Berini, 2010), Scales and Berini show that a further enhancement of this probability emission can be obtained in structures realized with thin metal film buried in a semiconductor and forming two Schottky barriers.

#### **2.1.3 IPE enhancement by optical cavity**

The enhancement of IPE in thin metal film lead to a resulting electrode so thin to be semitransparent, producing low metal absorbance. In order to increase the absorbance in

constant and T is the absolute temperature. As it is possible to see in Eq. 3, Fe is strongly

In order to study the quantum efficiency for thin metal films, the theory must be further extended, taking into account multiple reflections of the excited electrons from the surfaces of the metals film, in addition to collisions with phonons, imperfections and cold electrons. Assuming a thin metal film, a phenomenological, semiclassical, ballistic transport model for the effects of the scattering mechanisms resulting in a multiplicative factor for quantum efficiency was developed by Vickers (Vickers, 1971) and recently reviewed by Scales *et al.* (Scales & Berini, 2010). According to this model the accumulated probability PE that the electrons will have sufficient normal kinetic energy to overcome potential barrier is given by:

> 1 2

> > (4)

1 *<sup>e</sup> d e L*

where d is the metal thickness and Le the metal mean free path. For example, as it can be seen by plotting Eq. 4 in Fig. 2 (assuming a copper mean free path of 45 nm (Chan et al.,

**<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>0</sup>**

**Metal thickness [nm]**

It is worth noting that in a recent work (Scales & Berini, 2010), Scales and Berini show that a further enhancement of this probability emission can be obtained in structures realized with

The enhancement of IPE in thin metal film lead to a resulting electrode so thin to be semitransparent, producing low metal absorbance. In order to increase the absorbance in

 

depending from the potential barrier height of the metal-semiconductor interface.

*E*

*<sup>L</sup> P e d*

**2.1.2 IPE enhancement in thin metal films** 

1980a), the lower metal thickness, the higher PE.

**2.1.3 IPE enhancement by optical cavity** 

Fig. 2. Vicker's factor PE versus metal thickness for copper metal.

thin metal film buried in a semiconductor and forming two Schottky barriers.

**Vicker's Factor [adimensional]**

thin metal film, the use of a microcavity Fabry-Perot has been proposed. The sketch of proposed photodetector is shown in Fig. 3. The resonant cavity is a Fabry–Perot vertical tothe-surface structure. It is formed by a distributed Bragg reflector (DBR), a metallic top mirror and in between a Si cavity. A dielectric layer on top of the metal is generally considered for thin metal protection purpose.

Fig. 3. Sketch of IPE photodetector based on optical cavity.

The proposed structure of Fig. 4 can be modelled by the multilayer shown in Fig. 3 and absorbance calculation can be carried out by the Transfer Matrix Method (TMM) (Muriel & Carballar, 1997).

Fig. 4. Multilayer schematizing device in fig. 3 for absorbance calculation by TMM.

Normal incidence condition and the unidimensional variations of refractive index (n) along the propagation direction z, are taken into account. Let Einc and ER be the electric field complex amplitude of the incident and reflected waves at the interface between the air and the dielectric layer, and ET the waves transmitted by final plane of structure, by TMM we obtain:

Silicon Photodetectors Based on Internal

*T*

magnitude at 1550 nm.

et al., 2006a, 2006b).

**0.1**

**0.2**

**0.3**

**0.4**

**0.5**

**Absorbance**

**0.6**

**0.7**

**0.8**

**0.9**

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 57

0 0 0

2 2 2

*n n M M E*

0 0 0

24 24

2 2 2

*n n M M E*

2, 4, 2 4

2, 4, 2 4

*input output AB AB*

*P MM MM* 

In a work of 2006, Casalino *et al.* (Casalino et al., 2006a, 2006b) proposed a methodology in order to design an optimum device, in fact with a right choice of the multilayer thicknesses a maximum in absorbance can be obtained. In the device proposed by the authors DBR is considered formed by alternate layers of Si and SiO2 having refractive index 3.45 and 1.45, and thickness of 340 nm and 270 nm, respectively, while a metallic layer of gold, having refractive index nAu=0.174+j9.96 (Chan & Card, 1980b), mean free path Le=55 nm (Chan et al., 1980a), Fermi level EF=5.53 eV (Yeh, 1988) and thickness d=32 nm, was chosen as Schottky contact. We point out that in the simulations, semi-infinite first and last layers of air (n1=1) and silicon (n6=3.45), respectively, are considered. In Fig. 5 absorbance plotted against wavelengths is reported for the proposed device without and with DBR, showing a significant enhancement in absorbance, due to the optical cavity, of almost two order of

**<sup>1520</sup> <sup>1530</sup> <sup>1540</sup> <sup>1550</sup> <sup>1560</sup> <sup>1570</sup> <sup>1580</sup> <sup>0</sup>**

**Wavelength [nm]**

Fig. 5. Absorbance versus wavelengths for the device proposed by Casalino et al. (Casalino

*P P MM MM*

*A nn nn*

where n2 and n4 are the refractive indices of the 2th and 4th layer, respectively.

*inc TOT TOT TOT TOT*

2 2 2 4

*P E En n*

*P E En n*

*input F B*

*output B F*

2 2 2 4

and metal absorbance can be written as (Casalino, 2006a, 2006b):

  11 21 11 11

*TOT TOT*

*M M*

*M M*

2 2 2

*A B inc*

*A B inc*

2 2 2

(12)

(13)

 

21 11 11 11

22 22

**With optical cavity Without optical cavity**

*TOT TOT*

11 21 21 11 11 11 11 11

$$\begin{bmatrix} E\_{inc} \\ E\_R \end{bmatrix} = M\_{TOT} \begin{bmatrix} E\_T \\ 0 \end{bmatrix} \tag{5}$$
 
$$E\_T = \frac{1}{M\_{TOT\_{11}}} E\_{inc} \qquad E\_R = \frac{M\_{TOT\_{21}}}{M\_{TOT\_{11}}} E\_{inc}$$

where *MTOTij* are the element of the matrix MTOT representing the whole multilayer:

$$M\_{\rm TOT} = M\_{1/2} \cdot M\_2 \cdot M\_{2/3} \cdot M\_3 \cdot M\_{3/4} \cdot M\_4 \cdot M\_{4/5} \cdot M\_5 \cdot M\_{5/6} \tag{6}$$

where Mi/j and Mi are the interface matrix between i and j layer and the i-layer matrix, respectively, which can calculated by knowing refractive index (ni) and thickness (di) of every ith layer:

$$M\_{i/j} = \frac{1}{2n\_i} \begin{pmatrix} n\_i + n\_j & n\_i - n\_j \\ n\_i - n\_j & n\_i + n\_j \end{pmatrix} \qquad M\_i = \begin{pmatrix} e^{jk\_0 n\_i d\_i} & 0 \\ 0 & e^{-jk\_0 n\_i d\_i} \end{pmatrix} \tag{7}$$

Let E2,F (E4,F) and E2,B (E4,B) be the frequency domain electric field complex amplitudes of the forward and backward travelling plane waves in the layer 2 (4), by TMM we obtain:

$$\begin{bmatrix} E\_{2,F} \\ E\_{2,B} \end{bmatrix} = M\_A \begin{bmatrix} E\_T \\ 0 \end{bmatrix}$$

$$E\_{2,F} = \frac{M\_{A\_{11}}}{M\_{TOT\_{11}}} E\_{inc} \qquad E\_{2,B} = \frac{M\_{A\_{21}}}{M\_{TOT\_{11}}} E\_{inc}$$

and

$$\begin{bmatrix} E\_{4,F} \\ E\_{4,B} \end{bmatrix} = M\_B \begin{bmatrix} E\_T \\ 0 \end{bmatrix} \tag{9}$$

$$E\_{4,F} = \frac{M\_{B\_{11}}}{M\_{TOT\_{11}}} E\_{inc} \qquad E\_{4,B} = \frac{M\_{B\_{21}}}{M\_{TOT\_{11}}} E\_{inc}$$

where MA is the matrix calculated from interface between protection coating and metal to the final plane, while MB is the matrix calculated from interface between metal and silicon cavity to the final plane,

$$\begin{aligned} M\_A &= M\_{2/3} \cdot M\_3 \cdot M\_{3/4} \cdot M\_4 \cdot M\_{4/5} \cdot M\_5 \cdot M\_{5/6} \\ M\_B &= M\_4 \cdot M\_{4/5} \cdot M\_5 \cdot M\_{5/6} \end{aligned} \tag{10}$$

The optical power is linked at the electrical field by:

$$P = \frac{n}{2\eta\_0} |E|^2\tag{11}$$

where η0 is the vacuum characteristic impedance of electromagnetic waves. The total power going in (Pinput) and going out (Poutput) from the metal is given by:

*inc T TOT*

 

*E E M*

1

where *MTOTij*

every ith layer:

and

cavity to the final plane,

*R*

*E*

11 11

are the element of the matrix MTOT representing the whole multilayer:

*T inc R inc TOT TOT*

where Mi/j and Mi are the interface matrix between i and j layer and the i-layer matrix, respectively, which can calculated by knowing refractive index (ni) and thickness (di) of

> <sup>0</sup> / 1 0

*jk n d ijij i j <sup>i</sup> jk n d <sup>i</sup> iji j nnnn e*

 

> 11 21 11 11

11 21 11 11

2/3 3 3/4 4 4/5 5 5/6

2

(10)

(11)

*B B F inc B inc TOT TOT*

*M M E EE E M M*

where MA is the matrix calculated from interface between protection coating and metal to the final plane, while MB is the matrix calculated from interface between metal and silicon

*M M MM MM MM*

<sup>0</sup> 2 *<sup>n</sup> P E* 

where η0 is the vacuum characteristic impedance of electromagnetic waves. The total power

*F T B*

*E E M*

0

*A A F inc B inc TOT TOT*

*M M E EE E M M*

*F T A*

*E E M*

0

*n nnnn e*

Let E2,F (E4,F) and E2,B (E4,B) be the frequency domain electric field complex amplitudes of the

2 0

forward and backward travelling plane waves in the layer 2 (4), by TMM we obtain:

2, 2,

*E*

*B*

4, 4,

*E*

*B*

4 4/5 5 5/6

*M MM MM*

going in (Pinput) and going out (Poutput) from the metal is given by:

*A B*

The optical power is linked at the electrical field by:

4, 4,

2, 2,

*M M*

*E EE E M M*

21

0

*i i*

*i i*

(7)

(8)

(9)

(5)

*TOT*

*M M MM MM MM MM TOT* 1/2 2 2/3 3 3/4 4 4/5 5 5/6 (6)

0

*M*

$$\begin{aligned} P\_{\text{input}} &= \frac{n\_2}{2\eta\_0} \left| E\_{2,F} \right|^2 + \frac{n\_4}{2\eta\_0} \left| E\_{4,B} \right|^2 = \left( n\_2 \left| \frac{M\_{A\_{11}}}{M\_{TOT\_{11}}} \right|^2 + n\_4 \left| \frac{M\_{B\_{21}}}{M\_{TOT\_{11}}} \right|^2 \right) \frac{\left| E\_{inc} \right|^2}{2\eta\_0} \\\ P\_{\text{output}} &= \frac{n\_2}{2\eta\_0} \left| E\_{2,B} \right|^2 + \frac{n\_4}{2\eta\_0} \left| E\_{4,F} \right|^2 = \left( n\_2 \left| \frac{M\_{A\_{21}}}{M\_{TOT\_{11}}} \right|^2 + n\_4 \left| \frac{M\_{B\_{11}}}{M\_{TOT\_{11}}} \right|^2 \right) \frac{\left| E\_{inc} \right|^2}{2\eta\_0} \end{aligned} \tag{12}$$

and metal absorbance can be written as (Casalino, 2006a, 2006b):

$$A\_{T} = \frac{P\_{input} - P\_{output}}{P\_{inc}} = \left| \left( n\_{2} \left| \frac{M\_{A\_{11}}}{M\_{TOT\_{11}}} \right|^{2} + n\_{4} \left| \frac{M\_{B\_{21}}}{M\_{TOT\_{11}}} \right|^{2} \right) - \left( n\_{2} \left| \frac{M\_{A\_{21}}}{M\_{TOT\_{11}}} \right|^{2} + n\_{4} \left| \frac{M\_{B\_{11}}}{M\_{TOT\_{11}}} \right|^{2} \right) \right| \tag{13}$$

where n2 and n4 are the refractive indices of the 2th and 4th layer, respectively.

In a work of 2006, Casalino *et al.* (Casalino et al., 2006a, 2006b) proposed a methodology in order to design an optimum device, in fact with a right choice of the multilayer thicknesses a maximum in absorbance can be obtained. In the device proposed by the authors DBR is considered formed by alternate layers of Si and SiO2 having refractive index 3.45 and 1.45, and thickness of 340 nm and 270 nm, respectively, while a metallic layer of gold, having refractive index nAu=0.174+j9.96 (Chan & Card, 1980b), mean free path Le=55 nm (Chan et al., 1980a), Fermi level EF=5.53 eV (Yeh, 1988) and thickness d=32 nm, was chosen as Schottky contact. We point out that in the simulations, semi-infinite first and last layers of air (n1=1) and silicon (n6=3.45), respectively, are considered. In Fig. 5 absorbance plotted against wavelengths is reported for the proposed device without and with DBR, showing a significant enhancement in absorbance, due to the optical cavity, of almost two order of magnitude at 1550 nm.

Fig. 5. Absorbance versus wavelengths for the device proposed by Casalino et al. (Casalino et al., 2006a, 2006b).

Silicon Photodetectors Based on Internal

The RC limited bandwidth is given by:

forward I-V characteristic.

(Donati, 1999):

formula:

1999):

device to work at GHz range.

Fig. 6. Small-signal circuit associated to IPE-based device.

*f*

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 59

1 1

*R C* (18)

(19)

*<sup>v</sup> <sup>f</sup> t L* (20)

(17)

2 2 ( )//

 

*RC R C R R RC Tot <sup>j</sup> s load j j*

where Rj can be evaluated from the inverse derivative of the reverse current-voltage (I-V) electrical characteristic (tipical values are in the MΩ range) and Rs can be extracted from the

For high speed applications, the load resistance RL, tipically 50 Ω, is much lower than series resistance (Rs) and junction resistance (Rj). Therefore, the device 3-dB frequency, becomes

1

*Si Ph*

Where the junction capacitor is linked to the photodetector area (APh) from the following

with εSi is the silicon dielectric constant (10-12 C/cmV). It is worth noting that when the detector area is made sufficiently small, the influence due to the capacitance is reduced, the

The intrinsic carrier-transit time limited 3-dB bandwidth for the device is given by (Donati,

*d*

where vt is the effective carrier saturation velocity (107cm/s in Si) and L is the carrier transit distance. In Fig. 7, the frequency due to RC time constant and transit time are reported as a function of the detector area for RL= 50 an W=1 µm. It is worth noting that when the detector area is made sufficiently small, i.e. smaller than APh<65μm2, the influence due to the capacitance is reduced of one order to magnitude with respect to the transit time, enabling

*s j*

3

*f*

2 *dB*

*j <sup>A</sup> <sup>C</sup> W* 

effect of the transit time dominates and high speed operation can be reached.

0.44 0.44 *<sup>t</sup> tr*

#### **2.1.4 IPE enhancement by reverse voltage applied**

In order to complete the transaction of the IPE theory, the image force between an electron and the metal surface must be taken into account. Due to the image force effect a lowering (∆ΦB) and displacement (xm) of the metal-semiconductor interface potential barrier, is provided. These barrier lowering and displacement are given by (Sze, 1981):

$$
\Delta\phi\_{\rm B} = \sqrt{\frac{q}{4\pi\varepsilon\_{\rm Si}} \frac{\left|V\_{\rm Bins}\right|}{\mathcal{W}}} \qquad \propto\_m = \sqrt{\frac{q}{16\pi\varepsilon\_{\rm Si}} \frac{\mathcal{W}}{\left|V\_{\rm Bins}\right|}} \tag{14}
$$

where εSi is the permittivity of silicon (10-12 C/cmV), W is the depletion width and VBias the applied bias voltage.

It is worth noting that while the potential barrier lowering can be taken into account by Fowler's theory (Eq. 3), the potential barrier displacement influences the probability that an electron undergoes scattering phenomena in Si travelling from the metal-semiconductor interface to the Schottky barrier maximum. This probability can be taken into account by the barrier collection efficiency (ηc):

$$\eta\_c = e^{-\frac{\mathbf{x}\_m}{L\_s}} \tag{15}$$

where Ls is the electron scattering length in the silicon. It is worth noting that increasing the bias voltage, a shift of Schottky barrier closer to metal/semiconductor interface is obtained. Therefore, the barrier collection efficiency increases. The improvement of the device efficiency by increasing reverse voltage has been theoretically investigated by Casalino et al. (Casalino et al., 2008b).

#### **2.2 Bandwidth of IPE-based devices**

The electrical properties of diodes based on Schottky junctions are determined by majority carrier phenomena, while for p-n diodes they are primarily determined by minority carriers. Therefore, the Schottky diodes can be switched faster because there are no minority carrier storage effects. The response time of Schottky barrier photodiodes can be determined by three parameters: the diffusion time in the quasi neutral region; the electrical frequency response or RC time required to discharge the junction capacitance through the resistance and the transit time across the depletion region. By designing the diode in such a way that depleted region length (W) equals to the device length (L), under suitable revere bias applied, the diffusion time can be neglected and the total frequency (ftot) can be written as:

$$\frac{1}{f\_{\text{tot}}} = \frac{1}{f\_{\text{tr}}} + \frac{1}{f\_{\text{RC}}} \tag{16}$$

where ftr and fRC are the transit time and time constant limited 3-dB bandwidth, respectively.

According to the model for small signal shown in Fig. 6 (Casalino et al., 2010b), the photodetector can be schematized as current generator (Iph), resistance (Rj), capacitance (Cj) associated to the junction, series resistance (Rs) and load resistance (Rload)

Fig. 6. Small-signal circuit associated to IPE-based device.

The RC limited bandwidth is given by:

58 Photodetectors

In order to complete the transaction of the IPE theory, the image force between an electron and the metal surface must be taken into account. Due to the image force effect a lowering (∆ΦB) and displacement (xm) of the metal-semiconductor interface potential barrier, is

> 4 16 *Bias B m*

where εSi is the permittivity of silicon (10-12 C/cmV), W is the depletion width and VBias the

It is worth noting that while the potential barrier lowering can be taken into account by Fowler's theory (Eq. 3), the potential barrier displacement influences the probability that an electron undergoes scattering phenomena in Si travelling from the metal-semiconductor interface to the Schottky barrier maximum. This probability can be taken into account by the

> *m s x L*

*c e* 

where Ls is the electron scattering length in the silicon. It is worth noting that increasing the bias voltage, a shift of Schottky barrier closer to metal/semiconductor interface is obtained. Therefore, the barrier collection efficiency increases. The improvement of the device efficiency by increasing reverse voltage has been theoretically investigated by Casalino et al.

The electrical properties of diodes based on Schottky junctions are determined by majority carrier phenomena, while for p-n diodes they are primarily determined by minority carriers. Therefore, the Schottky diodes can be switched faster because there are no minority carrier storage effects. The response time of Schottky barrier photodiodes can be determined by three parameters: the diffusion time in the quasi neutral region; the electrical frequency response or RC time required to discharge the junction capacitance through the resistance and the transit time across the depletion region. By designing the diode in such a way that depleted region length (W) equals to the device length (L), under suitable revere bias applied, the diffusion time can be neglected and the total frequency (ftot) can be written as:

> 111 *tot tr RC f ff*

where ftr and fRC are the transit time and time constant limited 3-dB bandwidth,

According to the model for small signal shown in Fig. 6 (Casalino et al., 2010b), the photodetector can be schematized as current generator (Iph), resistance (Rj), capacitance (Cj)

associated to the junction, series resistance (Rs) and load resistance (Rload)

*Si Si Bias q q V W x W V*

(14)

(15)

(16)

provided. These barrier lowering and displacement are given by (Sze, 1981):

**2.1.4 IPE enhancement by reverse voltage applied** 

applied bias voltage.

barrier collection efficiency (ηc):

(Casalino et al., 2008b).

respectively.

**2.2 Bandwidth of IPE-based devices** 

$$\left\|f\right\|\_{\mathrm{RC}} = \frac{1}{2\pi \left[ \left( \mathcal{R}\_s + \mathcal{R}\_{load} \right) / \left( \mathcal{R}\_j \right) \right] \mathbb{C}\_j} = \frac{1}{2\pi \mathcal{R}\_{\mathrm{Tot}} \mathbb{C}\_j} \tag{17}$$

where Rj can be evaluated from the inverse derivative of the reverse current-voltage (I-V) electrical characteristic (tipical values are in the MΩ range) and Rs can be extracted from the forward I-V characteristic.

For high speed applications, the load resistance RL, tipically 50 Ω, is much lower than series resistance (Rs) and junction resistance (Rj). Therefore, the device 3-dB frequency, becomes (Donati, 1999):

$$f\_{3dB} = \frac{1}{2\pi R\_s C\_j} \tag{18}$$

Where the junction capacitor is linked to the photodetector area (APh) from the following formula:

$$C\_j = \frac{\varepsilon\_{Si} A\_{Pl}}{W} \tag{19}$$

with εSi is the silicon dielectric constant (10-12 C/cmV). It is worth noting that when the detector area is made sufficiently small, the influence due to the capacitance is reduced, the effect of the transit time dominates and high speed operation can be reached.

The intrinsic carrier-transit time limited 3-dB bandwidth for the device is given by (Donati, 1999):

$$f\_{tr} = \frac{0.44}{t\_d} = 0.44 \frac{v\_t}{L} \tag{20}$$

where vt is the effective carrier saturation velocity (107cm/s in Si) and L is the carrier transit distance. In Fig. 7, the frequency due to RC time constant and transit time are reported as a function of the detector area for RL= 50 an W=1 µm. It is worth noting that when the detector area is made sufficiently small, i.e. smaller than APh<65μm2, the influence due to the capacitance is reduced of one order to magnitude with respect to the transit time, enabling device to work at GHz range.

Silicon Photodetectors Based on Internal

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 61

2 3 2 1 ( )

where hν and λ are the photon energy and wavelength, respectively. It could be shown that this radiance is proportional to a dc background current (Ibg) limiting infrared photodetector sensitivity. In order to get higher sensitivity, is necessary to reduce the background current contribute by reducing the emission radiance, i.e., by lowering the photodetector temperature (see Plank law). For this reason infrared detector needs to work at cryogenic temperature. On the other hand, photodetectors working at visible or NIR wavelengths are not able to detect blackbody emission and the Schottky junction saturation current limits the detector sensitivity.

*h*

*s ph I A ATe*

where A\*\*=fpA\*, A\* is the Richardson constant (30 A/cm2K2 for p-type Si and 110 A/cm2K2 for n-type Si) (Sze, 1981), fp is the barrier escape probability which as a first approximation is given by fp(V)=exp(-xm(V)/Ls), Ls is the electron scattering length, T is the absolute temperature and ΦB(V)= ΦB0-ΔΦB(V) is the potential barrier which is the potential barrier at zero voltage minus the lowering due to the reverse bias. It is worth noting that both potential barrier and barrier escape probability are reverse voltage dependent due to force image effect. The plot of the saturation current density against reverse voltage applied for three different metal: gold (Au), silver (Ag) and copper (Cu), is reported in Fig. 8 (Casalino et al., 2008b). Schottky barrier used in the simulations for Au/p-Si, Ag/p-Si and Cu/p-Si interfaces are 0.78 eV, 0.78 eV and 0.58 eV, respectively. Due to the lowest potential barrier,

**0 5 10 15 20 25 30 35 40**

**Reverse Voltage [V]**

different metals: gold, silver and copper. Reprinted with permission from M. Casalino *et al.*,

Fig. 8. Dark current density plotted against reverse voltage (semi-log scale) for three

"A silicon compatible resonant cavity enhanced photodetector working at 1.55 μm," Semicond. Sci. Technol., 23, 075001, 2008 (doi: 10.1088/0268-1242/23/7/075001). IOP

*r*

It is given by the Richardson-Dushamann equation (Yuan & Perera, 1995):

copper shows the highest dark current density.

**10-1**

**100**

**Dark current density [pA/um2**

Publishing is acknowledged.

**]**

**101**

**102**

*h kT*

*e* 

1

( ) \*\* 2 *<sup>B</sup> <sup>V</sup> kT*

(25)

**Au Cu Ag**

(24)

Fig. 7. Frequency due to the RC time and the transit time plotted against area detector for RL=50 and W=1µm.

#### **2.3 Noise and sensitivity of IPE-based devices**

It is well known that the photodetector output is affected by a noise contribution which hinders the device sensitivity. The r.m.s. noise is due to two contributions: Johnson noise and shot noise. Johnson is the thermal noise associated to the resistance Rload and its r.m.s value can be defined as (Donati, 1999):

$$i\_f = \sqrt{\frac{4kBT}{R\_{load}}}\tag{21}$$

where Rload is the load resistance, k is the Boltzmann constant and T is the absolute temperature. On the other hand, the shot noise is associated to the discrete nature of the total current, i.e., the sum of the signal current and dark current (Donati, 1999):

$$\dot{\mathbf{u}}\_S = \sqrt{2q(I\_{\text{ph}} + I\_d)B} \tag{22}$$

where q is the electron charge.

Because the two contributions are statistically independent their m.s. values can be added in order to get the total noise:

$$\dot{\mathbf{u}}\_n = \sqrt{\dot{\mathbf{i}}\_S^2 + \dot{\mathbf{i}}\_f^2} = \sqrt{2q(I\_{ph} + I\_{cl})B + \frac{4kBT}{R}}\tag{23}$$

From Eq. 23, it should be clear that shot noise can always dominate the Johnson with a right choice of Rload. Dark current of infrared photodetectors based on the IPE effect is composed by the inverse saturation current (Is) and by a background current (Ibg). The background current is due to the fact that at first approximation an infrared photodetector at a temperature T, can be view as a blackbody emitting a radiance r(λ) according to the Planck law (Donati, 1999):

**<sup>0</sup> <sup>500</sup> <sup>1000</sup> <sup>1500</sup> <sup>2000</sup> <sup>0</sup>**

**]**

**45 GHz**

*<sup>R</sup>* (21)

2( ) *<sup>S</sup> ph d i q I IB* (22)

(23)

**f RC f tr**

**Photodetector Area [um2**

It is well known that the photodetector output is affected by a noise contribution which hinders the device sensitivity. The r.m.s. noise is due to two contributions: Johnson noise and shot noise. Johnson is the thermal noise associated to the resistance Rload and its r.m.s

4

where Rload is the load resistance, k is the Boltzmann constant and T is the absolute temperature. On the other hand, the shot noise is associated to the discrete nature of the

Because the two contributions are statistically independent their m.s. values can be added in

From Eq. 23, it should be clear that shot noise can always dominate the Johnson with a right choice of Rload. Dark current of infrared photodetectors based on the IPE effect is composed by the inverse saturation current (Is) and by a background current (Ibg). The background current is due to the fact that at first approximation an infrared photodetector at a temperature T, can be view as a blackbody emitting a radiance r(λ) according to the Planck law (Donati, 1999):

2 2 <sup>4</sup> 2( ) *n SJ ph d kBT i i i qI I B <sup>R</sup>*

*kBT <sup>i</sup>*

*load*

*J*

total current, i.e., the sum of the signal current and dark current (Donati, 1999):

Fig. 7. Frequency due to the RC time and the transit time plotted against area detector for

**2.3 Noise and sensitivity of IPE-based devices** 

value can be defined as (Donati, 1999):

where q is the electron charge.

order to get the total noise:

RL=50 and W=1µm.

