Electrical crosstalk in a multi-pixel superconducting nanowire single photon detector

B

T

A

P

Electrical crosstalk in a multi-pixel

superconducting nanowire single photon detector

Monique Gevers - 14119773

Supervisors:

N. Los

Dr. A.J. Lock

Abstract

Multiple scientific and industrial applications such as telecommunication and quantum cryptography causing a rise in the development of the superconducting nanowire single photons detectors (SNSPD). In the last decade, the characteristics of the detectors improved but run against the problem that the technology demands larger detectors with the same characteristics. However, the larger the detector the bigger the trade off with other properties of the detector. Therefore, multiple single detectors are fabricated and operate as one multi-pixel detector. This multi-pixel SNSPD has the advantage that the size of every single pixel can stay the same. Therefore, keep the optimized characteristics of the single pixel detector but the opening the possibilities for photon number resolving.

The downside of a multi-pixel SNSPD is that it is more sensitive to electrical crosstalk. when the nanowires of the detector are placed in close proximity to each other, electromagnetic wave propagation at high frequencies and the µm scale can become a problem. Four potential sources of crosstalk will be examined further, starting with the transmission line effects, following the capacitive and magnetic field coupling between nanowires of adjacent pixels. At last, the electric schema of the detectors is examined.

The measurements are performed by biasing one of the four detectors close to the critical current. The other three detectors are not biased and therefore, nonsensitive to photon absorption. The only electrical signal the three pixels will transport is noise and potential crosstalk. The noise is averaged out by measuring over 10,000 sweeps, therefore, only the crosstalk signal remains. To identify a potential source of crosstalk, an estimation of the amount of coupled signal is made using a simple model and simple electric scheme’s of the detector setup.

Measurements of crosstalk resulted in measuring an inverted voltage pulse compared to the signal measured due to photon absorption. In addition, the inverted signal is attenuated with 40-50 dB in voltage. The potential source of crosstalk is not due to the capacitive coupling, as the amount of coupled signal should be a positive pulse and is too weak to measure. The amount of magnetic field coupling is too small to be the source of crosstalk as the meander structure of the nanowire has a negative effect on the coupled signal. The source of crosstalk is due to the connection from the detector to the ground as the bonding wires have a resistance which causes a negative pulse to be measured in the other pixels.

The crosstalk source is identified and is caused by the connection from the detector to the ground. This is a positive result for Single Quantum as the signal is too small to trigger a false detection event. In addition, the signal will not become larger when more multi-pixels are used or when the detectors become larger.

Acknowledgements

I would like to thank Arjan Lock and Niels Los for supervising my final project for the degree of Bachelor of Science in Applied Physics at The Hague University. For the duration of seventeen weeks I worked with state of the art super conducting nanowire multi-pixel single photon detectors. I am grateful towards Single Quantum for the opportunity to fulfill my graduation project at their premises.

Most of all I would like to thank Andreas Fognini, because of his guidance and feedback I was able to stay on the right track. Furthermore, I would like to thank Victor Harting for the discussions and his knowledge. I also would like to thank Yves an Gijs for the hard, but fun times we had during bonding sessions. And off course everyone else at Single Quantum for the amazing time and lots of fun.

Contents

1 Introduction 1

1.1 Thesis overview . . . 2

2 Superconducting Nanowire Single Photon Detector 3 2.1 Superconductivity . . . 3 2.2 Operating principle of an SNSPD . . . 4 2.3 Multi-pixel SNSPD . . . 5 3 Crosstalk 7 3.1 Transmission lines . . . 7 3.2 Mutual Capacitance . . . 8 3.3 Magnetic Induction . . . 9 4 Methods 11 4.1 Experimental setup . . . 11 4.2 The Detector . . . 13 4.3 Data acquisition . . . 15

4.4 Modelling the Capacitance . . . 16

4.5 Modelling Crosstalk . . . 17

5 Results 18 5.1 Crosstalk Measurement . . . 18

5.3 Magnetic Field Coupling . . . 22 5.4 Connection to ground . . . 24

6 Conclusion 26

A Main pulses 27

1

Introduction

Single photon detectors work on the simple principle that an electric signal is generated when a photon is absorbed. The first single photon detector is a photomultiplier tube (PMT) invented in 1935 by H.E. Iams and B. Salzberg [1]. A photon is absorbed by the photocathode of the PMT. Electrons are then ejected as a result of the photoelectric effect and attracted by a dynode. Secondary electrons are ejected after reaching the dynode. The process of electron attraction and ejection by a dynode is repeated several times until the electric signal is high enough to be detected. As successor of the PMT, the avalanche photodiode (APD) became reality due to the efforts of McIntyre [2]. In this semiconductor, the electrons are excited after absorption of a photon. The excited electrons are then accelerated in the device and acquire enough energy to generate secondary electrons by impact ionization. Recently, new technological standards demand better detector characteristics, so in 2001 an even more sophisticated single photon detector was manufactured by Gol’tsman et al. [3]. This single photon detector works on the principle of superconductivity and operates from ultra violet to infra red wavelengths.