**Frequency [GHz]**

**450 GHz**

**65 um2**

$$r(\lambda) = \frac{2\hbar\nu^2}{\lambda^3} \left| \frac{1}{e^{\frac{\hbar\nu}{kT}} - 1} \right| \tag{24}$$

where hν and λ are the photon energy and wavelength, respectively. It could be shown that this radiance is proportional to a dc background current (Ibg) limiting infrared photodetector sensitivity. In order to get higher sensitivity, is necessary to reduce the background current contribute by reducing the emission radiance, i.e., by lowering the photodetector temperature (see Plank law). For this reason infrared detector needs to work at cryogenic temperature. On the other hand, photodetectors working at visible or NIR wavelengths are not able to detect blackbody emission and the Schottky junction saturation current limits the detector sensitivity. It is given by the Richardson-Dushamann equation (Yuan & Perera, 1995):

$$I\_s = A\_{ph} A^{\ast \ast} T^2 e^{-\frac{\phi\_b(V)}{kT}} \tag{25}$$

where A\*\*=fpA\*, A\* is the Richardson constant (30 A/cm2K2 for p-type Si and 110 A/cm2K2 for n-type Si) (Sze, 1981), fp is the barrier escape probability which as a first approximation is given by fp(V)=exp(-xm(V)/Ls), Ls is the electron scattering length, T is the absolute temperature and ΦB(V)= ΦB0-ΔΦB(V) is the potential barrier which is the potential barrier at zero voltage minus the lowering due to the reverse bias. It is worth noting that both potential barrier and barrier escape probability are reverse voltage dependent due to force image effect. The plot of the saturation current density against reverse voltage applied for three different metal: gold (Au), silver (Ag) and copper (Cu), is reported in Fig. 8 (Casalino et al., 2008b). Schottky barrier used in the simulations for Au/p-Si, Ag/p-Si and Cu/p-Si interfaces are 0.78 eV, 0.78 eV and 0.58 eV, respectively. Due to the lowest potential barrier, copper shows the highest dark current density.

Fig. 8. Dark current density plotted against reverse voltage (semi-log scale) for three different metals: gold, silver and copper. Reprinted with permission from M. Casalino *et al.*, "A silicon compatible resonant cavity enhanced photodetector working at 1.55 μm," Semicond. Sci. Technol., 23, 075001, 2008 (doi: 10.1088/0268-1242/23/7/075001). IOP Publishing is acknowledged.

Silicon Photodetectors Based on Internal

Electronics Review, 6, 1, 1998.

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 63

mainstream infrared systems and applications. (Shepherd, 1984; Shepherd, 1988; Kosonocky, 1991; Kosonocky, 1992; Shepherd, 1998). In the early years, the development of SBD FPA technology progressed from the demonstration of the initial concepts in the 1970's (Kohn et al., 1975; Capone et al., 1978; Kosonocky et al., 1978; Shepherd et al., 1979) to the development of high resolution scanning and staring devices in the 1980's and in the 1990's, that were at the basis of many applications for infrared imaging in the NIR, MIR and FIR bands. The first Schottky-barrier FPAs were made with thick Pd2Si or PtSi detectors using about 600 Å of deposited palladium or platinum. These FPAs exhibited relatively small photoresponse. More than an order of magnitude improvement in photoresponse was demonstrated in 1980 with 50×50-element FPAs constructed with thin PtSi SBDs at the David Sarnoff Research Center (Taylor et al, 1980; Kosonocky et al., 1980). The SBDs in this FPA had an optical cavity in the form of a thin (20 – 40 Å) PtSi layer separated from an aluminium reflector by a layer of deposited SiO2. The general concept of thin SBD with optical cavity was first described in 1973 by Archer and Cohen for SBDs in the form of Au on p-type Si (Archer & Cohen, 1973). The improved PtSi SBD structure was further developed and from 1980 to 1985 the fabrication process for PtSi SBDs was optimized at Sarnoff with the development of 32×63, 64×128 and 160×244 IR-CCD FPAs (Elabd et al., 1982; Kosonocky, 1985). Similar PtSi SBDs characteristics were also reported in the same years by Mitsubishi

Corporation, Fujitsu, NEC, EG&G Reticon, Hughes, Loral Fairchild and Kodak.

silicide is transferred to a CCD by the direct charge injection method (see Fig. 9).

Fig. 9. Typical construction and operation of PtSi Schottky-barrier IR FPA designed with interline transfer CCD readout architecture. (a) and (b) show the potential diagrams in the integration and readout operations, respectively (Kimata, 1995, 1998). Reprinted with permission from M. Kimata, "PtSi Schottky-barrier infrared focal plane arrays," Opto-

As already discussed, the most popular Schottky-barrier detector is the PtSi detector, which can be used for the detection in the 3–5 µm spectral range (Kimata & Tsubouchi, 1995; Kimata, 2000). Radiation is transmitted through the p-type silicon and is absorbed in the metal PtSi (not in the semiconductor), producing hot holes, which are then emitted over the potential barrier into the silicon, leaving the silicide charged negatively. Negative charge of

The photodetector sensitivity is directly linked to the r.m.s. noise current and can be represented in term of a well known parameter called NEP (Noise Equivalent Power) which is defined as the ratio of r.m.s. noise to responsivity (Donati, 1999):

$$\text{NEP} = \frac{i\_n}{\mathfrak{R}} \quad \text{[} \mathcal{W} \text{]} \tag{26}$$

NEP represents the lowest input power giving a unit signal/noise ratio. Because higher device performances correspond to smaller NEP, it is more convenient to define its inverse: the detectivity. It can be expected that the square of the r.m.s. noise (m.s. noise) is proportional to the electrical bandwidth (B) and detector area (APh), for this reason in order to get a figure of merit not depending on these parameters, the detectivity is normalized to the square root of detector area and bandwidth (Donati, 1999):

$$\mathbf{D} = \sqrt{A\_{Pl} \cdot B} \frac{\Re}{i\_n} \left[ \frac{cm\sqrt{Hz}}{\mathcal{W}} \right] \tag{27}$$

#### **3. Silicon photodetectors based on IPE**

This section reviews the history and the progresses of the main structures reported in literature, starting from historical devices for imaging application at infrared wavelengths, up to new NIR devices adapted for telecommunications and power monitoring applications.

#### **3.1 Infrared devices for imaging application**

Schottky-barrier focal plane arrays (FPAs) are infrared imagers, fabricated by wellestablished silicon very-large-scale-integration (VLSI) process, representing one of the most effective technology for large-area high-density focal plane arrays for many near infrared (NIR, 1 to 3 µm) and medium infrared (MIR, 3 to 5 µm) applications. PtSi Schottky Barrier detectors (SBDs) represent the most established SBD technology but they must operate at 77 °K in order to reduce the dark current density in the range of a few nA/cm2. Pd2Si SBDs were developed for operation with passive cooling at 120 °K in the NIR band, while IrSi SBDs have also been investigated to extend the application of Schottky-barrier focal plane arrays into the far infrared (FIR, 8 to 10 µm) spectral range.

In 1973 Shepherd and Yang (Shepherd & Yang, 1973) proposed the concept of silicide Schottky-barrier detectors as much more reproducible alternative to HgCdTe FPAs for infrared thermal imaging. After being dormant for about ten years, extrinsic Si was reconsidered as a material for infrared imaging, especially after the invention of chargecoupled devices (CCDs) by Boyle and Smith (Boyle & Smith, 1970). For the first time it became possible to have much more sophisticated readout schemes and both detection and readout could be implemented on one common silicon chip. Since then, the development of the Schottky-barrier technology progressed continuously and currently offers large IR image sensor formats. These trends in IR FPA development show that the IR community today prefers more producible technologies with higher uniformity to the technology based on the narrow gap semiconductors, which still have serious material problems. Such attributes as: monolithic construction, uniformity in responsivity and signal to noise (the performance of an IR system ultimately depends on the ability to compensate the non uniformity of an FPA using external electronics and a variety of temperature references), and absence of discernible 1/f noise; make Schottky-barrier devices a formidable contender to the

The photodetector sensitivity is directly linked to the r.m.s. noise current and can be represented in term of a well known parameter called NEP (Noise Equivalent Power) which

NEP [ ] *ni <sup>W</sup>*

NEP represents the lowest input power giving a unit signal/noise ratio. Because higher device performances correspond to smaller NEP, it is more convenient to define its inverse: the detectivity. It can be expected that the square of the r.m.s. noise (m.s. noise) is proportional to the electrical bandwidth (B) and detector area (APh), for this reason in order to get a figure of merit not depending on these parameters, the detectivity is normalized to

> *n cm Hz A B i W*

This section reviews the history and the progresses of the main structures reported in literature, starting from historical devices for imaging application at infrared wavelengths, up to new NIR devices adapted for telecommunications and power monitoring applications.

Schottky-barrier focal plane arrays (FPAs) are infrared imagers, fabricated by wellestablished silicon very-large-scale-integration (VLSI) process, representing one of the most effective technology for large-area high-density focal plane arrays for many near infrared (NIR, 1 to 3 µm) and medium infrared (MIR, 3 to 5 µm) applications. PtSi Schottky Barrier detectors (SBDs) represent the most established SBD technology but they must operate at 77 °K in order to reduce the dark current density in the range of a few nA/cm2. Pd2Si SBDs were developed for operation with passive cooling at 120 °K in the NIR band, while IrSi SBDs have also been investigated to extend the application of Schottky-barrier focal plane

In 1973 Shepherd and Yang (Shepherd & Yang, 1973) proposed the concept of silicide Schottky-barrier detectors as much more reproducible alternative to HgCdTe FPAs for infrared thermal imaging. After being dormant for about ten years, extrinsic Si was reconsidered as a material for infrared imaging, especially after the invention of chargecoupled devices (CCDs) by Boyle and Smith (Boyle & Smith, 1970). For the first time it became possible to have much more sophisticated readout schemes and both detection and readout could be implemented on one common silicon chip. Since then, the development of the Schottky-barrier technology progressed continuously and currently offers large IR image sensor formats. These trends in IR FPA development show that the IR community today prefers more producible technologies with higher uniformity to the technology based on the narrow gap semiconductors, which still have serious material problems. Such attributes as: monolithic construction, uniformity in responsivity and signal to noise (the performance of an IR system ultimately depends on the ability to compensate the non uniformity of an FPA using external electronics and a variety of temperature references), and absence of discernible 1/f noise; make Schottky-barrier devices a formidable contender to the

(26)

(27)

is defined as the ratio of r.m.s. noise to responsivity (Donati, 1999):

the square root of detector area and bandwidth (Donati, 1999):

D *Ph*

**3. Silicon photodetectors based on IPE** 

**3.1 Infrared devices for imaging application** 

arrays into the far infrared (FIR, 8 to 10 µm) spectral range.

mainstream infrared systems and applications. (Shepherd, 1984; Shepherd, 1988; Kosonocky, 1991; Kosonocky, 1992; Shepherd, 1998). In the early years, the development of SBD FPA technology progressed from the demonstration of the initial concepts in the 1970's (Kohn et al., 1975; Capone et al., 1978; Kosonocky et al., 1978; Shepherd et al., 1979) to the development of high resolution scanning and staring devices in the 1980's and in the 1990's, that were at the basis of many applications for infrared imaging in the NIR, MIR and FIR bands. The first Schottky-barrier FPAs were made with thick Pd2Si or PtSi detectors using about 600 Å of deposited palladium or platinum. These FPAs exhibited relatively small photoresponse. More than an order of magnitude improvement in photoresponse was demonstrated in 1980 with 50×50-element FPAs constructed with thin PtSi SBDs at the David Sarnoff Research Center (Taylor et al, 1980; Kosonocky et al., 1980). The SBDs in this FPA had an optical cavity in the form of a thin (20 – 40 Å) PtSi layer separated from an aluminium reflector by a layer of deposited SiO2. The general concept of thin SBD with optical cavity was first described in 1973 by Archer and Cohen for SBDs in the form of Au on p-type Si (Archer & Cohen, 1973). The improved PtSi SBD structure was further developed and from 1980 to 1985 the fabrication process for PtSi SBDs was optimized at Sarnoff with the development of 32×63, 64×128 and 160×244 IR-CCD FPAs (Elabd et al., 1982; Kosonocky, 1985). Similar PtSi SBDs characteristics were also reported in the same years by Mitsubishi Corporation, Fujitsu, NEC, EG&G Reticon, Hughes, Loral Fairchild and Kodak.

As already discussed, the most popular Schottky-barrier detector is the PtSi detector, which can be used for the detection in the 3–5 µm spectral range (Kimata & Tsubouchi, 1995; Kimata, 2000). Radiation is transmitted through the p-type silicon and is absorbed in the metal PtSi (not in the semiconductor), producing hot holes, which are then emitted over the potential barrier into the silicon, leaving the silicide charged negatively. Negative charge of silicide is transferred to a CCD by the direct charge injection method (see Fig. 9).

Fig. 9. Typical construction and operation of PtSi Schottky-barrier IR FPA designed with interline transfer CCD readout architecture. (a) and (b) show the potential diagrams in the integration and readout operations, respectively (Kimata, 1995, 1998). Reprinted with permission from M. Kimata, "PtSi Schottky-barrier infrared focal plane arrays," Opto-Electronics Review, 6, 1, 1998.

Silicon Photodetectors Based on Internal

Publishers, Boston, USA, 2000.

technology for medium wavelength applications (see Table 1).

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 65

surface to p-type <100> silicon (with resisitivity 30–50 /cm) at the Schottky-barrier detector location. In the case of PtSi detectors, a very thin layer of Pt (1–2 nm) is deposited and sintered (annealed at a temperature in the range of 300–600°C) to form PtSi and the unreacted Pt on the SiO2 surfaces is removed by dip etching in hot aqua regia. The Schottkybarrier structure is then completed by a deposition of a suitable dielectric (usually SiO2) for forming the resonant cavity, removing this dielectric outside the Schottky-barrier regions, and depositing and defining Al for the detector reflector and the interconnects of the Si readout multiplexer. In the case of the 10 µm IrSi Schottky-barrier detectors, the IrSi was formed by in situ vacuum annealing and the unreacted Ir was removed by reactive ion etching. The progress of the Schottky-barrier FPA technology has been constant (Kimata et al., 1998). At the present time Schottky-barrier FPAs represent the most advanced FPAs

Table 1. Specifications and performances of typical PtSi Schottky-barrier FPAs (Kimata, 2000). Reprinted with permission from M. Kimata, Metal silicide Schottky infrared detector arrays. In: Infrared Detectors and Emitters: Materials and Devices, Kluwer Academic

The details of the geometry, and the method of charge transfer differ for different manufacturers. The design of a staring Schottky-barrier FPAs for given pixel size and design rules, involves a trade-off between the charge handling capacity and the fill factor. Most of the reported Schottky-barrier FPAs have the interline transfer CCD architecture. The typical cross section view of the pixel and its operation in interline transfer CCD architecture is shown in Fig. 9. The pixel consists of a Schottky-barrier detector with an optical cavity, a transfer gate, and a stage of vertical CCD. The n-type guard ring on the periphery of the Schottky-barrier diode reduces the edge electric field and suppresses dark current. The effective detector area is determined by the inner edge of the guard ring. The transfer gate is an enhancement MOS transistor. The connection between detector and the transfer gate is made by an n+ diffusion. A buried-channel CCD is used for the vertical transfer. During the optical integration time the surface-channel transfer gate is biased into accumulation. The Schottky-barrier detector is isolated from the CCD register in this condition. The IR

The effective quantum efficiency in the 3–5 µm window is very low, of the order of 1%, but useful sensitivity is obtained by means of near full frame integration in area arrays. The quantum efficiency has been improved by thinning PtSi film and implementation of an optical cavity. Due to very low quantum efficiency, the operating temperature of Schottkybarrier photoemissive detectors is lower than another types of IR photon detectors (Rogalsky, 1999).

Schottky photoemission is independent of such factors as semiconductor doping, minority carrier lifetime, and alloy composition, and, as a result of this, has spatial uniformity characteristics that are far superior to those of other detector technologies. Uniformity is only limited by the geometric definition of the detectors. The fundamental source of dark current in the devices is thermionic emission of holes over the potential barrier and its magnitude is given by Richardson's equation (Sze, 1982). The cooling requirements of photoemissive detectors are comparable to the extrinsic devices, and while an extension of the technology to the long wavelength band is possible using IrSi (see Fig. 10) this will require cooling below 77 °K (Shepherd, 1988).

Fig. 10. Comparison of various infrared detectors based on IPE for PtSi, IrSi, PtSi/SiGe, PtSi doping spike, and SiGe/Si HIP (Kimata, 2000). Reprinted with permission from M. Kimata, Metal silicide Schottky infrared detector arrays. In: Infrared Detectors and Emitters: Materials and Devices, Kluwer Academic Publishers, Boston, USA, 2000.

The Schottky-barrier detector is typically operated in backside illumination mode. The quantum efficiency has been improved by thinning PtSi film. The thinning is effective down to the PtSi thickness of 2 nm. Another means of improving responsivity is implementation of an optical cavity. The optical cavity structure consists of the metal reflector and the dielectric film between the reflector and metal electrode of the Schottky-barrier diode. According to fundamental optical theory, the effect of the optical cavity depends on the thickness and refractive index of the dielectric films, and the wavelength (Kurianski et al., 1989). The main advantage of the Schottky-barrier detectors is that they can be fabricated as monolithic arrays in a standard silicon VLSI process. Typically, the silicon array is completed up to the Al metallization al step. A Schottky-contact mask is used to open SiO2

The effective quantum efficiency in the 3–5 µm window is very low, of the order of 1%, but useful sensitivity is obtained by means of near full frame integration in area arrays. The quantum efficiency has been improved by thinning PtSi film and implementation of an optical cavity. Due to very low quantum efficiency, the operating temperature of Schottkybarrier photoemissive detectors is lower than another types of IR photon detectors

Schottky photoemission is independent of such factors as semiconductor doping, minority carrier lifetime, and alloy composition, and, as a result of this, has spatial uniformity characteristics that are far superior to those of other detector technologies. Uniformity is only limited by the geometric definition of the detectors. The fundamental source of dark current in the devices is thermionic emission of holes over the potential barrier and its magnitude is given by Richardson's equation (Sze, 1982). The cooling requirements of photoemissive detectors are comparable to the extrinsic devices, and while an extension of the technology to the long wavelength band is possible using IrSi (see Fig. 10) this will

Fig. 10. Comparison of various infrared detectors based on IPE for PtSi, IrSi, PtSi/SiGe, PtSi doping spike, and SiGe/Si HIP (Kimata, 2000). Reprinted with permission from M. Kimata,

The Schottky-barrier detector is typically operated in backside illumination mode. The quantum efficiency has been improved by thinning PtSi film. The thinning is effective down to the PtSi thickness of 2 nm. Another means of improving responsivity is implementation of an optical cavity. The optical cavity structure consists of the metal reflector and the dielectric film between the reflector and metal electrode of the Schottky-barrier diode. According to fundamental optical theory, the effect of the optical cavity depends on the thickness and refractive index of the dielectric films, and the wavelength (Kurianski et al., 1989). The main advantage of the Schottky-barrier detectors is that they can be fabricated as monolithic arrays in a standard silicon VLSI process. Typically, the silicon array is completed up to the Al metallization al step. A Schottky-contact mask is used to open SiO2

Metal silicide Schottky infrared detector arrays. In: Infrared Detectors and Emitters:

Materials and Devices, Kluwer Academic Publishers, Boston, USA, 2000.

(Rogalsky, 1999).

require cooling below 77 °K (Shepherd, 1988).

surface to p-type <100> silicon (with resisitivity 30–50 /cm) at the Schottky-barrier detector location. In the case of PtSi detectors, a very thin layer of Pt (1–2 nm) is deposited and sintered (annealed at a temperature in the range of 300–600°C) to form PtSi and the unreacted Pt on the SiO2 surfaces is removed by dip etching in hot aqua regia. The Schottkybarrier structure is then completed by a deposition of a suitable dielectric (usually SiO2) for forming the resonant cavity, removing this dielectric outside the Schottky-barrier regions, and depositing and defining Al for the detector reflector and the interconnects of the Si readout multiplexer. In the case of the 10 µm IrSi Schottky-barrier detectors, the IrSi was formed by in situ vacuum annealing and the unreacted Ir was removed by reactive ion etching. The progress of the Schottky-barrier FPA technology has been constant (Kimata et al., 1998). At the present time Schottky-barrier FPAs represent the most advanced FPAs technology for medium wavelength applications (see Table 1).


Table 1. Specifications and performances of typical PtSi Schottky-barrier FPAs (Kimata, 2000). Reprinted with permission from M. Kimata, Metal silicide Schottky infrared detector arrays. In: Infrared Detectors and Emitters: Materials and Devices, Kluwer Academic Publishers, Boston, USA, 2000.

The details of the geometry, and the method of charge transfer differ for different manufacturers. The design of a staring Schottky-barrier FPAs for given pixel size and design rules, involves a trade-off between the charge handling capacity and the fill factor. Most of the reported Schottky-barrier FPAs have the interline transfer CCD architecture. The typical cross section view of the pixel and its operation in interline transfer CCD architecture is shown in Fig. 9. The pixel consists of a Schottky-barrier detector with an optical cavity, a transfer gate, and a stage of vertical CCD. The n-type guard ring on the periphery of the Schottky-barrier diode reduces the edge electric field and suppresses dark current. The effective detector area is determined by the inner edge of the guard ring. The transfer gate is an enhancement MOS transistor. The connection between detector and the transfer gate is made by an n+ diffusion. A buried-channel CCD is used for the vertical transfer. During the optical integration time the surface-channel transfer gate is biased into accumulation. The Schottky-barrier detector is isolated from the CCD register in this condition. The IR

Silicon Photodetectors Based on Internal

plateau, and a slow progress from now on is expected.

**3.2 IPE-based NIR devices** 

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 67

design rules and has 53% fill factor (Kimata et al., 1992). If the signal charges of 1040×1040 pixels are readout from one output port at the TV compatible frame rate, an unrealistic pixel rate of about 40 MHz is required. Therefore, a 4-output chip design was adopted (Shiraishi et al., 1996). The array of 1040×1040 pixels is divided into four blocks of 520×520 pixels. Each block has a horizontal CCD and a floating diffusion amplifier. One-million pixel data at a 30 Hz frame rate can be readout by operating each horizontal CCD at a 10 MHz clock frequency. The NEDT of 1040×1040 element FPA at 300 °K with a 30 Hz frame is 0.1 °K. More recently, a high-performance 801×512-element PtSi Schottky-barrier infrared image sensor has been developed with an enhanced CSD readout architecture (Inoue et al., 1997; Kimata et al., 1997). The developed image sensor has a large fill factor of 61% in spite of a small pixel size of 17×20 µm2. The NEDT was 0.037 °K at 300 °K. The total power consumption of the device was less than 50 mW. Current PtSi Schottky barrier FPAs are mainly manufactured in 150 mm wafer process lines with around 1 µm lithography technologies; the most advanced Si technology offers 200 mm wafers process with 0.25 µm design rules. Furthermore, 300 mm Si wafer processes with 0.15 µm fine patterns will soon available. However, the performance of monolithic PtSi Schottky-barrier FPAs has reached a

In this section, the main Near-Infrared silicon photodetectors structures based on the aforementioned IPE effect will be reviewed. The simplest detectors consist of a metal layer on a semiconductor forming a Schottky contact at the material interface, with the Schottky barrier energy B determined by the materials (Sze, 1981). If an infrared radiation reaches metal/semiconductor interface, the conduction electrons inside the metal can absorb photons gaining sufficient energy. These excited (hot) electrons are able to cross over the Schottky barrier, sweep out the depletion region of the semiconductor, and be collected as a photocurrent under reverse bias operation from an ohmic semiconductor-metal interface. In practice, p-type silicon is often used as the semiconductor because Schottky barriers are lower thereupon than on n-type silicon, allowing detection at longer wavelengths. The IPEeffect, as described in the section 2.1, is very fast allowing to reach high data rates. On the other hand, it is inherently weak so several photodetector designs have been proposed for achieving a suitable quantum efficiency. Elabd *et a*l. proposed to use a very thin metal film (about 2 nm) to increase the escape probability of hot carriers due to their multiple reflections inside the metal film (Elabd et al., 1982). In particular, the proposed Schottky photodetector was characterized by a responsivity of about 250 mA/W for a wavelength =1500 nm. However, since the volume in which the photons interact with electrons in the metal is very small, only a small fraction of the incident photons actually causes photoemission. Several solutions have been proposed to enhance the efficiency of the IPE process. For example, in the 2001 Wang's group (Lee et al., 2001) investigated the spectral responsivity of Al-porous silicon Schottky barrier photodetectors in the wavelength range 0.4-1.7 µm. The structure of the PS photodetector was Al (finger type)/PS/Si/Al (ohmic), and the active area was 18 mm2. The photodetectors show strong photoresponsivity in both the visible and the infrared bands, especially at 1.55 µm. The photocurrent can reach 1.8 mA at a reverse bias of 6 V under illumination by a 1.55-µm, 10-mW laser diode. The corresponding quantum efficiency is 14.4%; this high value comes from a very high surfacearea-to-volume ratio, of the order of 200-800 m2/cm3 of porous Si. The dark current is 5 µA

radiation generates hot holes in the PtSi film and some of the excited hot holes are emitted into the silicon substrate leaving excess electrons in the PtSi electrode. This lowers the electrical potential of the PtSi electrode. At the end of the integration time, the transfer gate is pulsed-on to read out the signal electrons from the detector to the CCD register. At the same time, the electrical potential of the PtSi electrode is reset to the channel level of the transfer gate. A unique feature of the Schottky-barrier IR FPAs is the built-in blooming control (blooming is a form of crosstalk in which a well saturates and the electrons spill over into neighbouring pixels). A strong illumination forward biases the detector and no further electrons are accumulated at the detector. The small negative voltage developed at the detector is not sufficient to forward bias the guard ring to the extent that electrons are injected to the CCD register through the silicon region under the transfer gate. Therefore, unless the vertical CCD has an insufficient charge handling capacity, blooming is suppressed perfectly in the Schottky-barrier IR FPA. The responsivity of the FPAs is proportional to their fill factor, and improvement in the fill factor has been one of the most important issues in the development of imagers. For improving the fill factor a readout architecture called the charge sweep device (CSD) developed by Mitsubishi Corporation is also used. Kimata and co-workers have developed a series of IR image sensors with the CSD readout architecture with array sizes from 256×256 to 1040×1040 elements. Specifications and performance of these devices are summarised in Table 2. The effectiveness of this readout architecture is enhanced as the design rule becomes finer. Using a 1.2 µm CSD technology, a large fill factor of 71% was achieved with a 26×20 µm2 pixel in the 512×512 monolithic structure (Yagi et al., 1994). The noise equivalent temperature difference (NETD) was estimated as 0.033 °K at 300 °K. The 1040×1040 element CSD FPA has the smallest pixel size (17×17 µm2) among two-dimensional IR FPAs. The pixel was constructed with 1.5 µm


Table 2. Specifications and performance of 2-D PtSi Schottky-barrier FPAs with CSD readout (Kimata, 1998). Reprinted with permission from M. Kimata, "PtSi Schottky-barrier infrared focal plane arrays," Opto-Electronics Review, 6, 1, 1998.

pixels are readout from one output port at the TV compatible frame rate, an unrealistic pixel rate of about 40 MHz is required. Therefore, a 4-output chip design was adopted (Shiraishi et al., 1996). The array of 1040×1040 pixels is divided into four blocks of 520×520 pixels. Each block has a horizontal CCD and a floating diffusion amplifier. One-million pixel data at a 30 Hz frame rate can be readout by operating each horizontal CCD at a 10 MHz clock frequency. The NEDT of 1040×1040 element FPA at 300 °K with a 30 Hz frame is 0.1 °K. More recently, a high-performance 801×512-element PtSi Schottky-barrier infrared image sensor has been developed with an enhanced CSD readout architecture (Inoue et al., 1997; Kimata et al., 1997). The developed image sensor has a large fill factor of 61% in spite of a small pixel size of 17×20 µm2. The NEDT was 0.037 °K at 300 °K. The total power consumption of the device was less than 50 mW. Current PtSi Schottky barrier FPAs are mainly manufactured in 150 mm wafer process lines with around 1 µm lithography technologies; the most advanced Si technology offers 200 mm wafers process with 0.25 µm design rules. Furthermore, 300 mm Si wafer processes with 0.15 µm fine patterns will soon available. However, the performance of monolithic PtSi Schottky-barrier FPAs has reached a plateau, and a slow progress from now on is expected.

### **3.2 IPE-based NIR devices**

66 Photodetectors

radiation generates hot holes in the PtSi film and some of the excited hot holes are emitted into the silicon substrate leaving excess electrons in the PtSi electrode. This lowers the electrical potential of the PtSi electrode. At the end of the integration time, the transfer gate is pulsed-on to read out the signal electrons from the detector to the CCD register. At the same time, the electrical potential of the PtSi electrode is reset to the channel level of the transfer gate. A unique feature of the Schottky-barrier IR FPAs is the built-in blooming control (blooming is a form of crosstalk in which a well saturates and the electrons spill over into neighbouring pixels). A strong illumination forward biases the detector and no further electrons are accumulated at the detector. The small negative voltage developed at the detector is not sufficient to forward bias the guard ring to the extent that electrons are injected to the CCD register through the silicon region under the transfer gate. Therefore, unless the vertical CCD has an insufficient charge handling capacity, blooming is suppressed perfectly in the Schottky-barrier IR FPA. The responsivity of the FPAs is proportional to their fill factor, and improvement in the fill factor has been one of the most important issues in the development of imagers. For improving the fill factor a readout architecture called the charge sweep device (CSD) developed by Mitsubishi Corporation is also used. Kimata and co-workers have developed a series of IR image sensors with the CSD readout architecture with array sizes from 256×256 to 1040×1040 elements. Specifications and performance of these devices are summarised in Table 2. The effectiveness of this readout architecture is enhanced as the design rule becomes finer. Using a 1.2 µm CSD technology, a large fill factor of 71% was achieved with a 26×20 µm2 pixel in the 512×512 monolithic structure (Yagi et al., 1994). The noise equivalent temperature difference (NETD) was estimated as 0.033 °K at 300 °K. The 1040×1040 element CSD FPA has the smallest pixel size (17×17 µm2) among two-dimensional IR FPAs. The pixel was constructed with 1.5 µm

Table 2. Specifications and performance of 2-D PtSi Schottky-barrier FPAs with CSD readout (Kimata, 1998). Reprinted with permission from M. Kimata, "PtSi Schottky-barrier infrared

focal plane arrays," Opto-Electronics Review, 6, 1, 1998.

In this section, the main Near-Infrared silicon photodetectors structures based on the aforementioned IPE effect will be reviewed. The simplest detectors consist of a metal layer on a semiconductor forming a Schottky contact at the material interface, with the Schottky barrier energy B determined by the materials (Sze, 1981). If an infrared radiation reaches metal/semiconductor interface, the conduction electrons inside the metal can absorb photons gaining sufficient energy. These excited (hot) electrons are able to cross over the Schottky barrier, sweep out the depletion region of the semiconductor, and be collected as a photocurrent under reverse bias operation from an ohmic semiconductor-metal interface. In practice, p-type silicon is often used as the semiconductor because Schottky barriers are lower thereupon than on n-type silicon, allowing detection at longer wavelengths. The IPEeffect, as described in the section 2.1, is very fast allowing to reach high data rates. On the other hand, it is inherently weak so several photodetector designs have been proposed for achieving a suitable quantum efficiency. Elabd *et a*l. proposed to use a very thin metal film (about 2 nm) to increase the escape probability of hot carriers due to their multiple reflections inside the metal film (Elabd et al., 1982). In particular, the proposed Schottky photodetector was characterized by a responsivity of about 250 mA/W for a wavelength =1500 nm. However, since the volume in which the photons interact with electrons in the metal is very small, only a small fraction of the incident photons actually causes photoemission. Several solutions have been proposed to enhance the efficiency of the IPE process. For example, in the 2001 Wang's group (Lee et al., 2001) investigated the spectral responsivity of Al-porous silicon Schottky barrier photodetectors in the wavelength range 0.4-1.7 µm. The structure of the PS photodetector was Al (finger type)/PS/Si/Al (ohmic), and the active area was 18 mm2. The photodetectors show strong photoresponsivity in both the visible and the infrared bands, especially at 1.55 µm. The photocurrent can reach 1.8 mA at a reverse bias of 6 V under illumination by a 1.55-µm, 10-mW laser diode. The corresponding quantum efficiency is 14.4%; this high value comes from a very high surfacearea-to-volume ratio, of the order of 200-800 m2/cm3 of porous Si. The dark current is 5 µA

Silicon Photodetectors Based on Internal

Institute of Physics.

of Physics.

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 69

Fig. 12. Schematic view of the Cu/p-Si Schottky barrier-based integrated photodetector proposed by Casalino et al. (Casalino et al., 2010c). Reprinted with permission from M. Casalino *et al.*, "Cu/p-Si Schottky barrier-based near infrared photodetector integrated with a silicon-on-insulator waveguide," Appl. Phys. Lett., 96, 241112. Copyright 2010 American

By means of this technological solution a very narrow semiconductor/metal barrier transverse to the optical field coming out from the waveguide has been achieved. The integrated photodetector was characterized by a responsivity of 0.08 mA/W at a wavelength of 1550 nm with an reverse bias of -1V. Measured dark current at -1 V was about 10 nA. Moreover, the authors assert that the thinness of Cu/p-Si Schottky barrier could enable a speed operation in the gigahertz range. An indirect evaluation of the bandwidth of the detector was reported to confirm the operation speed potentialities. A bandwidth of about 3 GHz was measured by Zhu *et al.* in (Zhu et al., 2008a) on a Schottky barrier based integrated photodetector, where the junction was achieved by a nickel silicide

Fig. 13. Schematic structure of waveguide-based silicide Schottky-barrier photodetector proposed by Zhu et al. (Zhu et al., 2008b, 2008a). Reprinted with permission from S. Zhu *et al.*, "Near-infrared waveguide-based nickel silicide Schottky-barrier photodetector for optical communications," Appl. Phys. Lett., 92, 081103. Copyright 2008 American Institute

In particular, the authors proposed the lengthening of a thin silicide layer on the surface of a SOI waveguide to achieve both a suitable optical absorption and an efficient photoexcitation of metal electrons or holes across the silicide/Si interface. A detailed analysis was reported

layer (NiSi2) on silicon (Zhu et al., 2008b, 2008c)(Fig. 13).

at -10 V. Recently, Casalino *et al.* (Casalino et al., 2008a, 2010b) propose to enhance the IPE absorption by a resonant cavity effect. The conceptual scheme of the proposed device in shown in Fig. 11.

Fig. 11. Schematic cross section of the photodetector proposed by Casalino *et al.* (Casalino et al., 2010b) in 2010.

The resonant cavity is a vertical-to-the-surface Fabry-Perot structure. It is formed by a buried reflector, a top mirror interface and, in the middle, a silicon cavity. The buried reflector is a Bragg mirror, realized by alternating layers of amorphous hydrogenated silicon (aSi:H) and silicon nitride (Si3N4). A Schottky metal layer (Cu), working both as active (absorbing) layer and as cavity mirror, is deposited above the silicon layer. The room temperature responsivity measurements on the device return a peak value of about 2.3 µA/W and 4.3 µA/W, respectively for 0 V and -10 mV of reverse bias applied (Casalino et al., 2008a). In (Casalino et al., 2010b) the authors propose both to scaling down and optimize the same device described in (Casalino et al., 2008a). In particular, the cavity finesse was increased allowing to reach a responsivity of 8 µA/W for a reverse bias of 100mV.

A substantial enhancement of the IPE process efficiency has been achieved increasing the interaction of light with the metal in the vicinity of the interface by the confinement of the infrared radiation into a silicon waveguide. This solution has effectively proven that Schottky diode photodetectors are good candidate for the highly integrated photonics circuits. An example of this approach has recently demonstrated by Coppola's group (Casalino et al., 2010c). The proposed device is schematically illustrated in Fig. 12. A rib waveguide was terminated on a deep trench that reaches down the buried oxide layer of the SOI wafer. A Cu/p-Si Schottky contact was fabricated on the vertical surface of the deep trench.

at -10 V. Recently, Casalino *et al.* (Casalino et al., 2008a, 2010b) propose to enhance the IPE absorption by a resonant cavity effect. The conceptual scheme of the proposed device in

Fig. 11. Schematic cross section of the photodetector proposed by Casalino *et al.* (Casalino et

The resonant cavity is a vertical-to-the-surface Fabry-Perot structure. It is formed by a buried reflector, a top mirror interface and, in the middle, a silicon cavity. The buried reflector is a Bragg mirror, realized by alternating layers of amorphous hydrogenated silicon (aSi:H) and silicon nitride (Si3N4). A Schottky metal layer (Cu), working both as active (absorbing) layer and as cavity mirror, is deposited above the silicon layer. The room temperature responsivity measurements on the device return a peak value of about 2.3 µA/W and 4.3 µA/W, respectively for 0 V and -10 mV of reverse bias applied (Casalino et al., 2008a). In (Casalino et al., 2010b) the authors propose both to scaling down and optimize the same device described in (Casalino et al., 2008a). In particular, the cavity finesse was

A substantial enhancement of the IPE process efficiency has been achieved increasing the interaction of light with the metal in the vicinity of the interface by the confinement of the infrared radiation into a silicon waveguide. This solution has effectively proven that Schottky diode photodetectors are good candidate for the highly integrated photonics circuits. An example of this approach has recently demonstrated by Coppola's group (Casalino et al., 2010c). The proposed device is schematically illustrated in Fig. 12. A rib waveguide was terminated on a deep trench that reaches down the buried oxide layer of the SOI wafer. A Cu/p-Si Schottky contact was fabricated on the vertical surface of the

increased allowing to reach a responsivity of 8 µA/W for a reverse bias of 100mV.

shown in Fig. 11.

al., 2010b) in 2010.

deep trench.

Fig. 12. Schematic view of the Cu/p-Si Schottky barrier-based integrated photodetector proposed by Casalino et al. (Casalino et al., 2010c). Reprinted with permission from M. Casalino *et al.*, "Cu/p-Si Schottky barrier-based near infrared photodetector integrated with a silicon-on-insulator waveguide," Appl. Phys. Lett., 96, 241112. Copyright 2010 American Institute of Physics.

By means of this technological solution a very narrow semiconductor/metal barrier transverse to the optical field coming out from the waveguide has been achieved. The integrated photodetector was characterized by a responsivity of 0.08 mA/W at a wavelength of 1550 nm with an reverse bias of -1V. Measured dark current at -1 V was about 10 nA. Moreover, the authors assert that the thinness of Cu/p-Si Schottky barrier could enable a speed operation in the gigahertz range. An indirect evaluation of the bandwidth of the detector was reported to confirm the operation speed potentialities. A bandwidth of about 3 GHz was measured by Zhu *et al.* in (Zhu et al., 2008a) on a Schottky barrier based integrated photodetector, where the junction was achieved by a nickel silicide layer (NiSi2) on silicon (Zhu et al., 2008b, 2008c)(Fig. 13).

Fig. 13. Schematic structure of waveguide-based silicide Schottky-barrier photodetector proposed by Zhu et al. (Zhu et al., 2008b, 2008a). Reprinted with permission from S. Zhu *et al.*, "Near-infrared waveguide-based nickel silicide Schottky-barrier photodetector for optical communications," Appl. Phys. Lett., 92, 081103. Copyright 2008 American Institute of Physics.

In particular, the authors proposed the lengthening of a thin silicide layer on the surface of a SOI waveguide to achieve both a suitable optical absorption and an efficient photoexcitation of metal electrons or holes across the silicide/Si interface. A detailed analysis was reported

Silicon Photodetectors Based on Internal

(Akbari et al., 2010).

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 71

barriers (Scales & Berini, 2010). In order to simplify the fabrication process and to obtain a very short device, Berini's group has proposed an asymmetric SPPs-based photodetector (Akbari et al., 2009, 2010). The proposed structure supports highly confined and highly attenuated SPP modes; this latter feature allows to fabricate shorter devices compared to the symmetric photodetectors. The device, sketched in Fig. 15, consists of a metal stripe cladded at the bottom by a layer of Silicon and covered by air and exhibits for a wavelength =1280nm a maximum value of responsivity of about 1 mA/W with a dark current of 6 μA

Fig. 15. Cross-section of the aymmetric SPPs Schottky detector proposed in (Akbari et al., 2010).

A detailed simulation analysis performed on asymmetric Schottky detector shows that a significant enhancement in the responsivity can be achieved for a thin metal stripe (about 5 nm) due to multiple internal reflections of excited carriers (Akbari et al., 2009). In particular, authors report that the enhancement is more noticeable for thin metal stripe on p-Si compared to device on n-Si. In this case, the energy range over which the hot electrons can experience multiple reflections is very small because of the larger Schottky barrier height. Finally, the same group has recently demonstrated that a considerable increase in responsivity can be reached with strong applied reverse bias (Olivieri et al., 2010). In particular, authors driving a no optimized detector like that shown in Fig. 15 into breakdown (V~–210 V), such that internal electronic gain is obtained by carrier

As mentioned above the possibility to achieve a strong light confinement implies both an improvement of responsivity and an important advancement in device miniaturization enabling the realization of on-chip photodetectors on the nanoscale. In this contest, Goykhman et al. in (Goykhmanet al., 2011) have been characterized a nanoscale silicon

The detector was fabricated employing a self-aligned approach of local-oxidation of silicon (LOCOS) on silicon on insulator substrate. This approach has been proved useful for fabricating in the same process both a low-loss bus photonic waveguide and the detector. Actually, the oxide spacers effectively define the nanometric area of metal-silicon interface and thus allow avoiding lateral misalignment between the silicon surface and the metal layer to form a Schottky contact. The so realized photodetector was characterized by a

multiplication, obtaining a responsivity of 2.35 mA/W.

surface-plasmon Schottky detector shown in Fig. 16.

on the influence of the silicide layer dimension on the performances of both NiSi2/p-Si and NiSi2/n-Si diodes. The overall performances of the NiSi2/p-Si structure have been resulted better than that relative to the NiSi2/n-Si interface, due to the lower Schottky barrier height of the latter structure. In particular, a responsivity of about 4.6 mA/W at a wavelength of 1550nm and a reverse bias of -1V was estimated for the NiSi2/p-Si diode against the value of 2.3 mA/W of the NiSi2/n-Si junction. Moreover, also the 3 nA of the measured dark current can be considered acceptable.

In order to increase further the capability of IPE-based silicon photodetector to detect longwavelength (infrared) photons, the possibility to use the concept of surface plasmon polaritons (SPPs) has been explored. SPP are TM-polarized electromagnetic waves trapped at or guided along a metal-dielectric interface. This effect was discovered several decades ago but attention has been renewed by the phenomena of enhanced optical transmission through metallic films with nanostructure (Reather, 1988; Ebbesen et al., 1998). In fact, SPPs are shorter in wavelength than the incident light providing a significant increase in spatial confinement and local field intensity. In other words, while optical systems are basically diffraction limited, surface plasmon polaritons allow a tight localization of optical field to strongly subwavelength dimensions at the metal-dielectric interface (Barnes et al., 2003; Maier, 2006). Such property means that the infrared light is directly guided toward the active area of the IPE detector, confining the optical power at the boundary between the materials forming the Schottky contact, thereby increasing the interaction of light with the metal in the vicinity of the interface where the photoemission process takes place. Berini's group at University of Ottawa has recently applied these concepts confirming that the high light confinement of the SPPs structures can significantly improve the detection capability of the IPE-based photodetector integrated on Si wafers. In particular, Scales *et al.* in (Scales et al., 2004, 2009) described a Schottky diode photodetector obtained embedding a metal stripe of finite width in a homogeneous dielectric cladding (symmetric structure). The proposed device (see Fig. 14) was characterized by a responsivity of about 0.1 A/W, a dark currents of 21 nA, and minimum detectable powers of -22 dBm at a wavelength =1550 nm.

Fig. 14. SPPs Schottky detector based on a metal stripe surrounded by silicon.

An accurate investigation of the performance of the same device for different wavelengths and for several metals forming Schottky contacts were carried out in (Scales et al., 2011). It is worth noting that in these symmetric structures the realization of a thin metal film buried in a semiconductor become the fabrication process more complicated, however, despite of this drawback, the IPE is enhanced due to emission carriers occurring through two Schottky

on the influence of the silicide layer dimension on the performances of both NiSi2/p-Si and NiSi2/n-Si diodes. The overall performances of the NiSi2/p-Si structure have been resulted better than that relative to the NiSi2/n-Si interface, due to the lower Schottky barrier height of the latter structure. In particular, a responsivity of about 4.6 mA/W at a wavelength of 1550nm and a reverse bias of -1V was estimated for the NiSi2/p-Si diode against the value of 2.3 mA/W of the NiSi2/n-Si junction. Moreover, also the 3 nA of the measured dark current

In order to increase further the capability of IPE-based silicon photodetector to detect longwavelength (infrared) photons, the possibility to use the concept of surface plasmon polaritons (SPPs) has been explored. SPP are TM-polarized electromagnetic waves trapped at or guided along a metal-dielectric interface. This effect was discovered several decades ago but attention has been renewed by the phenomena of enhanced optical transmission through metallic films with nanostructure (Reather, 1988; Ebbesen et al., 1998). In fact, SPPs are shorter in wavelength than the incident light providing a significant increase in spatial confinement and local field intensity. In other words, while optical systems are basically diffraction limited, surface plasmon polaritons allow a tight localization of optical field to strongly subwavelength dimensions at the metal-dielectric interface (Barnes et al., 2003; Maier, 2006). Such property means that the infrared light is directly guided toward the active area of the IPE detector, confining the optical power at the boundary between the materials forming the Schottky contact, thereby increasing the interaction of light with the metal in the vicinity of the interface where the photoemission process takes place. Berini's group at University of Ottawa has recently applied these concepts confirming that the high light confinement of the SPPs structures can significantly improve the detection capability of the IPE-based photodetector integrated on Si wafers. In particular, Scales *et al.* in (Scales et al., 2004, 2009) described a Schottky diode photodetector obtained embedding a metal stripe of finite width in a homogeneous dielectric cladding (symmetric structure). The proposed device (see Fig. 14) was characterized by a responsivity of about 0.1 A/W, a dark currents of

21 nA, and minimum detectable powers of -22 dBm at a wavelength =1550 nm.

Fig. 14. SPPs Schottky detector based on a metal stripe surrounded by silicon.

An accurate investigation of the performance of the same device for different wavelengths and for several metals forming Schottky contacts were carried out in (Scales et al., 2011). It is worth noting that in these symmetric structures the realization of a thin metal film buried in a semiconductor become the fabrication process more complicated, however, despite of this drawback, the IPE is enhanced due to emission carriers occurring through two Schottky

can be considered acceptable.

barriers (Scales & Berini, 2010). In order to simplify the fabrication process and to obtain a very short device, Berini's group has proposed an asymmetric SPPs-based photodetector (Akbari et al., 2009, 2010). The proposed structure supports highly confined and highly attenuated SPP modes; this latter feature allows to fabricate shorter devices compared to the symmetric photodetectors. The device, sketched in Fig. 15, consists of a metal stripe cladded at the bottom by a layer of Silicon and covered by air and exhibits for a wavelength =1280nm a maximum value of responsivity of about 1 mA/W with a dark current of 6 μA (Akbari et al., 2010).

Fig. 15. Cross-section of the aymmetric SPPs Schottky detector proposed in (Akbari et al., 2010).

A detailed simulation analysis performed on asymmetric Schottky detector shows that a significant enhancement in the responsivity can be achieved for a thin metal stripe (about 5 nm) due to multiple internal reflections of excited carriers (Akbari et al., 2009). In particular, authors report that the enhancement is more noticeable for thin metal stripe on p-Si compared to device on n-Si. In this case, the energy range over which the hot electrons can experience multiple reflections is very small because of the larger Schottky barrier height. Finally, the same group has recently demonstrated that a considerable increase in responsivity can be reached with strong applied reverse bias (Olivieri et al., 2010). In particular, authors driving a no optimized detector like that shown in Fig. 15 into breakdown (V~–210 V), such that internal electronic gain is obtained by carrier multiplication, obtaining a responsivity of 2.35 mA/W.

As mentioned above the possibility to achieve a strong light confinement implies both an improvement of responsivity and an important advancement in device miniaturization enabling the realization of on-chip photodetectors on the nanoscale. In this contest, Goykhman et al. in (Goykhmanet al., 2011) have been characterized a nanoscale silicon surface-plasmon Schottky detector shown in Fig. 16.

The detector was fabricated employing a self-aligned approach of local-oxidation of silicon (LOCOS) on silicon on insulator substrate. This approach has been proved useful for fabricating in the same process both a low-loss bus photonic waveguide and the detector. Actually, the oxide spacers effectively define the nanometric area of metal-silicon interface and thus allow avoiding lateral misalignment between the silicon surface and the metal layer to form a Schottky contact. The so realized photodetector was characterized by a

Silicon Photodetectors Based on Internal

Vol.49, pp. 587–593

No.24, pp. 241103.

Vol.121, pp. 399-402.

pp. 909-913.

No.7, pp. 075001

10571-10600

Jersey, USA

251104

Layer. *U.S. Patent 3,757,123*

*Opt. Express*, Vol.18, No.8, pp. 8505- 8514.

*Nature*, Vol.424, No.6950, pp. 824–830

*Engineers (SPIE)*, San Diego, Aug. 28-29

*Electronics,* Vol.QE-16, No.3, pp. 373-381

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 73

Akbari, A.; Tait, R. N. & Berini, P. (2010). Surface plasmon waveguide Schottky detector.

Archer, R.J. & Cohen, J. (1973). Schottky-Barrier Monolithic Detector Having Ultrathin Metal

Barnes, W. L.; Dereux, A. & Ebbesen, T.W. (2003) Surface plasmon subwavelength optics.

Boyle, W.S. & Smith, G.E. (1970). Charge-coupled semiconductor devices. *Bell Syst. Tech. J.*,

Bradley, J.D.B.; Jessop, P.E. & Knights, A.P. (2005). Silicon waveguide-integrated optical

Capone, B.R.; Skolnik, L.H.; Taylor, R.W.; Shepherd, F.D.; Roosild, S.A.; Ewing, W.;

Casalino, M. ; Sirleto, L. ; Moretti, L. ; Della Corte, F. & Rendina, I. (2006a). Design of a

Casalino, M. ; Sirleto, L. ; Moretti, L. ; Della Corte, F. & Rendina, I. (2006b). Design of a

Casalino, M.; Sirleto, L.; Moretti, L.; Gioffrè, M.; Coppola, G. & Rendina, I. (2008a). Silicon

Casalino, M.; Sirleto, L.; Moretti, L. & Rendina, I. (2008b). A silicon compatible resonant

Casalino, M. ; Coppola, G. ; Iodice, M. ; Rendina, I. & Sirleto, L. (2010a). Near Infrared All-

Casalino, M.; Coppola, G.; Gioffrè, M.; Iodice, M.; Moretti, L.; Rendina, I. & Sirleto, L.

Chan, E.Y. & Card, H.C. (1980b). Near IR interband transitions and optical parameters of

Donati, S. (1999). *Photodetectors: Devices, circuits, and applications*. Prentice Hall PTR, New

metal-germanium contacts. *Applied Optics*, Vol.19, No.8, pp. 1309

sub-bandgap detection. *J. Lightw. Technol.*, Vol. 28, No.22, pp. 3266-3272 Casalino, M.; Sirleto, L.; Iodice, M.; Saffioti, N.; Gioffrè, M.; Rendina, I & Coppola, G.

power monitor with enhanced sensitivity at 1550 nm. *Appl. Phys. Lett.,* Vol. 86,

Kosonocky, W.F. & Kohn, E.S. (1978). Evaluation of Schottky IRCCD Staring Mosaic Focal Plane. *22nd Int. Tech. Symp. Society of Photo-Optical Instrumentation* 

silicon RCE schottky photodetector working at 1.55 micron. *Journal of luminescence*,

silicon resonant cavity enhanced photodetector based on the internal photoemission effect at 1.55 µm. *Journal of Optics A: Pure and applied optics*, Vol.8,

resonant cavity enhanced photodetector based on the internal photoemission effect at 1.55 µm: Fabrication and characterization. *Appl. Phys. Lett.*, Vol.92, No.25, pp.

cavity enhanced photodetector working at 1.55 μm. *Semicond. Sci. Technol.*, Vol.23,

Silicon Photodetectors: State of the Art and Perspectives. *Sensors*, Vol.10, No.12, pp.