Since the invention of the superconducting nanowire single photon detector (SNSPD), the detectors have been optimized and are used in multiple scientific and industrial applications such as infrared spectroscopy, quantum communication, quantum cryptography and medical imaging. At this moment the characteristic properties such as the detection efficiency, dead time, timing jitter and dark counts of the detectors are developed significantly. Detector efficiency’s reaching over 90%. Small adjustments to optimize the detectors are still achievable, but a trade off is to be made between the different beneficial characteristics. For example, a larger detector area means a longer nanowire and, therefore, higher kinetic inductance. When a detector has a higher kinetic inductance, it takes a longer time before the detector is ready for detection again.

To be able to fulfill the demands and wishes of the scientific and industrial world an innovative way of thinking is required. The concept of multi-pixel SNSPD’s is proven as prototypes are developed and produced. However, the behaviour and properties of such a multi-pixel SNSPD still need to be characterized. An SNSPD can be characterized by its most important parameters as stated before. Multiple single SNSPD’s together can improve some of these characteristics and is called a multi-pixel SNSPD. The advantages of a multi-pixel SNSPD are explained: it increases the detection efficiency, when one of the pixels is out of the superconductive state, the other pixels are still detection ready. Therefore, a higher count rate can be achieved. In addition, a multi-pixel detector increases the timing jitter compared to a single pixel SNSPD of the same size, this is because of the reduced kinetic inductance. With a multi-pixel detector photon number resolution and the location of the absorbed photon become available.

The disadvantage of a multi-pixel SNSPD is that when multiple pixels are used, the electronics of the system become more complex as it increases with the increasing number of pixels. In addition, closely packed parallel nanowires of different pixels are potentially subjected to crosstalk, which is the topic of this thesis. Crosstalk can trigger false detection events in pixels of an SNSPD without the absorption of a photon. Therefore, this thesis will focus on finding the source of the potential crosstalk in a multi-pixel SNSPD.

1.1

Thesis overview

Chapter two will explain the principle of superconductivity by diving into the quantum behavior of free electrons at very low temperatures. Thereafter, the operating principle of an SNSPD is explained with the guidance of a figure in which the behaviour of a superconducting nanowire during a detection event is illustrated. To conclude with the advantages and disadvantages of the multi-pixel SNSPD.

Chapter three will introduce potential sources of crosstalk which are of interest during this project such as: the transmission line effects, capacitive coupling, magnetic induction and the connection from the multi-pixel detector to the ground. These potential sources are explained and discussed with figures and formulas to be able to interpret the measurement results.

Chapter four starts with explaining the experimental setup. Thereafter, two different multi-pixel SNSPD designs are presented. In which the dimensions of the pixels and the connections from the detector to ground become clear. Then, the methodology to data acquisition is discussed. Finally, explaining the models which are used to theoretically calculate potential crosstalk.

Chapter five presents the results of the crosstalk measurements. Analyzing the measured signals, starting with the voltage pulse due to absorption of a photon. Then, two different crosstalk signals measured in the multi-pixel SNSPD are shown and discussed. Thereafter, the results of the capacitive and magnetic field coupling are interpret and compared with the crosstalk signals measured. Concluding with the results of main source of crosstalk in the multi-pixel SNSPD’s.

Chapter six describes the conclusions made in the previous chapters and presents the answer to the question: what is the source of potential crosstalk in a multi-pixel SNSPD?

2

Superconducting Nanowire Single Photon Detector

In this section a theoretical understanding of the physics behind the superconducting nanowire single photon detector (SNSPD) is accomplished. Firstly, this section briefly explains the principle of supercon-ductivity, introducing the terms Cooper pair and critical temperature. Secondly, the working principle of the SNSPD will be explained. Finally, the multi-pixel SNSPD’s advantages and disadvantages are discussed.