(2010b). Cavity enhanced internal photoemission effect in silicon photodiode for

(2010c). Cu/p-Si Schottky barrier-based near infrared photodetector integrated with a silicon-on-insulator waveguide. *Appl. Phys. Lett.*, Vol.96, No.24, pp. 241112. Chan, E.Y; Card, H.C. & Teich, M.C. (1980a). Internal Photoemission Mechanism at

interfaces between Germanium and Thin Metal Films. *IEEE Journal of Quantum* 

responsivity 0.25, 1.4, and 13.3 mA/W for incident optical wavelengths of 1.55, 1.47, and 1.31 µm, respectively.

Fig. 16. SEM micrograph of the nanoscale Schottky contact proposed in (Goykhman et al., 2011). Reprinted with permission from I. Goykhmanet *al.*, "Locally Oxidized Silicon Surface-Plasmon Schottky Detector for Telecom Regime," Nano Lett. 11, 2219–2224. Copyright 2011 american Chemical Society.

### **4. Conclusion**

In this chapter an overview on the NIR all-Si photodetectors based on the IPE has been presented. First, we have attempted to elucidate the IPE effects allowing Si absorption at sub band-gap wavelengths and the main figures of merit of IPE-based devices. Then, a quantitative comparison of the photodetectors proposed in the scientific literature, including both bulk and integrated devices, have been reviewed. Unfortunately, the efficiency of devices is low compared to that of detectors based on inter-band absorption. This property is a direct result of many factors: 1) the low absorption due to high reflectivity of the metal layer at NIR wavelengths, 2) the conservation of momentum during carrier emission over the potential barrier which lowers the carriers emission probability into semiconductor, 3) the excitation of carriers lying in states far below the Fermi energy, which get very low probability to overcome the Schottky barrier. While the first and second points can be partially improved by using an optical cavity and a thin metal film, respectively, the third point remains the main limiting factor of IPE. In 1971 Sheperd, Vickers and Yang (Sheperd et al., 1971) recommended replacing the metal electrodes IPE-based photodiodes with degenerate semiconductors. They reasoned that reducing the Fermi energy of electrode by substitution of a degenerate semiconductor could improve emission efficiency as much as 20-fold. Even if NIR IPE-based devices efficiency is still only adapted for power monitoring application, in our opinion that new structures based on the aforementioned insights could play a key role in telecommunication field and could open new frontiers in the field of lowcost silicon photonic.

#### **5. References**

Akbari A. & Berini, P. (2009). Schottky contact surface-plasmon detector integrated with an asymmetric metal stripe waveguide. *Appl. Phys. Lett.,* Vol.95, No.2, pp. 021104.

responsivity 0.25, 1.4, and 13.3 mA/W for incident optical wavelengths of 1.55, 1.47, and

Fig. 16. SEM micrograph of the nanoscale Schottky contact proposed in (Goykhman et al., 2011). Reprinted with permission from I. Goykhmanet *al.*, "Locally Oxidized Silicon Surface-Plasmon Schottky Detector for Telecom Regime," Nano Lett. 11, 2219–2224. Copyright 2011

In this chapter an overview on the NIR all-Si photodetectors based on the IPE has been presented. First, we have attempted to elucidate the IPE effects allowing Si absorption at sub band-gap wavelengths and the main figures of merit of IPE-based devices. Then, a quantitative comparison of the photodetectors proposed in the scientific literature, including both bulk and integrated devices, have been reviewed. Unfortunately, the efficiency of devices is low compared to that of detectors based on inter-band absorption. This property is a direct result of many factors: 1) the low absorption due to high reflectivity of the metal layer at NIR wavelengths, 2) the conservation of momentum during carrier emission over the potential barrier which lowers the carriers emission probability into semiconductor, 3) the excitation of carriers lying in states far below the Fermi energy, which get very low probability to overcome the Schottky barrier. While the first and second points can be partially improved by using an optical cavity and a thin metal film, respectively, the third point remains the main limiting factor of IPE. In 1971 Sheperd, Vickers and Yang (Sheperd et al., 1971) recommended replacing the metal electrodes IPE-based photodiodes with degenerate semiconductors. They reasoned that reducing the Fermi energy of electrode by substitution of a degenerate semiconductor could improve emission efficiency as much as 20-fold. Even if NIR IPE-based devices efficiency is still only adapted for power monitoring application, in our opinion that new structures based on the aforementioned insights could play a key role in telecommunication field and could open new frontiers in the field of low-

Akbari A. & Berini, P. (2009). Schottky contact surface-plasmon detector integrated with an asymmetric metal stripe waveguide. *Appl. Phys. Lett.,* Vol.95, No.2, pp. 021104.

1.31 µm, respectively.

american Chemical Society.

**4. Conclusion** 

cost silicon photonic.

**5. References** 


Silicon Photodetectors Based on Internal

Vol.1682, pp. 2–19

No.3, pp. 1440-1445

Berlin, Germany

Vol.40, pp. 279–294

pp. 392-393

Vol.15, No. 8, pp. 4745-4752

*Electronics*, Vol.32, pp. 97–101

Photoemission Effect: The Challenge of Detecting Near-Infrared Light 75

Kosonocky, W.F. (1991). Review of infrared image sensors with Schottky–barrier detectors.

Kosonocky, W.F. (1992). State-of-the-art in Schottky-barrier IR image sensors. *SPIE Proc.*,

Kurianski, J.M.; Shanahan, S.T.; Theden, U.; Green, M.A. & Storey, J.W.V. (1989).

Lee, M. K.; Chu, C. H.; Wang & Y. H. (2001) 1.55-µm and infrared-band photoresponsivity of

Liang, T.K.; Tsang, H.K.; Day, I.E.; Drake, J.; Knights, A.P. & Asghari, M. (2002). Silicon

Liu, A.; Liao, L.; Rubin, D.; Nguyen, H.; Ciftcioglu, B.; Chetrit, Y.; Izhaky, N. & Paniccia, M.

Michael, C.P.; Borselli, M.; Johnson, T.J.; Chrystal, C. & Painter, O. (2007). An optical fiber-

Muriel, M.A . & Carballar, A. (1997). Internal field distributions in fiber Bragg gratings. *IEEE* 

Olivieri, A.; Akbari A. & Berini, P. (2010) Surface plasmon waveguide Schottky detectors operating near breakdown. *Phys. Status Solidi RRL*. Vol 4, No.10, pp. 283 – 285 Reather, H. (1988). *Surface Plasmons on Smooth and Rough Surfaces and on Gratings*. Springer,

Rogalski, A. (1999). Assessment of HgCdTe photodiodes and quantum well infrared

Rowe, L.K.; Elsey, M. ; Tarr, N.G. ; Knights, A.P. & Post, E. (2007). CMOS-compatible optical

Scales, C.; Breukelaar, I.; Charbonneau, R. & Berini, P. (2011). Infrared Performance of

Sheperd, F.D.; Vickers, V.E. & Yang, A.C. (1971). Schottky Barrier Photodiode with a Degenerate Semiconductor Active Region. U.S. Patent No. 3.603.847 Shepherd, F.D. & Yang, A.C. (1973). Silicon Schottky Retinas for Infrared Imaging. *IEDM* 

Scales, C. & Berini, P. (2004). Schottky Barrier Photodetectors, *U.S. Patent No. 7,026,701*. Scales, C.; Breukelaar, I. & Berini, P. (2009). Surface-plasmon Schottky contact detector based on a symmetric metal stripe in silicon. *Opt. Lett*., Vol. 35, No.4, pp. 529–531 Scales, C. & Berini, P. (2010). Thin-film Schottky barrier Photodetector Models. *IEEE Journal* 

photoconductors for long wavelength focal plane arrays. *Infrared Phys. Technol.*,

rib waveguides defined by local oxidation of silicon. *Electron. Lett.*, Vol.43, No.6,

Symmetric Surface-Plasmon Waveguide Schottky Detectors in Si. *IEEE J. Lightw.* 

autocorrelation measurements. *Appl. Phys. Lett.*, Vol.81, No.7, pp. 1323 Liu, A.; Jones, R.; Cohen, O.; Hak, D. & Paniccia M. (2006). Optical amplification and lasing

Maier A. (2006). *Plasmonics: Fundamentals and Applications.* Springer, New York, USA

Optimization of the cavity for silicide Schottky infrared detectors. *Solid-State* 

a Schottky barrier porous silicon photodetector. *Opt. Lett.*, Vol.26, No.3, pp. 160-162

waveguide two-photon absorption detector at 1.5 µm wavelength for

by stimulated Raman scattering in silicon waveguides. *J. Lightw. Technol.*, Vol.24,

(2007). High-speed optical modulation based on carrier depletion in a silicon

taper probe for wafer-scale microphotonic device characterization. *Opt. Express*,

*Optoelectronics Devices and Technologies*, Vol.6, pp. 173–203

waveguide. *Opt. Express,* Vol.15, No. 2, pp. 660-668

*Photonics technology letters*, Vol. 9, No.7, pp. 955, 1997

*of Quantum Electronics,* Vol.46, No.5, pp. 633-643

*Tech.*, Vol 29, No. 12, pp. 1852-1860

*Tech. Dig.*, pp. 310-313


Ebbesen, W. ; Lezec, H.J. ; Ghaemi, H.F. ; Thio, T. & Wolff, P.A. (1997). Extraordinary optical transmission through sub-wavelength hole arrays. *Nature, Vol.*391, pp 667-669 Elabd, H. & Kosonocky, W.F. (1982). Theory and measurements of photoresponse for thin

Elabd, H.; Villani, T. & Kosonocky, W.F. (1982). Palladium-Silicide Schottky-Barrier IR-CCD

Fowler, R.H. (1931). The Analysis of Photoelectric Sensitivity Curves for Clean Metals at

Goykhman, I.; Desiatov, B.; Khurgin, J.; Shappir, J. & Levy U. (2011). Locally Oxidized

Inoue, M.; Seto, T.; Takahashi, S.; Itoh, S.; Yagi, H.; Siraishi, T.; Endo, K. & Kimata, M. (1997).

Jalali, B & Fathpour, S. (2006). Silicon Photonics. *J. Lightwave Technol.*, Vol.24, No.12, pp.

Kimata, M.; Yutani, N.; Tsubouchi, N. & Seto, T. (1992). High performance 1040x1040 element PtSi Schottky-barrier image sensor. *SPIE Proc.*, Vol.1762, pp. 350–360 Kimata, M. & Tsubouchi, N. (1995). Schottky barrier photoemissive detectors. In: *Infrared* 

Kimata, M.; Ozeki, T.; Nunoshita, M. & Ito, S. (1997). PtSi Schottky-barrier infrared FPAs

Kimata, M.; Ueno, M.; Yagi, H.; Shiraishi, T.; Kawai, M.; Endo, K.; Kosasayama, Y.; Sone, T.;

Kimata, M. (2000). Metal silicide Schottky infrared detector arrays. In: *Infrared Detectors and Emitters: Materials and Devices*, Kluwer Academic Publishers, Boston, USA Kimerling, L.C.; Dal Negro, L.; Saini, S.; Yi, Y.; Ahn, D.; Akiyama, S.; Cannon, D.; Liu, J.;

Various Temperatures. *Physical Review*, Vol. 38, pp. 45-56

with CSD readout. *SPIE Proc.*, Vol.3179, pp. 212–223

*Opto-Electronics Review*, Vol.6, pp. 1–10

Sensors. *SPIE Proc.*, Vol. 225, pp. 69-71

*Review*, Vol. 43, pp. 569–589

*Lett.*, Vol. EDL-3, pp. 89-90

pp. 2219–2224.

4600-4615

Bellingham

Vol.3061, pp. 150–158

film Pd2Si and PtSi infrared Schottky-barrier detectors with optical cavity. *RCA* 

for SWIR Applications at Intermediate Temperatures. *IEEE Trans. Electron Devices* 

Silicon Surface-Plasmon Schottky Detector for Telecom Regime. Nano Lett. Vol. 11,

Portable high performance camera with 801x512 PtSi-SB IRCSD, *SPIE Proc.*,

*Photon Detectors*, A. Rogalski (Ed.), pp. 299-349, SPIE Optical Engineering Press,

Ozeki, T. & Tsubouchi, N. (1998). PtSi Schottky-barrier infrared focal plane arrays,

Sandland, J.G.; Sparacin, D.; Michel, J.; Wada, K. & Watts, M.R. (2004). *Silicon Photonics: Topics in Applied Physics,* Springer, ISBN 3-642-05909-0, Berlin, Germania Kohn, E.S.; Roosild, S.A.; Shepherd, F.D. & Yang, A.C. (1975). Infrared Imaging with

Monolithic CDD-Addressed Schottky-Barrier Detector Arrays, Theoretical and Experimental Results. *Int. Conf. on Application on CCD's*, San Diego, Oct. 29-31 Kosonocky, W.F.; Kohn, E.S.; Shalleross, F.V.; Sauer, D.J.; Shepherd, F.D.; Skolnik, L.H.; Taylor,

R.W.; Capone, B.R. & Roosild, S.A. (1978). Platinum Silicide Schottky-Barrier IR-CCD

Klein, J.; Skolnik, L.H.; Capone, B.R.; Taylor, R.W.; Ewing, W.; Shepherd, F.D. & Roosild, S.A. (1980). Advances in Platinum-Silicide Schottky-Barrier IR-CCD Image

Barrier IR-CCD Image Sensor. *IEEE Trans. Electron Dev.,* vol. ED-32, No.8, pp. 1564

Image Sensors. *Int. Conf. on Application on CCD's*, San Diego, Oct. 25-27

Kosonocky, W.F.; Erhardt, H.G.; Meray, G.M.; Shallcross, F.V.; Elabd, H.A.; Cantella, M.;

Kosonocky, W.F. ; Shallcross, F.V. & Villani, T.S. (1985). 160x244 Element PtSi Schottky-


**0**

**4**

*Italy*

**Silicon Photomultipliers:**

Maria Bondani2 and Massimo Caccia1

**Characterization and Applications**

<sup>1</sup>*Dipartimento di Fisica e Matematica - Università degli Studi dell'Insubria* <sup>2</sup>*Istituto di Fotonica e Nanotecnologie - Consiglio Nazionale delle Ricerche*

Silicon Photo-Multipliers (SiPMs henceover) are photo-detectors based on a technology originally invented in Russia (Akindinov et al., 1997). They essentially consist of an *array of p-n junctions* operated beyond the breakdown voltage (McKay, 1954), in a Geiger-M ¨uller (G-M) regime (Oldham et al., 1972), with typical gain of the order of 106 and on-cell integrated quenching mechanisms. Silicon photo-detectors with internal multiplication are in use since more than a decade (Lutz, 1995). Avalance Photo-Diodes (APDs) (Akindinov et al., 2005) are operated in a proportional regime, with typical gains of 104, Single-Photon Avalanche Diodes (SPADs) (Cova et al., 1996) are endowed with single-photon sensitivity and are tailored for high frequency counting with time resolutions down to 30 ps. However, being made of a single cell operated in binary mode, they do not carry any information about the intensity of

SiPMs complement the family of existing sensors: with a cell density of <sup>∼</sup> 103/mm2, areas up to 3 <sup>×</sup> 3 mm<sup>2</sup> and *a single output node*, they offer the possibility of measuring the intensity of the light field simply by counting the number of fired cells. The main features of SiPMs, due

• high gain, granted by G-M operating mode, comparable to the values achieved by standard

• enhanced linearity, owing to cell structure, with deviations that become relevant when the average number of detected photons approaches the same order of magnitude of the

• large dynamic range, provided by the pixelated structure with a common output, spanning

• operability in magnetic fields, compactness and relatively low cost, granted by

On the other hand, since G-M avalanches that are triggered by electron-hole pairs extracted by the impinging photons are obviously indistinguishable from the ones originated from other

processes, dealing with all possible sources of noise is far from being trivial:

**1. Introduction**

the incoming light field.

photomultipliers (PMTs);

silicon-based technology.

to their structural and operational characteristics are:

number of cells of the device (Tarolli et al., 2010);

from the single photon regime up to high intensities;

Marco Ramilli1, Alessia Allevi1, Luca Nardo1,


## **Silicon Photomultipliers: Characterization and Applications**

Marco Ramilli1, Alessia Allevi1, Luca Nardo1, Maria Bondani2 and Massimo Caccia1 <sup>1</sup>*Dipartimento di Fisica e Matematica - Università degli Studi dell'Insubria* <sup>2</sup>*Istituto di Fotonica e Nanotecnologie - Consiglio Nazionale delle Ricerche Italy*

#### **1. Introduction**

76 Photodetectors

Shepherd, F.D.; Taylor, R.W.; Skolnik, L.H.; Capone, B.R.; Roosild, S.A.; Kosonocky, W.F. &

Shepherd, F.D. (1984). Schottky diode based infrared sensors. *SPIE Proc.*, Vol. 443, pp. 42–49

Shepherd, F.D. (1998). *Platinum silicide internal emission infrared imaging arrays.* Academic

Shiraishi, T.; Yagi, H.; Endo, K.; Kimata, M.; Ozeki, T.; Kama, K. & Seto, T. (1996). PtSi FPA

Taylor, R.W.; Skolnik, L.H.; Capone, B.R.; Ewing, W.; Shepherd, F.D.; Roosild, S.A.;

Vickers, V.E. (1971). Model of Schottky Barrier Hot-electron-Mode Photodetection. *Applied* 

Vivien, L.; Pascal, D.; Lardenois, S.; Marris-Morini, D.; Cassan, E.; Grillot, F.; Laval, S.;

Yeh, P. (1988). *Optical Waves in Layerer Media*. Wiley Interscience Publication, New York, USA Yuan, H.X. & Perera, G.U. (1995). Dark current analysis of Si homojunction interfacial work

Zhu, S.; Yu, M.B.; Lo, G.Q. & Kwong, D.L. (2008a). Near-infrared waveguide-based nickel

Zhu, S.; Lo, G.Q. & Kwong, L. (2008b). Low-cost and high-gain silicide Schottky-barrier

Zhu, S.; Lo, G.Q. & Kwong, D.L. (2008c) Low-Cost and High-Speed SOI Waveguide-Based

Communications. *IEEE Phot. Tech. Lett.*, Vol. 20, No.16, pp. 1396-1398

Cochrum, B.; Cantella, M.; Klein, J.; Kosonocky, W.F. (1980). Improved Platinum

Fedeli, J.M. & El Melhaoui, L. (2006). Light injection in SOI microwaveguides using high-efficiency grating couplers. *J. Lightw. Technol*., Vol.24, No.10, pp. 3810-3815 Xu, Q.; Manipatruni, S.; Schmidt, B.; Shakya, J. & Lipson, M. (2007). 12.5 Gbit/s carrier-injectionbased silicon micro-ring silicon modulators. *Opt. Express,* Vol.15, No. 2,pp. 430-436 Yagi, H.; Yutani, N.; Nakanishi, J.; Kimata, M. & Nunoshita, M. (1994). A monolithic

Schottky-barrier infrared image sensor with 71% fill factor. *Optical Engineering*,

function internal photoemission far-infrared detectors. *Appl. Phys. Lett.*, Vol.66,

silicide Schottky-barrier photodetector for optical communications. *Appl. Phys. Lett.*,

collector phototransistor integrated on Si waveguide for infrared detection. *Appl.* 

Silicide Schottky-Barrier MSM Photodetectors for Broadband Optical

*7th Symp. Photo-Electronic Image Devices*, Vol. 22, pp. 495-512

Press, New York, USA

*Optics, Vol.* 10, No.9, pp. 219

Vol.33, pp. 1454–1460

No.17, pp. 2262-2264

Vol.92, No.8, pp. 081103

*Phys. Lett.*, Vol. 93, No.7, pp. 071108

Shepherd, F.D. (1988). Silicide infrared staring sensors. *SPIE Proc.*, Vol.930, pp. 2–10

with improved CSD operation. *SPIE Proc.*, Vol.2744, pp. 33–43 Sze, S.M. (1981). *Physics of Semiconductor Devices*. John Wiley & Sons, New York, USA

Silicide IRCCD Focal Plane. *SPIE Proc.*, Vol. 217, pp. 103-110

Kohn, E.S. (1979). Schottky IRCCD Thermal Imaging. *Adv. Electron. Electron Phys.,* 

Silicon Photo-Multipliers (SiPMs henceover) are photo-detectors based on a technology originally invented in Russia (Akindinov et al., 1997). They essentially consist of an *array of p-n junctions* operated beyond the breakdown voltage (McKay, 1954), in a Geiger-M ¨uller (G-M) regime (Oldham et al., 1972), with typical gain of the order of 106 and on-cell integrated quenching mechanisms. Silicon photo-detectors with internal multiplication are in use since more than a decade (Lutz, 1995). Avalance Photo-Diodes (APDs) (Akindinov et al., 2005) are operated in a proportional regime, with typical gains of 104, Single-Photon Avalanche Diodes (SPADs) (Cova et al., 1996) are endowed with single-photon sensitivity and are tailored for high frequency counting with time resolutions down to 30 ps. However, being made of a single cell operated in binary mode, they do not carry any information about the intensity of the incoming light field.

SiPMs complement the family of existing sensors: with a cell density of <sup>∼</sup> 103/mm2, areas up to 3 <sup>×</sup> 3 mm<sup>2</sup> and *a single output node*, they offer the possibility of measuring the intensity of the light field simply by counting the number of fired cells. The main features of SiPMs, due to their structural and operational characteristics are:


On the other hand, since G-M avalanches that are triggered by electron-hole pairs extracted by the impinging photons are obviously indistinguishable from the ones originated from other processes, dealing with all possible sources of noise is far from being trivial:

Characterization and Applications 3

Silicon Photomultipliers: Characterization and Applications 79

In particular, studies of the main SiPM parameters (such as gain, PDE, DCR and optical

In order to implement the characterization protocol, we have developed an experimental setup in which a green-emitting LED (*λ* = 510 nm), coupled to a fast pulse generator (PDL800-B PicoQuant), has been used as the light source. The SiPM output signal has been

**–** in experimental situations where the SiPM was operated under a continuous light flux, a transimpedance pre-amplifier provided by SensL has been used: this device converts the raw current from the SiPM into a voltage, with an amplification of 470 V/A; **–** when the SiPM was operated in the pulsed regime, a different pre-amplifier, called Pulse Amplifier and also provided by SensL, has been used. It allowed the fast rise of

• a leading edge discriminator (Lecroy 821) with a user-defined voltage threshold that the SiPM output has to exceed in order to provide a triggering signal has been then exploited

• in case of frequency measurements (e.g. DCR measurements), the discriminator output

**–** the board provides a charge measurement performing the integration of the input

**–** the integration gate is generated by a NIM timing unit and must precede the analog

The system composed by the investigated SiPM and the first-stage amplification board has been located in a metal box sealed with grease, in which air has been replaced with helium. The cooling fluid has been pumped into the box through a copper pipe, allowing controlled temperature variations. The temperature of the system has been measured by a thermistor

In order to quantitatively understand the effects of temperature changes on the SiPM main figures of merit, a suitable set of measurements must be accomplished. The procedure has been tested with SensL CSI 0747 015 A20 HD and the results regarding this sensor will be presented in more detail in the following. Once the procedure was tested, it has been repeated with different SiPM models: a SensL CSI 0740 001 A20 HD, a Hamamatsu S10362-11-100C and

Sensors have been placed in the cooling box described in Section 2.1 and several spectra have been acquired at different bias voltages and different temperatures in order to obtain the values of gain and PDE. Moreover, DCR measurements at different thresholds were used to evaluate both DCR and cross-talk contributions. To analyze the procedure in detail, a typical

cross-talk) as functions of temperature will be described in details.

• SiPM was directly connected to a first stage amplification board:

the detector to be exploited, providing an amplification factor of 20;

• the output of the SiPM has been integrated by a CAEN QDC V792N board:

• data have been stored in a PC via a USB-VME Bridge (CAEN).

placed in contact with the external packaging of the sensor.

voltage signal, with a conversion time of 2.8 *μ*s and a reset time of 4 *μ*s;

**2.1 Experimental setup**

elaborated in the following way:

at the output of the amplifier;

has been directly sent to a scaler;

input signal of at least 15 ns;

**2.2 Studying temperature behavior**

a ST Microelectronics TO-8 prototype.


These apparently huge disadvantages can be overridden by means of an exhaustive characterization of the SiPM performances. An example of this procedure will be given in the following Section, where we present the characterization of the devices as a function of temperature.

Another aspect of the characterization of SiPMs will be addressed by providing a description of the G-M avalanches probability distribution functions, in order to reconstruct the statistics of the impinging light (Ramilli et al., 2010).

Modeling the SiPM response is also preliminary to application of these sensors as detectors in Fluorescence Fluctuation Spectroscopy biophysical experiments (Chen et al., 1999; Schwille, 2001), where the parameters describing the system under investigation are inferred by the deviations of the fluorescence intensity of suitable probes around its mean value. A feasibility study on this topic will be presented in the final part of the Chapter.

### **2. Characterization**

Tests have been performed on existing devices with the main goal to define an exhaustive characterization protocol and to produce a comparative study. Three kinds of detectors from different manufacturers have been studied: SensL1, Hamamatsu Photonics2 and STMicroelectronics3. Detector characterization is a major task for all the applications of SiPMs and in particular for the identification of critical parameters: for example, DCR is an important parameter for low-rate-event applications, while thermal stability is essential for portable devices. For this reason, the following characterization protocol has been developed assessing:


<sup>1</sup> see http://sensl.com/.

<sup>2</sup> see http://www.hamamatsu.com/.

<sup>3</sup> see http://www.st.com/stonline/.

In particular, studies of the main SiPM parameters (such as gain, PDE, DCR and optical cross-talk) as functions of temperature will be described in details.

#### **2.1 Experimental setup**

2 Will-be-set-by-IN-TECH

• thermally extracted electron-hole pairs cause high Dark Count Rates (DCR), with values ranging from several hundreds of kHz up to the MHz level, depending on the total number

• spurious signals are also due to optical cross-talk (Sciacca et al., 2008): photons emitted by an avalanche can travel through silicon and reach the depleted region in a neighboring cell,

• carriers extracted during an avalanche process may be trapped in a false potential minimum in the depleted region: escaping from that trap they can originate an *afterpulse* avalanche, so called because it typically happens shortly after (or even during) the recovery

These apparently huge disadvantages can be overridden by means of an exhaustive characterization of the SiPM performances. An example of this procedure will be given in the following Section, where we present the characterization of the devices as a function of

Another aspect of the characterization of SiPMs will be addressed by providing a description of the G-M avalanches probability distribution functions, in order to reconstruct the statistics

Modeling the SiPM response is also preliminary to application of these sensors as detectors in Fluorescence Fluctuation Spectroscopy biophysical experiments (Chen et al., 1999; Schwille, 2001), where the parameters describing the system under investigation are inferred by the deviations of the fluorescence intensity of suitable probes around its mean value. A feasibility

Tests have been performed on existing devices with the main goal to define an exhaustive characterization protocol and to produce a comparative study. Three kinds of detectors from different manufacturers have been studied: SensL1, Hamamatsu Photonics2 and STMicroelectronics3. Detector characterization is a major task for all the applications of SiPMs and in particular for the identification of critical parameters: for example, DCR is an important parameter for low-rate-event applications, while thermal stability is essential for portable devices. For this reason, the following characterization protocol has been developed assessing:

• Geometrical parameters (number of cells, size of detectors and occupancy factor);

• Noise measurements: DCR, optical cross-talk, dependence on the environmental

• Analysis of photon spectra: resolution power, gain, working point optimization (at low and large flux), electronic noise measurement taking into account cell-to-cell variations,

of cells, the operating temperature and the overbias;

of the previous avalanche (Du et al., 2008; Eckert et al., 2010).

study on this topic will be presented in the final part of the Chapter.

dependence on the environmental parameters (temperature).

• Spectral response measurement: photon detection efficiency (PDE).

thus triggering another avalanche;

of the impinging light (Ramilli et al., 2010).

temperature.

**2. Characterization**

• I–V measurements;

<sup>1</sup> see http://sensl.com/.

• Linearity and dynamic range.

<sup>2</sup> see http://www.hamamatsu.com/. <sup>3</sup> see http://www.st.com/stonline/.

parameters.

In order to implement the characterization protocol, we have developed an experimental setup in which a green-emitting LED (*λ* = 510 nm), coupled to a fast pulse generator (PDL800-B PicoQuant), has been used as the light source. The SiPM output signal has been elaborated in the following way:

	- **–** in experimental situations where the SiPM was operated under a continuous light flux, a transimpedance pre-amplifier provided by SensL has been used: this device converts the raw current from the SiPM into a voltage, with an amplification of 470 V/A;
	- **–** when the SiPM was operated in the pulsed regime, a different pre-amplifier, called Pulse Amplifier and also provided by SensL, has been used. It allowed the fast rise of the detector to be exploited, providing an amplification factor of 20;
	- **–** the board provides a charge measurement performing the integration of the input voltage signal, with a conversion time of 2.8 *μ*s and a reset time of 4 *μ*s;
	- **–** the integration gate is generated by a NIM timing unit and must precede the analog input signal of at least 15 ns;

The system composed by the investigated SiPM and the first-stage amplification board has been located in a metal box sealed with grease, in which air has been replaced with helium. The cooling fluid has been pumped into the box through a copper pipe, allowing controlled temperature variations. The temperature of the system has been measured by a thermistor placed in contact with the external packaging of the sensor.

#### **2.2 Studying temperature behavior**

In order to quantitatively understand the effects of temperature changes on the SiPM main figures of merit, a suitable set of measurements must be accomplished. The procedure has been tested with SensL CSI 0747 015 A20 HD and the results regarding this sensor will be presented in more detail in the following. Once the procedure was tested, it has been repeated with different SiPM models: a SensL CSI 0740 001 A20 HD, a Hamamatsu S10362-11-100C and a ST Microelectronics TO-8 prototype.

Sensors have been placed in the cooling box described in Section 2.1 and several spectra have been acquired at different bias voltages and different temperatures in order to obtain the values of gain and PDE. Moreover, DCR measurements at different thresholds were used to evaluate both DCR and cross-talk contributions. To analyze the procedure in detail, a typical

Characterization and Applications 5

Silicon Photomultipliers: Characterization and Applications 81

SiPM Model m*BD* (mV/degree) SensL CSI 0747 015 A20 HD 23.2 ± 1.4 SensL CSI 0740 001 A20 HD 23.6 ± 0.9 Hamamatsu S10362-11-100C 61.9 ± 0.7 STM TO-8 prototype 31.5 ± 0.1

our range of interest, a linear dependence on temperature (Goetzberger et al., 1963), the over voltage, defined as the difference between the applied bias voltage and the breakdown one, is

where m*<sup>V</sup>* and m*<sup>T</sup>* are the slopes, *VBD*(*T*) is the breakdown voltage and *T*<sup>0</sup> is a reference temperature. Solving these equations for *VBD*(*T*), the rate of change of the breakdown voltage

<sup>m</sup>*BD* <sup>=</sup> <sup>−</sup> <sup>m</sup>*<sup>T</sup>*

The procedure has been applied to the different SiPMs. From the results in Table 1, we can see

By using the breakdown voltage value measured at room temperature (RT) as the reference

**<sup>3</sup>** ×**10 / ndf 2** χ **0.772 / 8**

−16 ° C −10 ° C +1 ° C +12 ° C +21 ° C

**29 29.5 30 30.5 31 31.5 32**

From Equation 6 it is possible to express the gain and all the other measured parameters as a function of the over voltage: in Fig. 4 the gain is plotted as a function of the over voltage, showing a global linear behavior independent of the temperature. Equation 6 can be used to fix the operational parameters of a SiPM regardless of the varying environmental conditions: if m*BD* is known with enough precision and frequent measurements of the SiPM temperature are performed, the over voltage across each cell can be maintained fixed by simply continuously adjusting the applied bias according to the temperature variations. This

Fig. 2. Gain as a function of bias voltage, for different temperatures; a linear fit has been

performed for each temperature set: the obtained fit slopes are in agreement.

value, the breakdown values for each temperature can be written as:

m*<sup>V</sup>*

**p0 −1.766e+07** ± **4.107e+05 p1 6.478e+05** ± **1.357e+04**

 **/ ndf 2** χ **0.06658 / 5**

**p0 −1.796e+07** ± **6.189e+05 p1 6.383e+05** ± **2.085e+04**

*G*(*V*, *T*) = m*V*(*V* − *VBD*(*T*)), (3) *G*(*V*, *T*) = m*TT* + *G*(*T*0, *V*), (4)

**bias voltage [V]**

*VBD*(*T*) = m*BD*(*T* − *TRT*) + *VBD*(*TRT*). (6)

. (5)

Table 1. Rate of change of the breakdown voltage with temperature.

thus a suitable candidate. The conditions can be summarized as follows

with temperature can be expressed as:

that the values of m*BD* are technology dependent.

**Gain**

Fig. 1. Example of a low flux spectrum of a SiPM; each histogram bin represents a single QDC channel corresponding to 0.11 pC.

low-flux spectrum is presented in Fig. 1 that shows the charge measurements performed by the CAEN QDC: a waveform generator triggered both the light source and the *gate* needed to perform the charge measurement. The structure of the histogram reflects the characteristics of the SiPM, since the *n*-th peak position represents the most probable output value (in released charge) of *n* cells firing simultaneously and its Gaussian broadening is due to stochastic noise sources. In this experimental situation, the very first peak represents the *zero-photon* peak, i.e. the output of the QDC board with no SiPM signal. The broadening of the *n*-th peak *σn* can be described with good approximation as:

$$
\sigma\_n^2 = \sigma\_0^2 + n\sigma\_{1'}^2 \tag{1}
$$

where *σ*<sup>0</sup> is the variance of the zero-photon peak (giving the noise contribution due to the electronic chain), and *σ*<sup>1</sup> is the variance of the first photon peak, providing an estimation of the SiPM noise.

#### **2.2.1 Gain**

The gain has been evaluated by illuminating the SiPM with a low photon flux, thus obtaining a spectrum whose peaks are clearly recognizable (see Fig. 1). As the *n*-th peak corresponds to the mean charge released by *n* G-M avalanches, the gain *G* can be computed in the following way:

$$G = \frac{\mathcal{Q}D\mathcal{C}\_{cal}}{e^{-}K\_{amp}}\Delta\_{PP\prime} \tag{2}$$

where *QDCcal* = 0.11 pC/channel is the charge corresponding to one QDC unit, *e*<sup>−</sup> is the elementary charge, *Kamp* is the global amplification factor of the electronic setup, and Δ*PP* is the distance in QDC units between two adjacent peaks of the collected spectra.

The gain behavior as a function of the bias voltage, at fixed temperatures, is shown in Fig. 2: a linear dependence in the range of interest is clearly observable with a slope independent of the temperature within the experimental errors. In Fig. 3 the same gain values are presented as functions of temperature, for fixed bias voltages: a linear behavior is still clearly recognizable; in this case as well the slope is not affected by the change of the applied bias.

This analysis suggests that the gain can be expressed as a linear function of a variable which can be re-scaled with temperature. Since it is well known that the breakdown voltage has, in 4 Will-be-set-by-IN-TECH

**100 150 200 250 300 350**

low-flux spectrum is presented in Fig. 1 that shows the charge measurements performed by the CAEN QDC: a waveform generator triggered both the light source and the *gate* needed to perform the charge measurement. The structure of the histogram reflects the characteristics of the SiPM, since the *n*-th peak position represents the most probable output value (in released charge) of *n* cells firing simultaneously and its Gaussian broadening is due to stochastic noise sources. In this experimental situation, the very first peak represents the *zero-photon* peak, i.e. the output of the QDC board with no SiPM signal. The broadening of the *n*-th peak *σn* can be

<sup>0</sup> <sup>+</sup> *<sup>n</sup>σ*<sup>2</sup>

where *σ*<sup>0</sup> is the variance of the zero-photon peak (giving the noise contribution due to the electronic chain), and *σ*<sup>1</sup> is the variance of the first photon peak, providing an estimation of

The gain has been evaluated by illuminating the SiPM with a low photon flux, thus obtaining a spectrum whose peaks are clearly recognizable (see Fig. 1). As the *n*-th peak corresponds to the mean charge released by *n* G-M avalanches, the gain *G* can be computed in the following

where *QDCcal* = 0.11 pC/channel is the charge corresponding to one QDC unit, *e*<sup>−</sup> is the elementary charge, *Kamp* is the global amplification factor of the electronic setup, and Δ*PP* is

The gain behavior as a function of the bias voltage, at fixed temperatures, is shown in Fig. 2: a linear dependence in the range of interest is clearly observable with a slope independent of the temperature within the experimental errors. In Fig. 3 the same gain values are presented as functions of temperature, for fixed bias voltages: a linear behavior is still clearly recognizable;

This analysis suggests that the gain can be expressed as a linear function of a variable which can be re-scaled with temperature. Since it is well known that the breakdown voltage has, in

*<sup>G</sup>* <sup>=</sup> *QDCcal e*−*Kamp*

the distance in QDC units between two adjacent peaks of the collected spectra.

in this case as well the slope is not affected by the change of the applied bias.

Fig. 1. Example of a low flux spectrum of a SiPM; each histogram bin represents a single

*σ*2 *<sup>n</sup>* = *<sup>σ</sup>*<sup>2</sup>

QDC channel corresponding to 0.11 pC.

described with good approximation as:

the SiPM noise.

**2.2.1 Gain**

way:

**QDC counts**

<sup>1</sup> , (1)

Δ*PP*, (2)


Table 1. Rate of change of the breakdown voltage with temperature.

our range of interest, a linear dependence on temperature (Goetzberger et al., 1963), the over voltage, defined as the difference between the applied bias voltage and the breakdown one, is thus a suitable candidate. The conditions can be summarized as follows

$$\mathbf{G}(V, T) = \mathbf{m}\_V(V - V\_{\rm BD}(T)),\tag{3}$$

$$G(V, T) = \mathbf{m}\_T T + G(T\_{0\prime} V)\_{\prime} \tag{4}$$

where m*<sup>V</sup>* and m*<sup>T</sup>* are the slopes, *VBD*(*T*) is the breakdown voltage and *T*<sup>0</sup> is a reference temperature. Solving these equations for *VBD*(*T*), the rate of change of the breakdown voltage with temperature can be expressed as:

$$\mathbf{m}\_{BD} = -\frac{\mathbf{m}\_T}{\mathbf{m}\_V}.\tag{5}$$

The procedure has been applied to the different SiPMs. From the results in Table 1, we can see that the values of m*BD* are technology dependent.

By using the breakdown voltage value measured at room temperature (RT) as the reference

Fig. 2. Gain as a function of bias voltage, for different temperatures; a linear fit has been performed for each temperature set: the obtained fit slopes are in agreement.

value, the breakdown values for each temperature can be written as:

$$V\_{BD}(T) = \mathbf{m}\_{BD}(T - T\_{RT}) + V\_{BD}(T\_{RT}).\tag{6}$$

From Equation 6 it is possible to express the gain and all the other measured parameters as a function of the over voltage: in Fig. 4 the gain is plotted as a function of the over voltage, showing a global linear behavior independent of the temperature. Equation 6 can be used to fix the operational parameters of a SiPM regardless of the varying environmental conditions: if m*BD* is known with enough precision and frequent measurements of the SiPM temperature are performed, the over voltage across each cell can be maintained fixed by simply continuously adjusting the applied bias according to the temperature variations. This

Characterization and Applications 7

Silicon Photomultipliers: Characterization and Applications 83

**DCR [kHz]**

**1**

**DCR [MHz]**

**0 1**

 **[%] talk X**

**35** −16 ° C −10 ° C +1 ° C +12 ° C +21 ° C

negligible, so that all the second photon events are due to cross-talk.

SiPM.

**2**

**3 4**

**<sup>5</sup> −16** °**<sup>C</sup> −10** ° **C +1** ° **C +12** ° **C +21** ° **C**

**10**

**<sup>2</sup> 10**

**50 100 150 200 250 300**

**1 1.5 2 2.5 3 3.5 4 4.5**

**1 1.5 2 2.5 3 3.5 4 4.5**

cell diodes thus triggering a second avalanche. The quantity named optical cross-talk (in short cross-talk, *XT*) is simply the percentage of avalanches triggered by such a mechanism.

Cross-talk has been calculated starting from DCR measurements by taking the ratio between the DCR frequencies with the discriminator threshold respectively at *"one-and-half photon"* and at *"half photon"*. This method is based on the assumption that the probability that two uncorrelated thermally-triggered avalanches are generated within the same rise time is

Fig. 7. Cross-talk as a function of the over voltage, at different temperatures, for a SensL

Fig. 6. DCR at different temperatures as a function of the over voltage, for a SensL SiPM.

Fig. 5. Example of *staircase curve*, obtained after a three-stage amplification.

**Threshold [mV]**

**over voltage [V]**

**over voltage [V]**

Fig. 3. Gain as a function of temperature (expressed in Celsius degrees), for fixed bias voltages; each set has been fitted using a linear law: the obtained slopes are in agreement.

Fig. 4. Gain as a function of the over voltage: values are obtained by calculating the breakdown voltage for each temperature; all the data acquired at different temperatures have been fitted with the same linear law.

idea has led to a collaboration between Università dell'Insubria and CAEN for the realization of CAEN SP5600 General Purpose Power Supply and Amplification Unit module with an integrated threshold discriminator.

#### **2.2.2 Dark count rate**

As explained in Section 1, DCR is the frequency of the G-M avalanches triggered by thermally extracted carriers. A scan of the DCR at different thresholds can be done, and the resulting plot (an example is shown in Fig. 5) is usually referred to as *staircase function*.

DCR has been measured at different voltages and different temperatures, also setting different discrimination thresholds; in Fig. 6, for example, the results for a *"half photon threshold"* are shown. A clear and expected dependence on temperature is recognizable.

An exhaustive knowledge of DCR behavior is of utmost importance for a complete characterization of all the noise sources of the detector. It can be a fundamental figure of merit for low flux applications.

#### **2.2.3 Optical cross-talk**

An electron avalanche is modeled as a microplasma (Oldham et al., 1972): photons emitted by the accelerated carriers during this event have a certain probability to reach the neighboring 6 Will-be-set-by-IN-TECH

 **/ ndf 2** χ **0.3038 / 4 p0 1.845e+06** ± **1.757e+04 p1 −1.467e+04** ± **998.5** bias 29 V bias 29.5 V bias 29.7 V bias 30 V bias 30.5 V

**over voltage [V]**

**/ndf = 5.8 / 43 <sup>2</sup>** χ **p0 −6.253e+04** ± **1.642e+04 p1 6.505e+05** ± **6.613e+03**

**Gain** 

**Gain** 

been fitted with the same linear law.

integrated threshold discriminator.

merit for low flux applications.

**2.2.3 Optical cross-talk**

**2.2.2 Dark count rate**

 **/ ndf** χ**2 0.5858 / 4 p0 8.721e+05** ± **1.651e+04 p1 −1.407e+04** ± **931.2**

> −16 ° C −10 ° C 1 ° C 12 ° C 21 ° C global fit

**temperature [**° **C] −15 −10 −5 0 5 10 15 20**

**1 1.5 2 2.5 3 3.5 4 4.5**

breakdown voltage for each temperature; all the data acquired at different temperatures have

idea has led to a collaboration between Università dell'Insubria and CAEN for the realization of CAEN SP5600 General Purpose Power Supply and Amplification Unit module with an

As explained in Section 1, DCR is the frequency of the G-M avalanches triggered by thermally extracted carriers. A scan of the DCR at different thresholds can be done, and the resulting

DCR has been measured at different voltages and different temperatures, also setting different discrimination thresholds; in Fig. 6, for example, the results for a *"half photon threshold"* are

An exhaustive knowledge of DCR behavior is of utmost importance for a complete characterization of all the noise sources of the detector. It can be a fundamental figure of

An electron avalanche is modeled as a microplasma (Oldham et al., 1972): photons emitted by the accelerated carriers during this event have a certain probability to reach the neighboring

plot (an example is shown in Fig. 5) is usually referred to as *staircase function*.

shown. A clear and expected dependence on temperature is recognizable.

Fig. 4. Gain as a function of the over voltage: values are obtained by calculating the

Fig. 3. Gain as a function of temperature (expressed in Celsius degrees), for fixed bias voltages; each set has been fitted using a linear law: the obtained slopes are in agreement.

Fig. 5. Example of *staircase curve*, obtained after a three-stage amplification.

Fig. 6. DCR at different temperatures as a function of the over voltage, for a SensL SiPM.

Fig. 7. Cross-talk as a function of the over voltage, at different temperatures, for a SensL SiPM.

cell diodes thus triggering a second avalanche. The quantity named optical cross-talk (in short cross-talk, *XT*) is simply the percentage of avalanches triggered by such a mechanism.

Cross-talk has been calculated starting from DCR measurements by taking the ratio between the DCR frequencies with the discriminator threshold respectively at *"one-and-half photon"* and at *"half photon"*. This method is based on the assumption that the probability that two uncorrelated thermally-triggered avalanches are generated within the same rise time is negligible, so that all the second photon events are due to cross-talk.

Characterization and Applications 9

Silicon Photomultipliers: Characterization and Applications 85

 **[%] 0 PDE**

−16 ° C −10 ° C +1 ° C +12 ° C +21 ° c

200-ns-long time window synchronized with the laser pulse.

wavelength of the impinging light is 510 nm.

room temperature.

**3.1 Detector response modeling**

(Mandel & Wolf, 1995):

**1 1.5 2 2.5 3 3.5 4 4.5**

Fig. 8. PDE values corrected taking into account cross-talk effects, for a SensL SiPM. The

Table 2. Main features of the SiPM MPPC S10362-11-100C (Hamamatsu). The data refer to

response is thus required in order to properly assess the statistical properties of the light field under investigation (Ramilli et al., 2010). In the following Section we present two different analysis methods, both validated by suitable measurements performed by using a SiPM produced by Hamamatsu Photonics. In particular, to keep DCR and cross-talk effects at reasonable levels and to optimize the PDE, a detector endowed with 100 cells was chosen (MPPC S10362-11-100C, see Table 2). To test the model, the sensor was illuminated by the pseudo-thermal light obtained by passing the second-harmonics (@ 523 nm, 5.4-ps pulse duration) of a mode-locked Nd:YLF laser amplified at 500 Hz (High Q Laser Production) through a rotating ground-glass diffuser (D in Fig.8). The light to be measured was delivered to the sensor by a multimode optical fiber (1 mm core diameter). The detector response was integrated by the charge digitizer V792 by Caen. The signal was typically integrated over a

The response of an ideal detector to a light field can be described as a Bernoullian process

being *n* the number of impinging photons over the integration time, *m* the number of detected photons and *η* < 1 the PDE. Actually, *η* is a single parameter quantifying detector effects and

*<sup>η</sup>m*(<sup>1</sup> <sup>−</sup> *<sup>η</sup>*)*n*−*<sup>m</sup>* (10)

*m* 

*Bm*,*n*(*η*) = *<sup>n</sup>*

Number of Diodes: 100 Area: 1 mm × 1 mm Diode dimension: 100 *μ*m × 100 *μ*m Breakdown Voltage: 69.23 V Dark Count Rate: 540 kHz at 70 V Optical Cross-Talk: 25 % at 70 V Gain: 3.3 · 106 at 70 V PDE (green): 15 % at 70 V

**over voltage [V]**

Hamamatsu MPPC S10362-11-100C

In Fig. 7 the results of the cross-talk evaluation are presented: it is worth noting that, within the experimental errors, cross-talk does not seem to suffer a strong temperature dependence.

#### **2.2.4 Photon detection efficiency**

Photon detection efficiency (PDE) is a key parameter for every light detector and is defined as the product of three terms:




The PDE value thus represents the fraction of impinging photons that are actually detected.

To measure this quantity, SiPMs have been illuminated with a light intensity Φ which has been previously measured with a calibrated PMT HAMAMATSU H5783 and the resulting spectra have been acquired. PDE has been evaluated by estimating the mean number of detected photons < *n* >*meas* with respect to the incoming light intensity:

$$PDE = \frac{\_{meas}}{\Phi} \tag{7}$$

In the case of spectra in which each photon peak could be resolved, < *n* >*meas* has been obtained by fitting the peak positions with a Poissonian curve and by evaluating the mean value of the fit distribution. In the case in which the peaks could not be resolved, < *n* >*meas* has been estimated as

$$\_{meas} = \frac{\text{QDC}\_{cal}}{e^- \text{ G K}\_{amp}} \Delta \text{QDC} \tag{8}$$

where *G* is the proper gain value, *e*<sup>−</sup> is the elementary charge, *QDCcal* = 0.11 pC is the charge per QDC unit, *Kamp* is the amplification factor of the electronics chain, and Δ*QDC* is the difference between the mean value of the obtained spectrum and the pedestal position.

However, this method provides a zero-order approximation of the PDE because it does not take into account cross-talk effects that can be as large as 40% (see Fig. 7 as a reference). To properly estimate the *"true"* number < *n* ><sup>0</sup> of impinging photons triggering an avalanche, the following relation has been used:

$$1 < n >\_{meas} = \frac{\_o}{1 - X\_T} \simeq \_o (1 + X\_T). \tag{9}$$

The approximation is valid for small values of the cross-talk; the previously obtained PDE values have thus been corrected using < *n* ><sup>0</sup> as the mean number of detected photons as shown in Fig. 8. Again, PDE does not seem to have any remarkable temperature dependance, thus confirming the agreement of our results with the definition of PDE.

#### **3. Photon-number statistics**

Due to their good linear response, SiPMs can be considered as the ideal candidates for the reconstruction of the statistics of any light state (Afek et al., 2009). Obviously, the presence of DCR and cross-talk effects must be taken into account. A proper model of the detector 8 Will-be-set-by-IN-TECH

In Fig. 7 the results of the cross-talk evaluation are presented: it is worth noting that, within the experimental errors, cross-talk does not seem to suffer a strong temperature dependence.

Photon detection efficiency (PDE) is a key parameter for every light detector and is defined as




The PDE value thus represents the fraction of impinging photons that are actually detected. To measure this quantity, SiPMs have been illuminated with a light intensity Φ which has been previously measured with a calibrated PMT HAMAMATSU H5783 and the resulting spectra have been acquired. PDE has been evaluated by estimating the mean number of detected

*PDE* <sup>=</sup> <sup>&</sup>lt; *<sup>n</sup>* <sup>&</sup>gt;*meas*

In the case of spectra in which each photon peak could be resolved, < *n* >*meas* has been obtained by fitting the peak positions with a Poissonian curve and by evaluating the mean value of the fit distribution. In the case in which the peaks could not be resolved, < *n* >*meas*

where *G* is the proper gain value, *e*<sup>−</sup> is the elementary charge, *QDCcal* = 0.11 pC is the charge per QDC unit, *Kamp* is the amplification factor of the electronics chain, and Δ*QDC* is the difference between the mean value of the obtained spectrum and the pedestal position. However, this method provides a zero-order approximation of the PDE because it does not take into account cross-talk effects that can be as large as 40% (see Fig. 7 as a reference). To properly estimate the *"true"* number < *n* ><sup>0</sup> of impinging photons triggering an avalanche,

*e*− *G Kamp*

<sup>&</sup>lt; *<sup>n</sup>* <sup>&</sup>gt;*meas* <sup>=</sup> *QDCcal*

1 − *XT*

The approximation is valid for small values of the cross-talk; the previously obtained PDE values have thus been corrected using < *n* ><sup>0</sup> as the mean number of detected photons as shown in Fig. 8. Again, PDE does not seem to have any remarkable temperature dependance,

Due to their good linear response, SiPMs can be considered as the ideal candidates for the reconstruction of the statistics of any light state (Afek et al., 2009). Obviously, the presence of DCR and cross-talk effects must be taken into account. A proper model of the detector

<sup>&</sup>lt; *<sup>n</sup>* <sup>&</sup>gt;*meas* <sup>=</sup> <sup>&</sup>lt; *<sup>n</sup>* <sup>&</sup>gt;*<sup>o</sup>*

thus confirming the agreement of our results with the definition of PDE.

<sup>Φ</sup> (7)

Δ*QDC* (8)

� < *n* >*<sup>o</sup>* (1 + *XT*). (9)

**2.2.4 Photon detection efficiency**

of the sensor with respect to the total area.

carrier, which is a function of the incoming light wavelength;

photons < *n* >*meas* with respect to the incoming light intensity:

the product of three terms:

over voltage;

has been estimated as

the following relation has been used:

**3. Photon-number statistics**

Fig. 8. PDE values corrected taking into account cross-talk effects, for a SensL SiPM. The wavelength of the impinging light is 510 nm.


Table 2. Main features of the SiPM MPPC S10362-11-100C (Hamamatsu). The data refer to room temperature.

response is thus required in order to properly assess the statistical properties of the light field under investigation (Ramilli et al., 2010). In the following Section we present two different analysis methods, both validated by suitable measurements performed by using a SiPM produced by Hamamatsu Photonics. In particular, to keep DCR and cross-talk effects at reasonable levels and to optimize the PDE, a detector endowed with 100 cells was chosen (MPPC S10362-11-100C, see Table 2). To test the model, the sensor was illuminated by the pseudo-thermal light obtained by passing the second-harmonics (@ 523 nm, 5.4-ps pulse duration) of a mode-locked Nd:YLF laser amplified at 500 Hz (High Q Laser Production) through a rotating ground-glass diffuser (D in Fig.8). The light to be measured was delivered to the sensor by a multimode optical fiber (1 mm core diameter). The detector response was integrated by the charge digitizer V792 by Caen. The signal was typically integrated over a 200-ns-long time window synchronized with the laser pulse.