2.1

Superconductivity

In 1911, the Dutch experimental physicist H. Kammerlingh Onnes discovered superconductivity [4]. Superconductivity is material state that occurs at very low temperatures in which a material looses all electric resistivity. At room temperature, when a current flows through a wire, the free electrons encounter a resistance because of scattering, for example at the lattice vibrations of the material. The electrons that flow through the conductor collide and scatter against the metal atoms of the crystal lattice, as is illustrated in figure 2.1 a). At low temperatures, certain materials make a transition from a normal resistive state to a superconductive state. These materials are called superconductors. For NbTiN this critical temperature Tc

at which the material becomes super conducting is ∼15 K and depends on the thickness of the material. The thinner the material the lower the critical temperature [5]. At this critical temperature the lattice vibrations are reduced such that the moving electrons attract the positive ions of the material, as is shown in figure 2.1 b). Because of the attraction a positive charge density is formed with an electron of opposite momentum. The two electrons form a weak bond and deplete the density of states close to the fermi level. Therefore, the electrons can not scatter and the current flows unperturbed. The electron pair is called a Cooper pair named after Bardeen Cooper who developed the theoretical model. Figure 2.1 c) illustrates a Cooper pair. In the Bardeen Cooper Schrieffer (BCS) theory it is explained that the bounded electrons have a quantum state with an energy gap ∆Egin [J] which is described at zero temperature in

equation 2.1. In this equation, k is Boltzmann’s constant in [JK−1] and Tcthe critical temperature in [K].

The quantum state is a lower energy level than the ground state of an electron in a non-superconductive state. The Cooper pair behave as a boson and therefore do not follow Pauli’s exclusion principle. As a result more Cooper pairs can occupy the same quantum state and flow undisturbed through the conductor because the scattering energies are to low to overcome the energy gap ∆Eg, as illustrated in figure 2.1 d).

This effect is known as superconductivity [6].

Figure 2.1: This figure describes the phenomenon of superconductivity. a) An illustration of the electrical resistance of a conductor at room temperature. b) At lower temperatures the lattice vibrations reduce and the positive ions are attracted to the electrons forming a positive charge density. c) A second electron is attracted by the positive charge density and forms a weak bond with the other electron, a Cooper pair is formed. The Cooper pair encounters less resistance than a single electron. d) More Cooper pairs can exist because the electron pair behave as a boson. The Cooper pairs can flow undisturbed, this is called superconductivity. [7]

2.2

Operating principle of an SNSPD

The operating principle of an SNSPD is based on the superconductive theory and is illustrated in figure 2.2. The superconducting nanowire’s temperature is sustained at a temperature far below its critical temperature. The nanowire is fabricated in a meander structure on a dielectric substrate. The nanowire is biased with a current IBjust below its critical current IC. A schematic view of this situation is shown

in figure 2.2 (i). Above its critical current the nanowire looses its superconductive properties. When an incident photon with enough energy is absorbed by the nanowire it breaks up Cooper pairs and creates a local hotspot, as shown in figure 2.2 (ii). The current is forced to flow around the hotspot and is thereby increasing the current density, as can been seen in figure 2.2 (iii). The increase in current density causes Joule heating and, therefore, the whole section of the nanowire is brought out of a superconductive state into a resistive state, which is illustrated in figure 2.2 (iv). The increase in resistance from zero to Rn> 1kΩ generates a voltage pulse, indicated by step (v) in figure 2.2. The bias current is then blocked

by the high resistance and shunted to an external load Z0<< Rn(t) which stops the heating. Therefore,

the nanowire is able to turn back into the superconductive state, shown in figure 2.2 (vi) [8] [9].

The process which is described above can be modelled by a simple electric circuit as is shown in figure 2.2 b). The kinetic inductance of a superconductor, is much larger than the inductance of a normal conductor, due to the unperturbed electron flow. When the nanowire with kinetic inductance Lkis in a

superconductive state the switch is closed. If a photon is absorbed, the nanowire temporary looses its superconductive state, the switch opens and a voltage pulse is measured across the resistance Z0. Figure

2.2 c) shows an example of the voltage measured across Z0. The rise time τ1, indicated in blue in this

figure, is the time that which is needed for the nanowire to heat up and is dependent on Lkand Rn(t).