#### **3.1 Detector response modeling**

The response of an ideal detector to a light field can be described as a Bernoullian process (Mandel & Wolf, 1995):

$$B\_{m,n}(\eta) = \binom{n}{m} \eta^m (1-\eta)^{n-m} \tag{10}$$

being *n* the number of impinging photons over the integration time, *m* the number of detected photons and *η* < 1 the PDE. Actually, *η* is a single parameter quantifying detector effects and

Characterization and Applications 11

Silicon Photomultipliers: Characterization and Applications 87

In Ref. (Bondani et al., 2009b) we presented a self-consistent method aimed at reconstructing the statistics of detected photons. According to this method, the analysis of the output of the detector is based on the assumptions that the detection process is described by a Bernoullian convolution and that the overall amplification-conversion process is given by a very precise constant factor *G*, which allows the shot-by-shot detector output to be converted into the number of detected photons. Experimentally, the value of *G* can be obtained by detecting a light field at different optical losses and keeping the detector parameters fixed. Once the amplification-conversion factor has been evaluated, the detected-photon distribution can be achieved by dividing the output values by *G* and re-binning the data in unitary bins. Here we present the extension of the method to detectors with a significant DCR and first order cross-talk effects. The second-order momentum of the recorded pulse distribution *Px*,out can

**3.1.1 Method I: an analytical evaluation of the second and third order momenta**

be used to evaluate the Fano factor:

where *<sup>Q</sup>*det+dc <sup>=</sup> *<sup>σ</sup>*(2)

of symmetry parameter

*Sx*,out <sup>=</sup> *<sup>σ</sup>*(3)

**SiPMs**

a two-step procedure:

where *Qs*,det+dc <sup>=</sup> *<sup>σ</sup>*(3)

*x*,out *x*out

coefficients (*Qs*,det+dc <sup>−</sup> <sup>3</sup>*Q*det+dc)/*m*<sup>2</sup>

the number of counts per peak;

ideality, such as DCR and cross-talk effects.

measured (Bondani et al., 2009a).

*Fx*,out <sup>=</sup> *<sup>σ</sup>*(2)

<sup>=</sup> *Qs*,det+dc <sup>−</sup> <sup>3</sup>*Q*det+dc *m*2 det+dc

*x*,out *x*out

<sup>=</sup> *<sup>Q</sup>*det+dc *m*det+dc

that, due to dark counts, the coefficient of *x*out in Equation 17 cannot be written as *Q*ph/*n* (Andreoni & Bondani, 2009; Bondani et al., 2009b), that is, the coefficient *Q*det<sup>+</sup>dc/*m*det+dc does not only depend on the light field to be measured. Similarly we can calculate a sort

*x*2

**3.1.2 Method II: a numerical evaluation based on the photon-number resolving properties of**

The above mentioned self-consistent method is very powerful, but requires the acquisition of several histograms at varying *η*, which might not always be easily performed in practical applications: from this point of view, the possibility to analyze each spectrum independently looks complementary to the self-consistent approach. This analysis has been performed with

• the areas of the spectrum peaks have been computed in order to obtain an estimation of

• the obtained data points have been fitted with a theoretical function, which takes into account the statistics of the light, the detection and all the deviations of the detectors from

To evaluate the area of each peak, we performed a multi-peak fit of the spectrum histogram

1 + <sup>3</sup>*hH*3(*w*) + <sup>4</sup>*hH*4(*w*)

modeling each peak with a Gauss-Hermite function (Van Der Marel et al., 1993):

GH <sup>=</sup> *Ne*−*w*2/2

out <sup>+</sup> *<sup>G</sup>* <sup>1</sup> <sup>+</sup> <sup>3</sup>*xt* 1 + *xt*

*<sup>x</sup>*out <sup>+</sup> *<sup>G</sup>* <sup>1</sup> <sup>+</sup> <sup>3</sup>*xt*

*<sup>m</sup>*,det+dc/*m*det+dc − 1 is the Mandel factor of the primary charges. Note

1 + *xt*

*Q*det+dc *m*det+dc

det+dc and *Q*det<sup>+</sup>dc/*m*det+dc depend on the light being

*<sup>m</sup>*,det+dc/*m*det+dc − 1. In the presence of dark counts both

, (17)

*<sup>x</sup>*out <sup>+</sup> *<sup>G</sup>*<sup>2</sup> <sup>1</sup> <sup>+</sup> <sup>7</sup>*xt*

1 + *xt*

, (18)

(19)

losses (intentional or accidental) due to the optical system. As a consequence, the distribution *Pm*,det of the number of detected photons has to be linked to the distribution *Pn*,ph of the number of photons in the light under measurement by (Agliati et al., 2005; Mandel & Wolf, 1995; Zambra et al., 2004):

$$P\_{m, \text{det}} = \sum\_{n=m}^{\infty} B\_{m,n}(\eta) P\_{n, \text{ph}} \tag{11}$$

It can be demonstrated (Bondani et al., 2009a) that for a combination of classical light states the statistics is preserved by the primary detection process. This simple description has to be further developed to link *Pm*,det to the probability density distribution of the G-M avalanches of any origin. First we must take into account spurious hits and cross-talk effects, not negligible in the detectors under study. The DCR results in a Poissonian process that can be described as:

$$P\_{m, \text{dc}} = \overline{m}\_{\text{dc}}^{\text{m}} / m! \exp(-\overline{m}\_{\text{dc}}) \tag{12}$$

where *<sup>m</sup>*dc is the mean number of dark counts during the gate window and *<sup>σ</sup>*(2) *<sup>m</sup>*,dc <sup>=</sup> *<sup>σ</sup>*(3) *<sup>m</sup>*,dc = *m*dc. As a consequence, the statistics of the recorded pulses may be described as:

$$P\_{m, \text{det} + \text{dc}} = \sum\_{i=0}^{m} P\_{i, \text{dc}} P\_{m-i, \text{det}} \tag{13}$$

obviously shifting the mean value and increasing variance and third-order central moment in the following way: *<sup>m</sup>*det+dc <sup>=</sup> *<sup>m</sup>*det <sup>+</sup> *<sup>m</sup>*dc, *<sup>σ</sup>*(2) *<sup>m</sup>*,det+dc <sup>=</sup> *<sup>σ</sup>*(2) *<sup>m</sup>*,det <sup>+</sup> *<sup>m</sup>*dc and *<sup>σ</sup>*(3) *<sup>m</sup>*,det+dc <sup>=</sup> *<sup>σ</sup>*(3) *<sup>m</sup>*,det + *m*dc. As a further step, cross-talk effects must be taken into account. Cross-talk is a genuine cascade phenomenon that can be described at first order as (Afek et al., 2009)

$$\mathbf{C}\_{k,l}(\mathbf{x}\_l) = \binom{l}{k-l} \mathbf{x}\_l^{k-l} (1-\mathbf{x}\_l)^{2l-k}.\tag{14}$$

being *xt* the (constant) probability that the G-M avalanche of a cell triggers a second cell (which becomes equivalent to the cross-talk probability *XT* in the limit of *XT* → 0), *l* the number of dark counts and photo-triggered avalanches and *k* the actual light signal amplitude. Within this first-order approximation, the real sensor response is described by

$$P\_{k, \text{cross}} = \sum\_{m=0}^{k} \mathbb{C}\_{k,m}(\mathbf{x}\_{l}) P\_{m, \text{det} + \text{dc}} \tag{15}$$

characterized by *<sup>k</sup>*cross = (<sup>1</sup> <sup>+</sup> *xt*)*m*det<sup>+</sup>dc, *<sup>σ</sup>*(2) *<sup>k</sup>*,cross = (<sup>1</sup> <sup>+</sup> *xt*)2*σ*(2) *<sup>m</sup>*,det+dc + *xt*(1 − *xt*)*mm*,det+dc and *<sup>σ</sup>*(3) *<sup>k</sup>*,cross = (<sup>1</sup> <sup>+</sup> *xt*)3*σ*(3) *<sup>m</sup>*,det+dc <sup>+</sup> <sup>3</sup>*xt*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup> *<sup>t</sup>* )*σ*(2) *<sup>m</sup>*,det+dc <sup>+</sup> *xt*(<sup>1</sup> <sup>−</sup> <sup>3</sup>*xt* <sup>+</sup> <sup>2</sup>*x*<sup>2</sup> *<sup>t</sup>*)*mm*,det<sup>+</sup>dc.

In the following we refer to this analytical model as *Method I* and compare it with a better refined model (*Method II*). This second method offers, in principle, an extended range of application even if it is limited to a numerical rather than an analytical solution. Irrespective of the model, the amplification and digitization processes that produce the output *x* can be simply described as a multiplicative parameter *G*:

$$P\_{\rm x,out} = GP\_{\rm Gk,cross} \tag{16}$$

whose moments are *<sup>x</sup>*out <sup>=</sup> *Gk*cross, *<sup>σ</sup>*(2) *<sup>x</sup>*,out <sup>=</sup> *<sup>G</sup>*2*σ*(2) *k*,cross and *<sup>σ</sup>*(3) *<sup>x</sup>*,out <sup>=</sup> *<sup>G</sup>*3*σ*(3) *<sup>k</sup>*,cross. 10 Will-be-set-by-IN-TECH

losses (intentional or accidental) due to the optical system. As a consequence, the distribution *Pm*,det of the number of detected photons has to be linked to the distribution *Pn*,ph of the number of photons in the light under measurement by (Agliati et al., 2005; Mandel & Wolf,

It can be demonstrated (Bondani et al., 2009a) that for a combination of classical light states the statistics is preserved by the primary detection process. This simple description has to be further developed to link *Pm*,det to the probability density distribution of the G-M avalanches of any origin. First we must take into account spurious hits and cross-talk effects, not negligible in the detectors under study. The DCR results in a Poissonian process that can

> *m* ∑ *i*=0

obviously shifting the mean value and increasing variance and third-order central moment in

*m*dc. As a further step, cross-talk effects must be taken into account. Cross-talk is a genuine

 *xk*−*<sup>l</sup>*

being *xt* the (constant) probability that the G-M avalanche of a cell triggers a second cell (which becomes equivalent to the cross-talk probability *XT* in the limit of *XT* → 0), *l* the number of dark counts and photo-triggered avalanches and *k* the actual light signal amplitude. Within this first-order approximation, the real sensor response is described by

*<sup>t</sup>* )*σ*(2)

In the following we refer to this analytical model as *Method I* and compare it with a better refined model (*Method II*). This second method offers, in principle, an extended range of application even if it is limited to a numerical rather than an analytical solution. Irrespective of the model, the amplification and digitization processes that produce the output *x* can be

*<sup>x</sup>*,out <sup>=</sup> *<sup>G</sup>*2*σ*(2)

*k*,cross

*<sup>m</sup>*,det+dc <sup>=</sup> *<sup>σ</sup>*(2)

*<sup>k</sup>*,cross = (<sup>1</sup> <sup>+</sup> *xt*)2*σ*(2)

*<sup>m</sup>*,det+dc <sup>+</sup> *xt*(<sup>1</sup> <sup>−</sup> <sup>3</sup>*xt* <sup>+</sup> <sup>2</sup>*x*<sup>2</sup>

and *<sup>σ</sup>*(3)

*Px*,out = *GPGk*,cross (16)

*<sup>x</sup>*,out <sup>=</sup> *<sup>G</sup>*3*σ*(3)

*<sup>k</sup>*,cross.

*Bm*,*n*(*η*)*Pn*,ph (11)

dc/*m*! exp(−*m*dc) (12)

*Pi*,dc*Pm*−*i*,det (13)

*<sup>t</sup>* (<sup>1</sup> <sup>−</sup> *xt*)2*l*−*<sup>k</sup>* . (14)

*Ck*,*m*(*xt*)*Pm*,det+dc (15)

*<sup>m</sup>*,det+dc + *xt*(1 − *xt*)*mm*,det+dc

*<sup>t</sup>*)*mm*,det<sup>+</sup>dc.

*<sup>m</sup>*,det <sup>+</sup> *<sup>m</sup>*dc and *<sup>σ</sup>*(3)

*<sup>m</sup>*,dc <sup>=</sup> *<sup>σ</sup>*(3)

*<sup>m</sup>*,det+dc <sup>=</sup> *<sup>σ</sup>*(3)

*<sup>m</sup>*,dc =

*<sup>m</sup>*,det +

∞ ∑*n*=*m*

*Pm*,det =

*Pm*,dc = *<sup>m</sup><sup>m</sup>*

where *<sup>m</sup>*dc is the mean number of dark counts during the gate window and *<sup>σ</sup>*(2)

*m*dc. As a consequence, the statistics of the recorded pulses may be described as:

*Pm*,det+dc =

cascade phenomenon that can be described at first order as (Afek et al., 2009)

 *l k* − *l*

> *k* ∑ *m*=0

*Ck*,*l*(*xt*) =

*Pk*,cross =

*<sup>m</sup>*,det+dc <sup>+</sup> <sup>3</sup>*xt*(<sup>1</sup> <sup>−</sup> *<sup>x</sup>*<sup>2</sup>

the following way: *<sup>m</sup>*det+dc <sup>=</sup> *<sup>m</sup>*det <sup>+</sup> *<sup>m</sup>*dc, *<sup>σ</sup>*(2)

characterized by *<sup>k</sup>*cross = (<sup>1</sup> <sup>+</sup> *xt*)*m*det<sup>+</sup>dc, *<sup>σ</sup>*(2)

simply described as a multiplicative parameter *G*:

whose moments are *<sup>x</sup>*out <sup>=</sup> *Gk*cross, *<sup>σ</sup>*(2)

*<sup>k</sup>*,cross = (<sup>1</sup> <sup>+</sup> *xt*)3*σ*(3)

1995; Zambra et al., 2004):

be described as:

and *<sup>σ</sup>*(3)

#### **3.1.1 Method I: an analytical evaluation of the second and third order momenta**

In Ref. (Bondani et al., 2009b) we presented a self-consistent method aimed at reconstructing the statistics of detected photons. According to this method, the analysis of the output of the detector is based on the assumptions that the detection process is described by a Bernoullian convolution and that the overall amplification-conversion process is given by a very precise constant factor *G*, which allows the shot-by-shot detector output to be converted into the number of detected photons. Experimentally, the value of *G* can be obtained by detecting a light field at different optical losses and keeping the detector parameters fixed. Once the amplification-conversion factor has been evaluated, the detected-photon distribution can be achieved by dividing the output values by *G* and re-binning the data in unitary bins. Here we present the extension of the method to detectors with a significant DCR and first order cross-talk effects. The second-order momentum of the recorded pulse distribution *Px*,out can be used to evaluate the Fano factor:

$$F\_{\rm x,out} = \frac{\sigma\_{\rm x,out}^{(2)}}{\overline{\pi}\_{\rm out}} = \frac{Q\_{\rm det+dc}}{\overline{m}\_{\rm det+dc}} \overline{\chi}\_{\rm out} + G \frac{1+3\chi\_{\rm t}}{1+\chi\_{\rm t}} \, , \tag{17}$$

where *<sup>Q</sup>*det+dc <sup>=</sup> *<sup>σ</sup>*(2) *<sup>m</sup>*,det+dc/*m*det+dc − 1 is the Mandel factor of the primary charges. Note that, due to dark counts, the coefficient of *x*out in Equation 17 cannot be written as *Q*ph/*n* (Andreoni & Bondani, 2009; Bondani et al., 2009b), that is, the coefficient *Q*det<sup>+</sup>dc/*m*det+dc does not only depend on the light field to be measured. Similarly we can calculate a sort of symmetry parameter

$$\mathcal{S}\_{\text{x,out}} = \frac{\sigma\_{\text{x,out}}^{(3)}}{\overline{\mathbf{x}}\_{\text{out}}} = \frac{Q\_{\text{s,det+dc}} - 3Q\_{\text{det+dc}}}{\overline{m}\_{\text{det+dc}}^2} \overline{\mathbf{x}}\_{\text{out}}^2 + G \frac{1 + 3\mathbf{x}\_t}{1 + \mathbf{x}\_t} \frac{Q\_{\text{det+dc}}}{\overline{m}\_{\text{det+dc}}} \overline{\mathbf{x}}\_{\text{out}} + G^2 \frac{1 + 7\mathbf{x}\_t}{1 + \mathbf{x}\_t}, \tag{18}$$

where *Qs*,det+dc <sup>=</sup> *<sup>σ</sup>*(3) *<sup>m</sup>*,det+dc/*m*det+dc − 1. In the presence of dark counts both coefficients (*Qs*,det+dc <sup>−</sup> <sup>3</sup>*Q*det+dc)/*m*<sup>2</sup> det+dc and *Q*det<sup>+</sup>dc/*m*det+dc depend on the light being measured (Bondani et al., 2009a).

#### **3.1.2 Method II: a numerical evaluation based on the photon-number resolving properties of SiPMs**

The above mentioned self-consistent method is very powerful, but requires the acquisition of several histograms at varying *η*, which might not always be easily performed in practical applications: from this point of view, the possibility to analyze each spectrum independently looks complementary to the self-consistent approach. This analysis has been performed with a two-step procedure:


To evaluate the area of each peak, we performed a multi-peak fit of the spectrum histogram modeling each peak with a Gauss-Hermite function (Van Der Marel et al., 1993):

$$\text{GH} = Ne^{-w^2/2} \left[ 1 + ^3hH\_3(w) + ^4hH\_4(w) \right] \tag{19}$$

Characterization and Applications 13

Silicon Photomultipliers: Characterization and Applications 89

Fig. 9. Experimental setup. Nd:YLF: laser source, ND: variable neutral density filter, SiPM:

which is described by a Poissonian distribution and is thus indistinguishable from the light under study. So we think that, within this context, it is more exhaustive and intriguing to show the results obtained with thermal light. The photon-number distribution of a field made

We reproduced a single mode of this pseudo-thermal light field by passing the coherent light of our laser source through an inhomogeneous diffusing medium and then selecting a single speckle with an aperture (∼ 150 *μ*m diameter), much smaller than the coherence area of the speckle pattern produced (Arecchi, 1965). By delivering this pseudo-thermal light to the SiPM sensor, we measured the values of the output, *x*, at 50000 subsequent laser shots and at 10 different mean values, obtained by means of a variable neutral-density filter (ND in Fig. 9). In Fig. 10 the plot of the experimental values of *Fx*,out and *Sx*,out is shown as a function of *x*out,

*x*out + *B*

*x*2 out + 3*B*

where *μ*, the number of independent thermal modes, has been set equal to one and, for simplicity of notation, the parameters *A* = 2, *B* = *G*(1 + 3*xt*)/(1 + *xt*) and *C* = *G*2(1 + 7*xt*)/(1 + *xt*) have been introduced. We fitted the data to *Fx*,out thus obtaining the values of

Then the data for *Sx*,out have been fitted substituting the above values of *x*dc ch and *B* in order to obtain *<sup>A</sup>* = (2.34754 <sup>±</sup> 0.091576) and *<sup>C</sup>* = (8531.48 <sup>±</sup> 419.571) ch2. These values were then used to evaluate *G* and *xt*, obtaining *G* = (74.2785 ± 1.76982) ch and *xt* = (0.100174 ± 0.050529). The *x*-values were then divided by *G* and re-binned in unitary bins (Bondani et al., 2009b) to obtain the *Pk*,cross distribution of the actual light signal amplitude measured in the presence of dark counts and cross-talk. In Fig. 11 six different *Pk*,cross distributions are plotted as bars at different mean values. Superimposed to the experimental values, two theoretical distributions are shown: the first one (white circles) is evaluated including the dark-count contribution that modifies the statistics of a single-mode thermal

<sup>1</sup> <sup>−</sup> *<sup>x</sup>*dc *<sup>x</sup>*out <sup>2</sup>

*<sup>μ</sup>* (*μ*/*Nth* + <sup>1</sup>)

*<sup>n</sup>* , (24)

*x*out + *C* , (25)

*Pn*,ph <sup>=</sup> (*<sup>n</sup>* <sup>+</sup> *<sup>μ</sup>* <sup>−</sup> <sup>1</sup>)! *n*!(*μ* − 1)!(*Nth*/*μ* + 1)

with *<sup>n</sup>* <sup>=</sup> *Nth*, *<sup>σ</sup>*(2) *<sup>n</sup>* <sup>=</sup> *Nth* (*Nth*/*<sup>μ</sup>* <sup>+</sup> <sup>1</sup>) and *<sup>σ</sup>*(3) *<sup>n</sup>* <sup>=</sup> *Nth* (*Nth*/*<sup>μ</sup>* <sup>+</sup> <sup>1</sup>) (2*Nth*/*<sup>μ</sup>* <sup>+</sup> <sup>1</sup>).

along with the fitting curves evaluated from Equations 17 and 18:

<sup>1</sup> <sup>−</sup> *<sup>x</sup>*dc *<sup>x</sup>*out <sup>2</sup>

> <sup>1</sup> <sup>−</sup> *<sup>x</sup>*dc *<sup>x</sup>*out <sup>3</sup>

*x*dc = (5.82028 ± 1.34015) and *B* = (87.805 ± 2.09009) ch.

detector.

of *μ* independent thermal modes is given by:

*Fx*,out =

*Sx*,out = *A*

where *w* = (*x* − *x*¯)/*σ*, *N* is a normalization factor, *x*¯ the peak position and *σ* the variance of the Gaussian function. *H*3(*w*) and *H*4(*w*) are the third and the fourth normalized Hermite polynomials and their contribution gives the asymmetry of the peak shape, whose entity is regulated by the pre-factors <sup>3</sup>*<sup>h</sup>* and <sup>4</sup>*h*, with values in the range [−1, 1]. The global fit function of the spectrum is a sum of as many Gauss-Hermite functions as the number of resolved peaks.

By using Equation 19 it is possible to calculate the area *An* of the *n*-th peak as:

$$A\_{\rm n} = N\_{\rm n} \sigma\_{\rm n} \left( \sqrt{2\pi} + {}^4 h\_{\rm n} \right). \tag{20}$$

The error *σAn* has been calculated by propagating the errors on the fit parameters. This analysis is also useful to estimate the system gain *G*: in fact, from the fitted values of the peak positions *x*¯*n*, the peak-to-peak distance Δ for all the resolved peaks is:

$$
\Delta\_{n,n+1} = \mathfrak{x}\_{n+1} - \mathfrak{x}\_n. \tag{21}
$$

The associated error *σ*Δ*n*,*n*+<sup>1</sup> can be once again obtained by propagating the fit errors; furthermore, the value of *G* can be derived as a weighted average on all the peak-to-peak values obtained from the analysis of the histograms.

The effect of detection, DCR and amplification have been modeled as described in the previous Sections (see Equations 10-13 and Equation 16).

The effect of cross-talk has been described by using a Bernoullian process, in a way analogue to what has been done in Equation 14. However, as the cross-talk process is intrinsically a cascade phenomenon, its contribution has been calculated by adding higher order effects:

$$P\_{\mathbf{k}, \text{cross}} = \sum\_{m=0}^{k} \sum\_{n=0}^{m} \sum\_{j=0}^{n} P\_{\mathbf{k}-m-n-j, \text{det}+\text{dc}} B\_{m, \text{k}-m-n-j}(\mathbf{x}\_{t}) B\_{\mathbf{n},m}(\mathbf{x}\_{t}) \, B\_{j,\text{n}}(\mathbf{x}\_{t});\tag{22}$$

where the terms like *Bj*,*n*(*xt*) stand for the Bernoullian distribution

$$B\_{j,n}(\mathbf{x}\_t) = \binom{n}{j} \mathbf{x}\_t^j (1 - \mathbf{x}\_t)^{n-j}.\tag{23}$$

Such a higher order expansion is not trivial to be achieved by the self-consistent approach of *Method I*, in which an explicit analytic expression of *Px*,out is needed in order to calculate its moments. Here, as all the elements of interest (*mel*, *mdc*, *xt*, the number of modes *μ*) will be obtained as parameters of a fit, this is not necessary and therefore *Px*,out can be just numerically computed as the fitting function.

The major limit of this approach is obvious: as all the information on the statistics of the system is obtained from the peak areas, this method can only be applied to peak-resolving histograms with a number of peaks larger than the number of free parameters of the fitting function, which, in the present analysis, can rise up to five.

#### **3.2 Experimental results**

In Ref. (Ramilli et al., 2010), we presented the experimental validation of the two complementary methods by using both coherent and pseudo-thermal light. In particular, we have demonstrated that it is not possible to use coherent light to derive the DCR contribution, 12 Will-be-set-by-IN-TECH

where *w* = (*x* − *x*¯)/*σ*, *N* is a normalization factor, *x*¯ the peak position and *σ* the variance of the Gaussian function. *H*3(*w*) and *H*4(*w*) are the third and the fourth normalized Hermite polynomials and their contribution gives the asymmetry of the peak shape, whose entity is regulated by the pre-factors <sup>3</sup>*<sup>h</sup>* and <sup>4</sup>*h*, with values in the range [−1, 1]. The global fit function of the spectrum is a sum of as many Gauss-Hermite functions as the number of resolved peaks.

√

This analysis is also useful to estimate the system gain *G*: in fact, from the fitted values of the

The associated error *σ*Δ*n*,*n*+<sup>1</sup> can be once again obtained by propagating the fit errors; furthermore, the value of *G* can be derived as a weighted average on all the peak-to-peak

The effect of detection, DCR and amplification have been modeled as described in the

The effect of cross-talk has been described by using a Bernoullian process, in a way analogue to what has been done in Equation 14. However, as the cross-talk process is intrinsically a cascade phenomenon, its contribution has been calculated by adding higher order effects:

2*π* + <sup>4</sup>*hn*). (20)

Δ*n*,*n*+<sup>1</sup> = *x*¯*n*+<sup>1</sup> − *x*¯*n*. (21)

*Pk*−*m*−*n*−*j*,*det*+*dc Bm*,*k*−*m*−*n*−*j*(*xt*) *Bn*,*m*(*xt*) *Bj*,*n*(*xt*); (22)

*<sup>t</sup>*(<sup>1</sup> <sup>−</sup> *xt*)*n*−*<sup>j</sup>* . (23)

By using Equation 19 it is possible to calculate the area *An* of the *n*-th peak as:

*An* = *Nn σ<sup>n</sup>* (

The error *σAn* has been calculated by propagating the errors on the fit parameters.

peak positions *x*¯*n*, the peak-to-peak distance Δ for all the resolved peaks is:

values obtained from the analysis of the histograms.

*Pk*,*cross* =

computed as the fitting function.

**3.2 Experimental results**

*k* ∑ *m*=0

previous Sections (see Equations 10-13 and Equation 16).

*m* ∑ *n*=0

*n* ∑ *j*=0

function, which, in the present analysis, can rise up to five.

where the terms like *Bj*,*n*(*xt*) stand for the Bernoullian distribution

*Bj*,*n*(*xt*) =

 *n j x j*

Such a higher order expansion is not trivial to be achieved by the self-consistent approach of *Method I*, in which an explicit analytic expression of *Px*,out is needed in order to calculate its moments. Here, as all the elements of interest (*mel*, *mdc*, *xt*, the number of modes *μ*) will be obtained as parameters of a fit, this is not necessary and therefore *Px*,out can be just numerically

The major limit of this approach is obvious: as all the information on the statistics of the system is obtained from the peak areas, this method can only be applied to peak-resolving histograms with a number of peaks larger than the number of free parameters of the fitting

In Ref. (Ramilli et al., 2010), we presented the experimental validation of the two complementary methods by using both coherent and pseudo-thermal light. In particular, we have demonstrated that it is not possible to use coherent light to derive the DCR contribution,

Fig. 9. Experimental setup. Nd:YLF: laser source, ND: variable neutral density filter, SiPM: detector.

which is described by a Poissonian distribution and is thus indistinguishable from the light under study. So we think that, within this context, it is more exhaustive and intriguing to show the results obtained with thermal light. The photon-number distribution of a field made of *μ* independent thermal modes is given by:

$$P\_{n, \text{ph}} = \frac{(n + \mu - 1)!}{n! \left(\mu - 1\right)! \left(N\_{\text{th}}/\mu + 1\right)^{\mu} \left(\mu/N\_{\text{th}} + 1\right)^{n}}\tag{24}$$
 
$$P\_{n, \text{ph}}^{(2)} = N\_{\text{th}} \cdot \left(N\_{\text{th}}/\mu + 1\right) \cdot \text{mult}\_{\text{eff}}^{(3)} \cdot N\_{\text{th}} \cdot \left(N\_{\text{th}}/\mu + 1\right) \left(\text{DM} \cdot /\mu + 1\right)$$

with *<sup>n</sup>* <sup>=</sup> *Nth*, *<sup>σ</sup>*(2) *<sup>n</sup>* <sup>=</sup> *Nth* (*Nth*/*<sup>μ</sup>* <sup>+</sup> <sup>1</sup>) and *<sup>σ</sup>*(3) *<sup>n</sup>* <sup>=</sup> *Nth* (*Nth*/*<sup>μ</sup>* <sup>+</sup> <sup>1</sup>) (2*Nth*/*<sup>μ</sup>* <sup>+</sup> <sup>1</sup>).

We reproduced a single mode of this pseudo-thermal light field by passing the coherent light of our laser source through an inhomogeneous diffusing medium and then selecting a single speckle with an aperture (∼ 150 *μ*m diameter), much smaller than the coherence area of the speckle pattern produced (Arecchi, 1965). By delivering this pseudo-thermal light to the SiPM sensor, we measured the values of the output, *x*, at 50000 subsequent laser shots and at 10 different mean values, obtained by means of a variable neutral-density filter (ND in Fig. 9). In Fig. 10 the plot of the experimental values of *Fx*,out and *Sx*,out is shown as a function of *x*out, along with the fitting curves evaluated from Equations 17 and 18:

$$F\_{\rm x,out} = \left(1 - \frac{\overline{\chi}\_{\rm dc}}{\overline{\varpi}\_{\rm out}}\right)^2 \overline{\chi}\_{\rm out} + B$$

$$S\_{\rm x,out} = A \left(1 - \frac{\overline{\chi}\_{\rm dc}}{\overline{\varpi}\_{\rm out}}\right)^3 \overline{\chi}\_{\rm out}^2 + 3B \left(1 - \frac{\overline{\chi}\_{\rm dc}}{\overline{\varpi}\_{\rm out}}\right)^2 \overline{\chi}\_{\rm out} + C\tag{25}$$

where *μ*, the number of independent thermal modes, has been set equal to one and, for simplicity of notation, the parameters *A* = 2, *B* = *G*(1 + 3*xt*)/(1 + *xt*) and *C* = *G*2(1 + 7*xt*)/(1 + *xt*) have been introduced. We fitted the data to *Fx*,out thus obtaining the values of *x*dc = (5.82028 ± 1.34015) and *B* = (87.805 ± 2.09009) ch.

Then the data for *Sx*,out have been fitted substituting the above values of *x*dc ch and *B* in order to obtain *<sup>A</sup>* = (2.34754 <sup>±</sup> 0.091576) and *<sup>C</sup>* = (8531.48 <sup>±</sup> 419.571) ch2. These values were then used to evaluate *G* and *xt*, obtaining *G* = (74.2785 ± 1.76982) ch and *xt* = (0.100174 ± 0.050529). The *x*-values were then divided by *G* and re-binned in unitary bins (Bondani et al., 2009b) to obtain the *Pk*,cross distribution of the actual light signal amplitude measured in the presence of dark counts and cross-talk. In Fig. 11 six different *Pk*,cross distributions are plotted as bars at different mean values. Superimposed to the experimental values, two theoretical distributions are shown: the first one (white circles) is evaluated including the dark-count contribution that modifies the statistics of a single-mode thermal

Characterization and Applications 15

Silicon Photomultipliers: Characterization and Applications 91

Fig. 11. Experimental *Pk*,cross distributions at different mean values (bars) and theoretical distributions evaluated according to *Method I*: thermal modified by dark count distribution (white circles), thermal modified by dark counts and cross-talk effect (full circles). The

Fig. 12. Experimental results for *Method II* applied on two of the histograms acquired with thermal light. Upper panels: result of the multi-peak fit procedure; lower panels: fitted theoretical function. The corresponding fidelity values of the reconstruction are also shown.

corresponding fidelity values of the reconstruction are also shown.

Fig. 10. Plot of *Fx*,out and *Sx*,out as functions of *x*out for pseudo-thermal light.

distribution (see Equation 24) into

$$\begin{split} P\_{m, \text{det} + \text{dc}} &= \sum\_{k=0}^{m} P\_{k, \text{dc}} P\_{m-k, \text{det}} = \\ &= \frac{e^{-\overline{m}\_{\text{dc}}}}{ (\mu - 1)!} \left( 1 + \frac{\mu}{\overline{m}\_{\text{det}}} \right)^{-m} \left( 1 + \frac{\overline{m}\_{\text{det}}}{\mu} \right)^{-\mu} \\ &\quad \text{II} \left[ -m, 1 - m - \mu, \overline{m}\_{\text{dc}} \left( 1 + \frac{\mu}{\overline{m}\_{\text{det}}} \right) \right] \end{split} \tag{26}$$

where *U*(*a*, *b*, *z*) is the confluent hypergeometric function. The parameters are evaluated as *m*dc = *x*dc/(*G*(1 + *xt*)) and *m*det = (*x*out − *x*dc)/(*G*(1 + *xt*)). The second curve (full circles) is evaluated from Equation 15 to take into account the cross-talk. Unfortunately, the calculation does not yield an easy analytical result, and hence it has been evaluated numerically.

The comparison between the data and the theoretical functions can be estimated through the evaluation of the fidelity *f* = ∑*<sup>m</sup> k*=0 *Pk*,exp*Pk*,theo (see Fig. 11).

Turning now to the other approach, it is worth noting that the number of fit parameters is large: the probability distribution is described by the expectation value *m*det of the avalanches generated by detection, the expectation value *m*dc of DCR contribution, the number of modes *μ*, the probability *xt* of triggering a cross-talk event (up to three "iterations") and again a global normalization factor, for a total of 5 fit parameters: obviously, this puts a severe limit on the applicability of this method, needing at least 6 resolved peaks.

As it can be noted from the fit results in Fig. 12, the results obtained by using *Method II* are compatible within errors with what we found by applying *Method I*. However, even if the global fits present a very low *χ*<sup>2</sup> value for degree of freedom, the obtained fit parameters present large uncertainties, probably indicating the presence of very large off-diagonal elements in the minimization matrix and suggesting a strong correlation between the various parameters. This problem can be avoided by fixing some of the fit parameters (such as DCR or cross-talk), once their values have been retrieved from an accurate direct measurement (see Section 2).

14 Will-be-set-by-IN-TECH

Fig. 10. Plot of *Fx*,out and *Sx*,out as functions of *x*out for pseudo-thermal light.

*Pk*,dc*Pm*−*k*,det =

−*m*, 1 − *m* − *μ*, *m*dc

where *U*(*a*, *b*, *z*) is the confluent hypergeometric function. The parameters are evaluated as *m*dc = *x*dc/(*G*(1 + *xt*)) and *m*det = (*x*out − *x*dc)/(*G*(1 + *xt*)). The second curve (full circles) is evaluated from Equation 15 to take into account the cross-talk. Unfortunately, the calculation

The comparison between the data and the theoretical functions can be estimated through the

Turning now to the other approach, it is worth noting that the number of fit parameters is large: the probability distribution is described by the expectation value *m*det of the avalanches generated by detection, the expectation value *m*dc of DCR contribution, the number of modes *μ*, the probability *xt* of triggering a cross-talk event (up to three "iterations") and again a global normalization factor, for a total of 5 fit parameters: obviously, this puts a severe limit on the

As it can be noted from the fit results in Fig. 12, the results obtained by using *Method II* are compatible within errors with what we found by applying *Method I*. However, even if the global fits present a very low *χ*<sup>2</sup> value for degree of freedom, the obtained fit parameters present large uncertainties, probably indicating the presence of very large off-diagonal elements in the minimization matrix and suggesting a strong correlation between the various parameters. This problem can be avoided by fixing some of the fit parameters (such as DCR or cross-talk), once their values have been retrieved from an accurate direct measurement (see

*Pk*,exp*Pk*,theo (see Fig. 11).

−*<sup>m</sup>*

 <sup>1</sup> <sup>+</sup> *<sup>μ</sup> m*det

1 +

*m*det *μ*

−*μ*

, (26)

 <sup>1</sup> <sup>+</sup> *<sup>μ</sup> m*det

does not yield an easy analytical result, and hence it has been evaluated numerically.

*m* ∑ *k*=0

<sup>=</sup> *<sup>e</sup>*−*m*dc (*μ* − 1)!

*U* 

*k*=0 

applicability of this method, needing at least 6 resolved peaks.

distribution (see Equation 24) into

evaluation of the fidelity *f* = ∑*<sup>m</sup>*

Section 2).

*Pm*,det+dc =

Fig. 11. Experimental *Pk*,cross distributions at different mean values (bars) and theoretical distributions evaluated according to *Method I*: thermal modified by dark count distribution (white circles), thermal modified by dark counts and cross-talk effect (full circles). The corresponding fidelity values of the reconstruction are also shown.

Fig. 12. Experimental results for *Method II* applied on two of the histograms acquired with thermal light. Upper panels: result of the multi-peak fit procedure; lower panels: fitted theoretical function. The corresponding fidelity values of the reconstruction are also shown.

Characterization and Applications 17

Silicon Photomultipliers: Characterization and Applications 93

Another instance in which the capability of resolving pulse-by-pulse the number of photons in high-repetition-rate periodic light signals is the analysis of fluorescence fluctuations. Traditionally, fluorescence techniques are applied to the study of systems constituted by huge ensembles of particles. For this reason, the probability distribution of the photon emission rate is peaked around its average value and fluctuations in the fluorescence signal are negligibly small. Nevertheless, if a microscopic system in which only few fluorophores are excited at a time is considered, the fluctuations in the fluorescence intensity become comparable to the average fluorescence intensity value, and can thus be measured. As these fluctuations are due to either the diffusion of the fluorophores in and out from the part of the specimen from which the fluorescence is collected and conveyed to the fluorescence detector, the so-called observation value, or to physical/chemical reactions inducing changes in the emission rate, their analysis can be applied both to characterize the diffusion kinetics and to study the physical/chemical reactions affecting emission. Such techniques are collectively called fluorescence fluctuation spectroscopy (FFS) techniques. This sort of analysis is particularly useful in probing biological systems both in vitro and in vivo. Indeed, many biomolecules (e.g. enzymes) and substances (such as minerals or ions) that are present in traces in cells have a determinant role in the cellular metabolism, and techniques allowing to study their action at physiological concentrations are most desirable. The most popular FFS technique is fluorescence correlation spectroscopy (FCS), which was developed in the

s (Gosch et al., 2005; Schwille, 2001), and consists in the analysis of the temporal

correlations of fluorescence fluctuations in small molecular ensembles, combining maximum sensitivity with high statistical confidence. Among a multitude of physical parameters that are in principle accessible by FCS, it most conveniently allows to determine characteristic rate constants of fast reversible and slow irreversible reactions of fluorescently labelled biomolecules at very low (nanomolar and sub-nanomolar) concentrations. Moreover, the technique is a non-invasive one, which means that the above parameters can be assessed without perturbing the system equilibrium and, possibly, under physiological conditions. The technique also allows the determination of the mobility coefficients and local concentrations of fluorescent (or fluorescently labelled) molecules in their natural environments, provided that the exact value of the observation volume is known. More recently, at the end of the

s, another FFS technique, called photon counting histogram (PCH), was developed by Chen et al. (Chen et al., 1999; 2005). This technique is based on the analysis of the statistical distribution of the fluctuations in amplitude of the fluorescence rate and is thus suited to yield steady state parameters of the system to be investigated, such as the number of different fluorescent species present in a solution, the average number of molecules of each species in the observation volume, and their molecular brightness, rather than dynamic parameters. This technique is thus to a certain extent complementary to FCS. Until now, both FCS and PCH measurements have been made by using APDs or other single-photon counters as the light detectors. Information on either the time-correlations or the statistical distribution of the fluorescence intensity fluctuations have been derived by counting the detection events in consecutive and equally long time intervals. As single-photon counters must be used only in the single photon regime (that is keeping the overall count rate very low in order to be sure that impingement on the detector sensitive area of photons during their dead time is a negligibly rare event), this sampling procedure is extremely time consuming and the sampling interval width must not be too small. Namely, in the most frequent case of pulsed excitation, it has to correspond to tens or even hundreds of pulse periods, which

**4. Photon counting histogram with SiPM**

early 1970

1990

#### **3.3 Discussion**

It is worth comparing the results of the two analysis methods on the same data sets (see Fig. 13). It is possible to notice that the agreement is better for the mean values of detected photons (panel (a)) and for the DCR (panel (b)), while the estimated values of *xt* with the two *Methods* definitely disagree. This can be due to the different approximations adopted by the two *Methods* (first order *vs* third order).

In Table 3 we summarize the results of the two *Methods*. Both *Methods* work in a self-consistent way, even if they have two definitely different approaches. *Method I* does not need peak resolving capability, but requires the acquisition of several histograms at varying *η*. Once determined the parameters *xt* and DCR, all the datasets in a series can be analyzed, independent of the number of distinguishable peaks in the pulse-height spectrum. *Method II* works analyzing each histogram independently, but, as the G-M-avalanches distribution is obtained with a fit of the data, it requires at least a number of resolved peaks larger than the number of free parameters.

The fact that the two *Methods* give very similar results for mean photon numbers is particularly important as in most applications this is the only important parameter. Merging the two *Methods* we can develop an optimal strategy based on a self-consistent calibration performed by measuring a known light and analyzing the data with *Method I*: once *xt* and DCR are known, the determination of the mean photon number is independent of the specific light statistics. Hence the information on the mean photon number can be obtained from each single measurement, even when the fitting procedure of *Method II* cannot be applied.

Fig. 13. Left panel: Cross- talk values of *xt* obtained by applying *Method II* to thermal light (full circles) and their weighted average (full line). Dashed line: value of *xt* obtained by *Method I*. Central panel: Values of DCR evaluated for the same data as in the left panel by applying *Method II* (full circles) and their weighted average (full line). Dashed line: value of DCR obtained by *Method I*. Right panel: Values of mean photon numbers evaluated by applying *Method II* (full circles) and by applying *Method I* (white circles).


Table 3. Comparison between the global DCR and cross-talk values obtained with *Method I* and weighted average of the values obtained with *Method II*.

#### **4. Photon counting histogram with SiPM**

16 Will-be-set-by-IN-TECH

It is worth comparing the results of the two analysis methods on the same data sets (see Fig. 13). It is possible to notice that the agreement is better for the mean values of detected photons (panel (a)) and for the DCR (panel (b)), while the estimated values of *xt* with the two *Methods* definitely disagree. This can be due to the different approximations adopted by the

In Table 3 we summarize the results of the two *Methods*. Both *Methods* work in a self-consistent way, even if they have two definitely different approaches. *Method I* does not need peak resolving capability, but requires the acquisition of several histograms at varying *η*. Once determined the parameters *xt* and DCR, all the datasets in a series can be analyzed, independent of the number of distinguishable peaks in the pulse-height spectrum. *Method II* works analyzing each histogram independently, but, as the G-M-avalanches distribution is obtained with a fit of the data, it requires at least a number of resolved peaks larger than the

The fact that the two *Methods* give very similar results for mean photon numbers is particularly important as in most applications this is the only important parameter. Merging the two *Methods* we can develop an optimal strategy based on a self-consistent calibration performed by measuring a known light and analyzing the data with *Method I*: once *xt* and DCR are known, the determination of the mean photon number is independent of the specific light statistics. Hence the information on the mean photon number can be obtained from each

single measurement, even when the fitting procedure of *Method II* cannot be applied.

Fig. 13. Left panel: Cross- talk values of *xt* obtained by applying *Method II* to thermal light (full circles) and their weighted average (full line). Dashed line: value of *xt* obtained by *Method I*. Central panel: Values of DCR evaluated for the same data as in the left panel by applying *Method II* (full circles) and their weighted average (full line). Dashed line: value of DCR obtained by *Method I*. Right panel: Values of mean photon numbers evaluated by

> Pseudo-thermal *Method I Method II*

DCR 0.071 ± 0.017 0.094 ± 0.035 Cross-talk 0.091 ± 0.042 0.035 ± 0.004

Table 3. Comparison between the global DCR and cross-talk values obtained with *Method I*

applying *Method II* (full circles) and by applying *Method I* (white circles).

and weighted average of the values obtained with *Method II*.

**3.3 Discussion**

two *Methods* (first order *vs* third order).

number of free parameters.

Another instance in which the capability of resolving pulse-by-pulse the number of photons in high-repetition-rate periodic light signals is the analysis of fluorescence fluctuations. Traditionally, fluorescence techniques are applied to the study of systems constituted by huge ensembles of particles. For this reason, the probability distribution of the photon emission rate is peaked around its average value and fluctuations in the fluorescence signal are negligibly small. Nevertheless, if a microscopic system in which only few fluorophores are excited at a time is considered, the fluctuations in the fluorescence intensity become comparable to the average fluorescence intensity value, and can thus be measured. As these fluctuations are due to either the diffusion of the fluorophores in and out from the part of the specimen from which the fluorescence is collected and conveyed to the fluorescence detector, the so-called observation value, or to physical/chemical reactions inducing changes in the emission rate, their analysis can be applied both to characterize the diffusion kinetics and to study the physical/chemical reactions affecting emission. Such techniques are collectively called fluorescence fluctuation spectroscopy (FFS) techniques. This sort of analysis is particularly useful in probing biological systems both in vitro and in vivo. Indeed, many biomolecules (e.g. enzymes) and substances (such as minerals or ions) that are present in traces in cells have a determinant role in the cellular metabolism, and techniques allowing to study their action at physiological concentrations are most desirable. The most popular FFS technique is fluorescence correlation spectroscopy (FCS), which was developed in the early 1970 s (Gosch et al., 2005; Schwille, 2001), and consists in the analysis of the temporal correlations of fluorescence fluctuations in small molecular ensembles, combining maximum sensitivity with high statistical confidence. Among a multitude of physical parameters that are in principle accessible by FCS, it most conveniently allows to determine characteristic rate constants of fast reversible and slow irreversible reactions of fluorescently labelled biomolecules at very low (nanomolar and sub-nanomolar) concentrations. Moreover, the technique is a non-invasive one, which means that the above parameters can be assessed without perturbing the system equilibrium and, possibly, under physiological conditions. The technique also allows the determination of the mobility coefficients and local concentrations of fluorescent (or fluorescently labelled) molecules in their natural environments, provided that the exact value of the observation volume is known. More recently, at the end of the 1990 s, another FFS technique, called photon counting histogram (PCH), was developed by Chen et al. (Chen et al., 1999; 2005). This technique is based on the analysis of the statistical distribution of the fluctuations in amplitude of the fluorescence rate and is thus suited to yield steady state parameters of the system to be investigated, such as the number of different fluorescent species present in a solution, the average number of molecules of each species in the observation volume, and their molecular brightness, rather than dynamic parameters. This technique is thus to a certain extent complementary to FCS. Until now, both FCS and PCH measurements have been made by using APDs or other single-photon counters as the light detectors. Information on either the time-correlations or the statistical distribution of the fluorescence intensity fluctuations have been derived by counting the detection events in consecutive and equally long time intervals. As single-photon counters must be used only in the single photon regime (that is keeping the overall count rate very low in order to be sure that impingement on the detector sensitive area of photons during their dead time is a negligibly rare event), this sampling procedure is extremely time consuming and the sampling interval width must not be too small. Namely, in the most frequent case of pulsed excitation, it has to correspond to tens or even hundreds of pulse periods, which

Characterization and Applications 19

Silicon Photomultipliers: Characterization and Applications 95

in turn would lead to difficulties in reconstructing the photon statistics. In the second case, the DT5720A Desktop Digitizer at our disposal samples the output signal every 4 ns, which means that the signal integration would be performed using only 5 points. For these reasons, a completely different approach to process the detector signal was followed. Namely, the coincidence frequency *ν*(*x*) between a threshold scan of the SiPM output and a trigger signal *ν<sup>T</sup>* synchronous with the laser pulses (but with a threefold reduced frequency) has been

 +∞ *x*

d*ν*(*x*)

where *P*(*y*) is the probability of measuring a value *y* of the detector output signal, *ν*(*x*) is the threshold value, *ν<sup>S</sup>* is the laser repetition rate and *tC* is the coincidence time window. Equation 28 is based on the fact that a coincidence occurs both if after a trigger signal at least a SiPM signal over the threshold *x* is triggered within the coincidence time window or if the trigger signal rises within the coincidence time after the SiPM signal crosses the threshold. For small

Because of the Fundamental Theorem of Integration the detector output signal probability

2*νTνStC*

which can be processed in the same way as the experimental detector output signal histograms experimentally obtained in the case of quantum-light statistics measurements to yield the probability distribution of the number of detected photons (see Section 3). In this set of measurements, the detector was controlled by the CAEN SP5600 General Purpose Power Supply and Amplification Unit module with an integrated threshold discriminator. The digital NIM output of the discriminator was put in coincidence with a synchronism signal triggered by the laser by using an EG&G CO4010 module. The coincidence frequency was sampled by a CAEN VME V1718. The fluorescence emission of two solutions of rhodamine B in water, with concentrations of 2 *μ*M and 200 nM, respectively, have been analyzed with the threshold scan method described above. For the 2 *μ*M concentrated solution a 1 optical density neutral filter was placed in front of the fiber, to reduce the mean detected fluorescence intensity to the value of the 200 nM concentrated solution. This makes the comparison between the two resulting distributions easier. Indeed, for equal expectation value, the 200 nM solution PCH should feature larger deviations from Poisson distribution, therefore showing a higher tail. Hereafter, the pertaining results are briefly summarized and their significance on the way towards implementation of PCH with SiPMs is outlined. Using Equation 30 the multi-peak spectra reported in Fig. 14 were obtained from the staircases for the 200 nM and the 2 *μ*M solution, respectively. The PCH were derived from the spectra by applying the same procedure implemented in the quantum-light statistics data analysis. The zero-photons

+ *νTtCν<sup>S</sup>*

 +∞ *x*

d*yP*(*y*); (28)

d*y P*(*y*). (29)

<sup>d</sup>*<sup>x</sup>* (30)

Π(*k*) (31)

acquired. The measured frequency can be described by the following relation:

*ν*(*x*) ≈ 2*νTνStC*

frequency was determined by imposing the normalization condition:

<sup>Π</sup>(0) = <sup>1</sup> <sup>−</sup>

*L* ∑ *k*=1

*<sup>P</sup>*(*x*) = <sup>−</sup><sup>1</sup>

 <sup>+</sup><sup>∞</sup> *<sup>x</sup>* d*yP*(*y*)

<sup>1</sup> <sup>−</sup> <sup>e</sup>−*tCν<sup>S</sup>*

*ν*(*x*) = *ν<sup>T</sup>*

distribution can be written as

expectation values, Equation 28 can be approximated to

means tens of microseconds or more. Obviously, this puts an inferior limit on the sensibility to fluorescence fluctuations occurring on time scales comparable, or shorter than, the sampling time. Conversely, many biological reactions occur on time scales of hundreds of nanoseconds to few microseconds (Nettels et al., 2007), that is slightly beyond the detection limit with the current procedures. We believe that the usage, for FFS techniques, of detectors able to count how many photons are contained in each fluorescence pulse could allow avoiding sampling. This will significantly and inherently enhance the temporal resolution of both techniques, virtually pulling it on the scale of the excitation source pulse period. Another advantage connected to the release of the single-photon regime operation condition is the possibility of performing statistically reliable measurements in much shorter times, which is notably beneficial for in-vivo analyses, as the samples quickly deteriorate. Recently, we have addressed the task of performing PCH analysis with SiPM detectors. Firstly, the standard PCH model developed for fitting data acquired with APDs (Chen et al., 1999; 2005) is not suitable to properly analyze the SiPM output; thus, the phenomenological model for describing the SiPM response presented above has been used to modify the PCH equations so as to take into account the deviations for ideal detection caused by the presence of optical cros-talk and sizeable DCR, leading to the following fitting model:

$$\Pi^{\star}\left(k; \vec{N}, \varepsilon, \mathbf{x}\_{t\prime}\left\_{dcr}\right) = $$

$$= \sum\_{N=0}^{\infty} p\_{\#}(N; \vec{N}) \sum\_{m=0}^{k} \mathcal{C}\_{k,m}(\mathbf{x}\_{t}) \sum\_{n=0}^{\infty} P\_{n}(\langle n\rangle\_{dcr}) p^{(N)}(m - n; \varepsilon) \tag{27}$$

where *xt* is the parameter representing the probability for a cell of triggering a neighbour cell by cross-talk and *Ck*,*m*(*xt*) is the Bernoullian-like distribution defined in Section 3.1 describing the probability of measuring *k* G-M avalanches out of *n* previously triggered, with *k* > *m*. Secondly, a two-photon excitation setup has been commissioned and tested by performing standard PCH measurements on Rodamine B water solutions with a single-photon avalanche diode (PDM50, MPD) and a PC board allowing for real-time PCH reconstruction (SPC150, Becker & Hickl GmbH). The setup relied on fluorescence excitation by means of a continuous-wave SESAM mode-locked Ti:sapphire laser (Tiger-ps SHG, Time Bandwidth Product) delivering 3.9 ps pulses of 840 nm wavelength at a repetition rate of 48 Mhz. A 1.25 NA, 100X oil-immersion microscope objective (Nikon) was used both to focus the excitation beam onto the sample and to collect the emitted light in epifluorescence configuration. The fluorescence signal was separated from the back-scattered excitation light by using a dichroic mirror. Residual stray light was removed by using a short-wavelength-pass filter with cut at 700 nm. The fluorescence was delivered to the detector, through a 1 mm diameter multimode fiber. With this setup, appreciably superPoissonian PCH distributions were obtained for <300 nM dye solutions. Finally, a series of feasibility studies on the same dye were performed by using as the detector a Hamamatsu SiPM (MPPC S10362-11-100C). Given the 48 MHz repetition rate of the laser light source, the SiPM is expected to detect a fluorescence light pulse every 20 ns. In such experimental conditions, the usual detector signal processing seemed unfeasible with the instrumentation at hand by both the integration method and the free-time signal acquisition method. Indeed, in the first case the CAEN V792N QDC used for the quantum-light statistics measurements presented above requires an integration gate starting at least 15 ns before the rise of the signal, meaning that the board would not be able to process every pulse; moreover, with a detector recovery time of ∼ 60 ns, the output signal may exhibit pile-up features, which 18 Will-be-set-by-IN-TECH

means tens of microseconds or more. Obviously, this puts an inferior limit on the sensibility to fluorescence fluctuations occurring on time scales comparable, or shorter than, the sampling time. Conversely, many biological reactions occur on time scales of hundreds of nanoseconds to few microseconds (Nettels et al., 2007), that is slightly beyond the detection limit with the current procedures. We believe that the usage, for FFS techniques, of detectors able to count how many photons are contained in each fluorescence pulse could allow avoiding sampling. This will significantly and inherently enhance the temporal resolution of both techniques, virtually pulling it on the scale of the excitation source pulse period. Another advantage connected to the release of the single-photon regime operation condition is the possibility of performing statistically reliable measurements in much shorter times, which is notably beneficial for in-vivo analyses, as the samples quickly deteriorate. Recently, we have addressed the task of performing PCH analysis with SiPM detectors. Firstly, the standard PCH model developed for fitting data acquired with APDs (Chen et al., 1999; 2005) is not suitable to properly analyze the SiPM output; thus, the phenomenological model for describing the SiPM response presented above has been used to modify the PCH equations so as to take into account the deviations for ideal detection caused by the presence of optical cros-talk and

sizeable DCR, leading to the following fitting model:

*p*#(*N*; *N*¯ )

= ∞ ∑ *N*=0 <sup>Π</sup>� (*k*; *<sup>N</sup>*¯ , , *xt*,�*n*�*dcr*) <sup>=</sup>

*Ck*,*m*(*xt*)

∞ ∑ *n*=0

where *xt* is the parameter representing the probability for a cell of triggering a neighbour cell by cross-talk and *Ck*,*m*(*xt*) is the Bernoullian-like distribution defined in Section 3.1 describing the probability of measuring *k* G-M avalanches out of *n* previously triggered, with *k* > *m*. Secondly, a two-photon excitation setup has been commissioned and tested by performing standard PCH measurements on Rodamine B water solutions with a single-photon avalanche diode (PDM50, MPD) and a PC board allowing for real-time PCH reconstruction (SPC150, Becker & Hickl GmbH). The setup relied on fluorescence excitation by means of a continuous-wave SESAM mode-locked Ti:sapphire laser (Tiger-ps SHG, Time Bandwidth Product) delivering 3.9 ps pulses of 840 nm wavelength at a repetition rate of 48 Mhz. A 1.25 NA, 100X oil-immersion microscope objective (Nikon) was used both to focus the excitation beam onto the sample and to collect the emitted light in epifluorescence configuration. The fluorescence signal was separated from the back-scattered excitation light by using a dichroic mirror. Residual stray light was removed by using a short-wavelength-pass filter with cut at 700 nm. The fluorescence was delivered to the detector, through a 1 mm diameter multimode fiber. With this setup, appreciably superPoissonian PCH distributions were obtained for <300 nM dye solutions. Finally, a series of feasibility studies on the same dye were performed by using as the detector a Hamamatsu SiPM (MPPC S10362-11-100C). Given the 48 MHz repetition rate of the laser light source, the SiPM is expected to detect a fluorescence light pulse every 20 ns. In such experimental conditions, the usual detector signal processing seemed unfeasible with the instrumentation at hand by both the integration method and the free-time signal acquisition method. Indeed, in the first case the CAEN V792N QDC used for the quantum-light statistics measurements presented above requires an integration gate starting at least 15 ns before the rise of the signal, meaning that the board would not be able to process every pulse; moreover, with a detector recovery time of ∼ 60 ns, the output signal may exhibit pile-up features, which

*Pn*(�*n*�*dcr*)*p*(*N*)(*<sup>m</sup>* <sup>−</sup> *<sup>n</sup>*; ) (27)

*k* ∑ *m*=0 in turn would lead to difficulties in reconstructing the photon statistics. In the second case, the DT5720A Desktop Digitizer at our disposal samples the output signal every 4 ns, which means that the signal integration would be performed using only 5 points. For these reasons, a completely different approach to process the detector signal was followed. Namely, the coincidence frequency *ν*(*x*) between a threshold scan of the SiPM output and a trigger signal *ν<sup>T</sup>* synchronous with the laser pulses (but with a threefold reduced frequency) has been acquired. The measured frequency can be described by the following relation:

$$\nu(\mathbf{x}) = \nu\_T \left( 1 - \mathbf{e}^{-t\_{\mathbb{C}} \nu\_{\mathbb{S}} \int\_{\mathfrak{x}}^{+\infty} \mathrm{d}y P(y)} \right) + \nu\_T t\_{\mathbb{C}} \nu\_{\mathbb{S}} \int\_{\mathfrak{x}}^{+\infty} \mathrm{d}y P(y);\tag{28}$$

where *P*(*y*) is the probability of measuring a value *y* of the detector output signal, *ν*(*x*) is the threshold value, *ν<sup>S</sup>* is the laser repetition rate and *tC* is the coincidence time window. Equation 28 is based on the fact that a coincidence occurs both if after a trigger signal at least a SiPM signal over the threshold *x* is triggered within the coincidence time window or if the trigger signal rises within the coincidence time after the SiPM signal crosses the threshold. For small expectation values, Equation 28 can be approximated to

$$\nu(\mathbf{x}) \approx 2\nu\_T \nu\_S t\_\mathbb{C} \int\_{\mathbf{x}}^{+\infty} \mathbf{d}y \, P(y). \tag{29}$$

Because of the Fundamental Theorem of Integration the detector output signal probability distribution can be written as

$$P(\mathbf{x}) = \frac{-1}{2\nu\_T \nu\_S t\_\mathbb{C}} \frac{\mathbf{d}\nu(\mathbf{x})}{\mathbf{d}\mathbf{x}} \tag{30}$$

which can be processed in the same way as the experimental detector output signal histograms experimentally obtained in the case of quantum-light statistics measurements to yield the probability distribution of the number of detected photons (see Section 3). In this set of measurements, the detector was controlled by the CAEN SP5600 General Purpose Power Supply and Amplification Unit module with an integrated threshold discriminator. The digital NIM output of the discriminator was put in coincidence with a synchronism signal triggered by the laser by using an EG&G CO4010 module. The coincidence frequency was sampled by a CAEN VME V1718. The fluorescence emission of two solutions of rhodamine B in water, with concentrations of 2 *μ*M and 200 nM, respectively, have been analyzed with the threshold scan method described above. For the 2 *μ*M concentrated solution a 1 optical density neutral filter was placed in front of the fiber, to reduce the mean detected fluorescence intensity to the value of the 200 nM concentrated solution. This makes the comparison between the two resulting distributions easier. Indeed, for equal expectation value, the 200 nM solution PCH should feature larger deviations from Poisson distribution, therefore showing a higher tail. Hereafter, the pertaining results are briefly summarized and their significance on the way towards implementation of PCH with SiPMs is outlined. Using Equation 30 the multi-peak spectra reported in Fig. 14 were obtained from the staircases for the 200 nM and the 2 *μ*M solution, respectively. The PCH were derived from the spectra by applying the same procedure implemented in the quantum-light statistics data analysis. The zero-photons frequency was determined by imposing the normalization condition:

$$\Pi^\star(0) = 1 - \sum\_{k=1}^L \Pi^\star(k) \tag{31}$$

Characterization and Applications 21

Silicon Photomultipliers: Characterization and Applications 97

**data**

**-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5**

**-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5**

Fig. 15. Fit (red dashed line) of the experimental PCH data (blue continuous line) for 200 nM

The brightness and average particle number values are fairly consistent with the experimental conditions. Namely, the mean numbers of particles in the excitation volume differ by approximately one order of magnitude, reflecting the 10:1 concentration ratio between the two samples. To this purpose it should be noted that, while the concentration of the 2 *μ*M solution could have been determined spectrophotometrically, 200 nM represents only a nominal value of the concentration of the less concentrated sample, obtained by dilution of the first, as the absorbance of it is below the sensitivity of our spectrophotometer (≈ 0.001) in this case. Also the difference in molecular brightness is roughly consistent with the insertion of the 1-optical-density filter. Indeed, the fact that the molecular brightness measured for the 2 *μ*M concentrated sample is found to be more than twenty times lower could be ascribed to a slight misalignment of either the laser or the microscope objective during the measurement session. The latter lasted for many hours due to the necessity of determining with high statistical reliability the very low frequencies of coincidences occurring at a high threshold value. Even a slight degeneration in the quality of the optical setup alignment would determine a notable decrease in the two-photon absorption cross section, which scales as the square power of the excitation intensity. This would be reflected in the brightness value. A broader focus is also consistent with the mean number of molecules in the excitation volume detected for the 2 *μ*M solution, which is slightly larger than expected. Finally, the difference between the cross-talk values is consistent with a lower over-biasing condition for the 2 *μ*M set, suggesting that a worsening of the photon detection efficiency could also have contributed to the lower molecular brightness. It must be admitted that the above presented procedure is very far from ideal pulse-by-pulse PCH acquisition. However, these preliminary data demonstrate that despite the relatively high dark count rates with respect to APDs

**-4 10**

**data**

**-4 10**

concentration (upper panel) and 2 *μ*M concentration (lower panel).

**-3 10**

**-2 10**

**-1 10**

**1**

**-3 10**

**-2 10**

**-1 10**

**1**

**Photons**

**data Mean 0.01519 RMS 0.1313**

**Photons**

**data Entries 4 Mean 0.02617 RMS 0.1748**

Fig. 14. Multipeak spectra of the fluorescence obtained from the threshold scans using Equation 30; in the upper panel spectrum from the 200 nM solution is shown, while in the lower panel the one obtained from the 2 *μ*M solution is presented. Since the staircases has been acquired with a 2 mV step, histograms have 2 mV large bins. Spectra have been fitted with a multi-Gaussian function and the areas of each peak have been used as the PCH entries.

where *L* is the last detectable peak. The PCH are displayed in Figs. 15, for the 200 nM and the 2 *μ*M solution, respectively. Fitting to Equation 27, with the average number of dark counts fixed to the experimental value preliminarily determined by performing a coincidence measurement of the dark counts, yields the fitting curves reported as dashed lines in Figs. 15.


The fitting parameters are summarized in Table 4.

Table 4. Results for the fit of the PCH distributions. The errors listed are obtained from the fit procedure. The results for the mean number of particles and the molecular brightness are consistent with what expected, even though there is not exactly a factor 10 between the parameters of the two data sets.

20 Will-be-set-by-IN-TECH

**histo 200 nM** 

**Threshold (mV) 20 30 40 50 60 70 80**

Fig. 14. Multipeak spectra of the fluorescence obtained from the threshold scans using Equation 30; in the upper panel spectrum from the 200 nM solution is shown, while in the lower panel the one obtained from the 2 *μ*M solution is presented. Since the staircases has been acquired with a 2 mV step, histograms have 2 mV large bins. Spectra have been fitted with a multi-Gaussian function and the areas of each peak have been used as the PCH

where *L* is the last detectable peak. The PCH are displayed in Figs. 15, for the 200 nM and the 2 *μ*M solution, respectively. Fitting to Equation 27, with the average number of dark counts fixed to the experimental value preliminarily determined by performing a coincidence measurement of the dark counts, yields the fitting curves reported as dashed lines in Figs. 15.

Parameters Concentrations (nM)

*<sup>N</sup>*¯ 5.24 <sup>±</sup> 0.03 71.70 <sup>±</sup> 0.03 *�* 3.99 <sup>×</sup>10−<sup>3</sup> <sup>±</sup> 2.4 <sup>×</sup>10−<sup>5</sup> 1.536 <sup>×</sup>10−<sup>4</sup> <sup>±</sup> <sup>7</sup> <sup>×</sup>10−<sup>7</sup> *xt* 0.08085 <sup>±</sup> <sup>5</sup>×10−<sup>5</sup> 0.0637 <sup>±</sup> <sup>1</sup> <sup>×</sup>10−<sup>4</sup> �*n*�*dcr* 3.24 <sup>×</sup>10−<sup>3</sup> 3.24 <sup>×</sup>10−<sup>3</sup>

Table 4. Results for the fit of the PCH distributions. The errors listed are obtained from the fit procedure. The results for the mean number of particles and the molecular brightness are consistent with what expected, even though there is not exactly a factor 10 between the

200 2000

**-6 10**

The fitting parameters are summarized in Table 4.

parameters of the two data sets.

entries.

**-5 10**

**-4 10**

**-3 10**

**histo Entries 38 Mean 26.62 RMS 5.97**

Fig. 15. Fit (red dashed line) of the experimental PCH data (blue continuous line) for 200 nM concentration (upper panel) and 2 *μ*M concentration (lower panel).

The brightness and average particle number values are fairly consistent with the experimental conditions. Namely, the mean numbers of particles in the excitation volume differ by approximately one order of magnitude, reflecting the 10:1 concentration ratio between the two samples. To this purpose it should be noted that, while the concentration of the 2 *μ*M solution could have been determined spectrophotometrically, 200 nM represents only a nominal value of the concentration of the less concentrated sample, obtained by dilution of the first, as the absorbance of it is below the sensitivity of our spectrophotometer (≈ 0.001) in this case. Also the difference in molecular brightness is roughly consistent with the insertion of the 1-optical-density filter. Indeed, the fact that the molecular brightness measured for the 2 *μ*M concentrated sample is found to be more than twenty times lower could be ascribed to a slight misalignment of either the laser or the microscope objective during the measurement session. The latter lasted for many hours due to the necessity of determining with high statistical reliability the very low frequencies of coincidences occurring at a high threshold value. Even a slight degeneration in the quality of the optical setup alignment would determine a notable decrease in the two-photon absorption cross section, which scales as the square power of the excitation intensity. This would be reflected in the brightness value. A broader focus is also consistent with the mean number of molecules in the excitation volume detected for the 2 *μ*M solution, which is slightly larger than expected. Finally, the difference between the cross-talk values is consistent with a lower over-biasing condition for the 2 *μ*M set, suggesting that a worsening of the photon detection efficiency could also have contributed to the lower molecular brightness. It must be admitted that the above presented procedure is very far from ideal pulse-by-pulse PCH acquisition. However, these preliminary data demonstrate that despite the relatively high dark count rates with respect to APDs

Characterization and Applications 23

Silicon Photomultipliers: Characterization and Applications 99

Agliati, A.; Bondani, M.; Andreoni, A.; De Cillis, G.; & Paris, M. G. A. (2005). Quantum and

Akindinov, A. V.; Martemianov, A. N.; Polozov, P. A.; Golovin, V. M.; & Grigoriev, E. A. (1997).

Akindinov, A; Bondarenko, G.; Golovin, V.; Grigoriev, E.; Grishuk, Yu.; Malkevich, D.;

Andreoni, A. & Bondani, M. (2009). Photon statistics in the macroscopic realm measured

Arecchi, F. T. (1965). Measurement of the Statistical Distribution of Gaussian and Laser Sources. *Phys. Rev. Lett.*, Vol. 15, No. 24, December 1965, 912–916, ISSN 0031-9007. Bondani, M.; Allevi, A.; & Andreoni, A. (2009). Light Statistics by Non-Calibrated Linear

Bondani, M.; Allevi, A.; Agliati, A.; & Andreoni, A. (2009). Self-consistent characterization

Chen, Y.; M ¨uller, J. D.; So, P. T. C.; & Gratton, E. (1999). The Photon Counting Histogram in

Chen, Y.; Tekmen, M.; Hillesheim, L.; Skinner, J.; Wu, B., & Muller, J. D. (2005). Dual-color

Cova, S; Ghioni, M.; Lacaita, A.; Samori, C.; & Zappa, F. (1996). Avalanche photodiodes and

Du Y. & Reti'ere, F. (2008). After-pulsing and cross-talk in multi-pixel photon counters. *Nuclear*

Eckert, P.; Schultz-Coulon, H.-C.; Shen, W.; Stamen, R.; & Tadday, A. (2010). Characterisation

Goetzberger, A.; McDonald, B.; Haitz, R. H.; & Scarlett, R. M. (1963). Avalanche Effects in

G ¨osch, M. & Rigler, R. (2005). Fluorescence correlation spectroscopy of molecular motions and

1464-4266.

1050-2947.

print/ 1362-3044 online.

ISSN 0006-3495.

0006-3495.

0168-9002.

March 1997, 231–234, ISSN 0168-9002.

539, No. 1-2, February 2005, 172–176, ISSN 0168-9002.

1956–1976, ISSN 1559-128X print/ 2155-3165 online.

620, No. 2-3, August 2010, 217–226, ISSN 0168-9002.

34, No. 6, June 1963, 1591–1600, ISSN 0021-8979.

classical correlations of intense beams of light investigated via joint photodetection. *J. Opt. B Quantum Semiclass. Opt.*, Vol. 7, No. 12, November 2005, S652–S663, ISSN

New results on MRS APDs. *Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment*, Vol. 387, No. 1-2,

Martemiyanov, A.; Ryabinin, M.; Smirnitskiy, A.; & Voloshin, K. (2005). Scintillation counter with MRS APD light readout. *Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment*, Vol.

without photon counters. *Phys. Rev. A*, Vol. 80, No. 1, July 2009, 013819, ISSN

Photodetectors. *Adv. Sci. Lett.*, Vol. 2, No. 4, December 2009, 463–468, ISSN 1936-6612.

of light statistics. *J. Mod. Opt.*, Vol. 56, No. 2, January 2009, 226–231, ISSN 0950-0340

Fluorescence Fluctuation Spectroscopy. *Biophys. J.*, Vol. 77, No. 1, July 1999, 553–567,

photon counting histogram. *Biophys. J.*, Vol. 88, No. 3, March 2005, 2177–2192, ISSN

quenching circuits for single-photon detection. *Appl. Opt.*, Vol. 35, No. 12, April 1996,

*Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment*, Vol. 596, No. 3, November 2008, 396–401, ISSN

studies of silicon photomultipliers. *Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment*, Vol.

Silicon p-n Junctions. II. Structurally Perfect Junctions. *Journal of Applied Physics*, Vol.

kinetics. *Adv. Drug. Del. Rev.*, Vol. 57, No. 1, January 2005, 169–190, ISSN 0169-409X.

and the additional artifice constituted by optical cross-talk, SiPMs are detectors capable of discriminating the tiny deviations from Poisson distribution typically displayed by the PCH of diffusing fluorophores in solution at concentrations relevant for biophysical applications. Moreover, we notice that our fitting model yields consistent values of brightness and average fluorophore number. The above conclusions suggest that pulse-by-pulse reconstruction of PCH should be straightforwardly obtained with our detector and electronics apparatus for excitation rates <10 MHz, according to both the recovery time of the SiPM detector and the requirements of the CAEN integrator. An optical pulse-picker may be used to customize the Ti:sapphire repetition rate. Even working at such rates would allow to speed up PCH reconstruction by more than two orders of magnitude with respect to state-of-art APD-based methods. Alternatively, high-repetition rate measurements might be performed by reconstruction methods based on the analysis of the detector free-run output signal, as recorded by fast-sampling-rate digitizers. In such a case, the issues connected with detector pile-up effects would be overcome, thus confining the limits on the acquisition rate to the excitation rate and to the intrinsic decay time of the fluorophores under analysis.

#### **5. Conclusions**

We have presented the characterization of a class of photodetectors, namely the Silicon Photomultipliers, that has acquired great importance in the last decades because of its good photon-counting capability. Besides this feature, the compactness, robustness, low cost, low operating voltage and power consumption are added value that would make these detectors suitable for many applications. To this aim, we have implemented an exhaustive characterization protocol providing a quantitative evaluation of the main figures of merit these detectors are endowed with. In particular, we have studied the dependence of the photon detection efficiency, gain, dark count rate and optical cross-talk on temperature. The protocol has been applied to three SiPM detectors produced by different manufactures and the results have been then employed in a partnership with CAEN to develop the SP5600 General Purpose Power Supply and Amplification Unit module.

The versatility of SiPMs has been investigated in the reconstruction of the statistics of light states: two complementary models that take into account the DCR and the cross-talk effects have been developed in order to describe the output signal of the detectors. The correctness of these methods has been experimentally validated by illuminating the SiPM sensor with a single-mode pseudo-thermal light. However, further improvements could be achieved by including in the models the afterpulsing effect, which would become important in high-rate events.

The applicability of SiPMs in Fluorescence Fluctuation Spectroscopy studies has been preliminary evaluated by analyzing Rhodamine solutions by means of a Two-Photon Excitation setup: the capability of the sensors to reconstruct Photon Counting Histogram and in particular to provide the estimation of the mean number of fluorophores in the excitation volume and their molecular brightness has been proved.

#### **6. References**

Afek, I.; Natan, A.; Ambar, O.; & Silberberg, Y. (2009). Quantum state measurements using multipixel photon detectors. *Phys. Rev. A.*, Vol. 79, No. 4, April 2009, 043830, ISSN 1050-2947.

22 Will-be-set-by-IN-TECH

and the additional artifice constituted by optical cross-talk, SiPMs are detectors capable of discriminating the tiny deviations from Poisson distribution typically displayed by the PCH of diffusing fluorophores in solution at concentrations relevant for biophysical applications. Moreover, we notice that our fitting model yields consistent values of brightness and average fluorophore number. The above conclusions suggest that pulse-by-pulse reconstruction of PCH should be straightforwardly obtained with our detector and electronics apparatus for excitation rates <10 MHz, according to both the recovery time of the SiPM detector and the requirements of the CAEN integrator. An optical pulse-picker may be used to customize the Ti:sapphire repetition rate. Even working at such rates would allow to speed up PCH reconstruction by more than two orders of magnitude with respect to state-of-art APD-based methods. Alternatively, high-repetition rate measurements might be performed by reconstruction methods based on the analysis of the detector free-run output signal, as recorded by fast-sampling-rate digitizers. In such a case, the issues connected with detector pile-up effects would be overcome, thus confining the limits on the acquisition rate to the

excitation rate and to the intrinsic decay time of the fluorophores under analysis.

Purpose Power Supply and Amplification Unit module.

volume and their molecular brightness has been proved.

We have presented the characterization of a class of photodetectors, namely the Silicon Photomultipliers, that has acquired great importance in the last decades because of its good photon-counting capability. Besides this feature, the compactness, robustness, low cost, low operating voltage and power consumption are added value that would make these detectors suitable for many applications. To this aim, we have implemented an exhaustive characterization protocol providing a quantitative evaluation of the main figures of merit these detectors are endowed with. In particular, we have studied the dependence of the photon detection efficiency, gain, dark count rate and optical cross-talk on temperature. The protocol has been applied to three SiPM detectors produced by different manufactures and the results have been then employed in a partnership with CAEN to develop the SP5600 General

The versatility of SiPMs has been investigated in the reconstruction of the statistics of light states: two complementary models that take into account the DCR and the cross-talk effects have been developed in order to describe the output signal of the detectors. The correctness of these methods has been experimentally validated by illuminating the SiPM sensor with a single-mode pseudo-thermal light. However, further improvements could be achieved by including in the models the afterpulsing effect, which would become important in high-rate

The applicability of SiPMs in Fluorescence Fluctuation Spectroscopy studies has been preliminary evaluated by analyzing Rhodamine solutions by means of a Two-Photon Excitation setup: the capability of the sensors to reconstruct Photon Counting Histogram and in particular to provide the estimation of the mean number of fluorophores in the excitation

Afek, I.; Natan, A.; Ambar, O.; & Silberberg, Y. (2009). Quantum state measurements using

multipixel photon detectors. *Phys. Rev. A.*, Vol. 79, No. 4, April 2009, 043830, ISSN

**5. Conclusions**

events.

**6. References**

1050-2947.


**5** 

Yu Zhu Gao

 *China* 

**High Sensitivity Uncooled InAsSb** 

**Photoconductors with Long Wavelength** 

*College of Electronics and Information Engineering, Tongji University, Shanghai* 

It is very attractive to research infrared (IR) materials and detectors in the 8-12 μm wavelength range for photodetection applications. Currently, HgCdTe is the dominant material system in this wavelength range. However, it suffers from chemical instability and nonuniformity due to the high Hg vapor pressure during its growth (Kim et al., 1996). Among III-V compound semiconductors, InAsSb alloy with a band gap as small as 0.1 eV has the advantages of high electron and hole mobilities, good operating characteristics at high temperatures, and high chemical stability. Therefore, the InAsSb system is a very

However, the lattice mismatch between InAsSb epilayers and the binary compound substrates is rather large (for InAs is larger than 6%, for GaAs is 7.2 ~ 14.6%). Thus it is very difficult to grow high-quality InAsSb single crystals with cutoff wavelengths of 8-12 m using conventional technologies (Kumar et al., 2006). Narrow-gap InAsSb epilayers have been grown by molecular beam epitaxy (MBE) (Chyi et al., 1988), metalorganic chemical vapor deposition (MOCVD) (Kim et al., 1996), and liquid phase epitaxy (LPE) (Dixit et al., 2004). The thicknesses of these epilayers are about 2-10 m. The dislocation densities observed in these thin films are as high as the order of 107 cm-2 that are caused by a large lattice mismatch (Kumar et al., 2006).

A melt epitaxy (ME) method for growth of InAsSb single crystals on InAs and GaAs substrates with the wavelengths longer than 8 μm was proposed for a first time by Gao et al (Gao et al., 2002, 2006). The thickness of InAsSb epilayers reaches several decades ~ 100 m. This thickness effectively eliminates the effect of lattice mismatch and results in a low dislocation density (the order of 104 cm-2) in epilayers with a lattice mismatch larger than 6%. Based on the thick InAsSb epilayers grown by ME, high-sensitivity uncooled photoconductors with long wavelength were successfully fabricated (Gao et al., 2011). The IR photodetectors operating at room temperature need not coolers, thus have the important advantages of high speed, small volume, and good reliability. The response speed of them is

We prepared InAsSb epilayers in a standard horizontal LPE growth system with a sliding fused silica boat in high-purity hydrogen ambient. Fig. 1 (a) shows the slideboat schematic

promising alternative long-wavelength IR material to HgCdTe.

It markedly lowers the terminal performance of the detectors.

more than three orders of magnitude faster than that of thermal detectors.

**1. Introduction** 

**2. Melt epitaxy** 