The time τ2, indicated in red, is the time it takes for the nanowire to turn back into the superconducting

state and is dependent on Lkand Z0. An important parameter of an SNSPD is the deadtime, which is the

time interval in which it is unable to detect an addional photon. The total dead time is giving in equation 2.2, [9]. τ = τ1+ τ2 = Lk Rn(t) +Lk Z0 . (2.2)

Figure 2.2: The operating principle of an SNSPD. a) An illustration of a detection event in a supercon-ducting nanowire. An incident photon is absorbed which breaks up the Cooper pairs. The release of energy causes a local resistive hotspot. The bias current is forced to flow around the hotspot. This causes a temporary loss of superconductivity in the full cross section of the nanowire due to Joule heating. b) An electrical circuit of the SNSPD is presented. The nanowire is in a superconducting state when the switch is closed and all current can flow directly to ground. When a photon is absorbed and the full cross section of the nanowire is out of the superconducting state, the switch opens and the bias current flows through load impedance Z0generating a voltage pulse. c) The voltage pulse generated due to absorption

of a photon is shown. The blue line indicates τ1which is the time it takes for the nanowire to go out of the

superconductive state. The red line indicates τ2 which is the time it takes for the nanowire to turn back

into the superconductive state. [9].

2.3

Multi-pixel SNSPD

An SNSPD can be characterized by its most important parameters: the system detection efficiency, dark counts, dead time, timing jitter and count rate. The system detection efficiency is the ratio of incident photons to the photons that are actually detected. Dark counts are false detection events which can have several sources. The timing jitter is the time between the absorption of a photon and the resulting generated electric pulse. The count rate is the amount of detection events per second. Multiple single SNSPD’s together can improve some of these characteristics and are called a multi-pixel SNSPD, when installed adjacent to each other. In this project a 2x2 and a 4x1 multi-pixel detectors are used. These detectors are shown in figure 2.3. One of the advantages of a multi-pixel SNSPD is the increase of the system detection efficiency. When one of the pixels is out of the superconductive state during the dead time, the other pixels are still operational. The same effect holds for the count rate. In addition, a multi-pixel detector decreases the timing jitter compared to a single SNSPD of the same size, due to the reduction in inductance, as can be explained by equation 2.2. When using a multi-pixel detector, more knowledge is available on the number of absorbed photons. Besides, the location of the absorbed photon on the detector is known with increased accuracy [8].

The disadvantage of a multi-pixel SNSPD is that closely packed parallel nanowires of different pixels are potentially experiencing crosstalk, which is the topic of this thesis. Crosstalk can trigger pixels of the SNSPD without an absorbed photon. In figure 2.4, a schematic 2x1 pixel detector is shown with the meander structure of the nanowire. The red lines indicates the part which is most sensitive to crosstalk.

Figure 2.3: In this figure two designs of a multi-pixel SNSPD’s are shown. a) A 2x2 multi-pixel SNSPD with lengths of 7 µm. b) A 4x1 multi-pixel SNSPD with adjacent lengths of 14 µm.

Figure 2.4: Illustrates a 2x1 multi-pixel SNSPD with a meander structure nanowire. The red part of the nanowires is the adjacent parallel part and is most sensitive to crosstalk.

3

Crosstalk

This section describes different sources of crosstalk. Crosstalk is the electro magnetic interference (EMI) or in other words, unwanted coupling of a signal from one conductor to another. Specifically, from one nanowire to another one. This phenomenon has many different sources but due to the small scale (∼ µm) this project is mainly interested in crosstalk as a result of capacitance and magnetic induction between the nanowires. Crosstalk occurs when an electric or magnetic field from one nanowire comes in close proximity with an other nanowire. The coupling can therefore be capacitive or inductive [10].

3.1

Transmission lines

The first source of crosstalk could be the effects of a transmission line, which is a pair of parallel conductors through which an electro magnetic (EM) wave propagates. The parallel nanowires of the SNSPD are conductive traces through which an EM wave travels. Therefore, the geometry of parallel nanowires resembles the one of a transmission line. As a rule of thumb, a pair of conductors should be treated as a transmission line when the length of the line is about 10% of the wavelength of the propagating signal. This rule of thumb is only valid for non-superconductors because in the case of superconductors the kinetic inductance is much larger than in non-superconducting materials. To understand the behavior of the electric signal that is travelling through the nanowire, the transmission line parameters are studied. In general, a transmission line can be described by the electric circuit shown in figure 3.1. The transmission line’s parameters as shown in the figure are the inductance per length L’ and the capacitance per length C’ of the conductors [11].

Figure 3.1: Shown are the lumped elements of a section ∆z of the transmission line

A two wire transmission line is a good approximation of the adjacent nanowires in the multi-pixel SNSPD. In comparison to a regular inductor, a superconducting inductor has a much larger inductance than what is calculated from its geometry. The reason is the unperturbed flow of the Cooper pairs. The kinetic inductance per length for a typical NbTiN based superconductor is 1 nHµm−1([12]).

The capacitance of a two wire transmission line is described by equation 3.1, in which  is the electrical permittivity in [Fm−1] of insulating material between the conductors, D is the distance between the centers of the wires in [m], and d the diameter of the wire in [m].

C0= π ln " D d + r  D d 2 − 1 # . (3.1)

A fundamental parameter of a transmission line is the characteristic impedance Z0 in [Ω], which is

described by transmission line parameters in equation 3.2,

Z0=

r L0

C0 . (3.2)

And estimation of the characteristic impedance is made by calculating C’. An approximation of D is taken to be 300 nm as it is the distance between the two centers of the conducting nanowires and d is taken to be 100 nm as it is the width of a nanowire. The relative electric permittivity of the insulating material SiO2is 3.9, the total electric permittivity is the permittivity of vacuum times the relative permittivity ε0εr

in Fm−1. Using equation 3.1 to calculate the capacitance,

C0= π 8.85 · 10 −12_{· 3.9} ln " 300·10−9 100·10−9 + r  300·10−9 100·10−9 2 − 1 # , C0 = 6.15 · 10−11Fm−1.

The characteristic impedance of the nanowire can now be estimated by filling in equation 3.2,

Z0 =

r

1 · 10−3

6.15 · 10−11 ≈ 4, 000 Ω .

The characteristic impedance becomes important in section 5.4, when the electric schema of the multi-pixel detector is further examined.

3.2

Mutual Capacitance

When a photon is absorbed, a hotspot having a resistance of ∼3 kΩ is formed. Therefore, a part of the nanowire will have a potential [V] that is not equal to ground, forcing a current to flow through the other nanowire. The mutual capacitance is the proportionality constant between the amount of current that is forced to flow in the other nanowire and the rate of change of electric potential between the nanowires. This relationship is shown in equation 3.3, where is i(t) the electric charge in [A], Cmthe

mutual capacitance, V the electric potential difference [V]. The mutual capacitance is dependent on geometry and the electric permittivity of the dielectric substrate separating the nanowires ([13]).

i(t) = Cm·

dV (t)

dt (3.3)

Figure 3.2: This figure shows a schematic view of two adjacent nanowires placed on a dielectric substrate. The nanowires are conductors and therefore act as a capacitor with a mutual capacitance between them. When an electric pulse travel through one of the nanowires an electric field arises. The electric potential difference forces a current to flow in the second nanowire ([14]).

3.3

Magnetic Induction

The last potential source of crosstalk of interest for this project is the magnetic induction. The signal in one nanowire produces a changing magnetic field which overlaps the adjacent nanowire. The rate of change in magnetic flux φBpassing through the entire meander of the other nanowire (i.e. the loop)

induces a voltage U as is shown in figure 3.3. This is called Faraday’s law of induction and is shown in equation 3.4. In which U is the induced voltage, φBthe magnetic flux in [T·m2] and t the rise time of the

electric pulse travelling through the first nanowire. The direction of the current that starts flowing due to the induced voltage opposes the magnetic field and explains the negative relation in the equation. This is called Lenz’s law [13].

U = −dφ

dt (3.4)

The magnetic flux can be determined by calculating the magnetic field passing through the total area of the meander of the other nanowire, which is described in equation 3.5. In this equation, B is the magnetic field strength in [T] and S the area of the meander in [m2],

φB =

Z Z

B dL dr. (3.5)

The magnetic field strength around a straight wire at distance r from the nanowire is given by equation 3.6. In this equation, the constant µ0 is the magnetic permeability of vacuum in [Hm−1], Ib the bias

Figure 3.3: A schematic view of two adjacent nanowires placed on a dielectric substrate. An electric pulse travelling through one nanowire produes a changing magnetic flux through the meander of the other nanowire inducing a voltage. [15]

B = µ0Ib

2πr. (3.6)

To find the magnetic flux in the meander of the second nanowire, the magnetic field strength B is integrated over the full length of the adjacent part of the wire according to,

φB = L

Z

B dr , (3.7)

which is integrated over the total width w in [m] of the loop from the nanowire in which d in [m] is the distance between the centers of the adjacent nanowires,

L Z d+w d Bdr = Lµ0I 2π h ln(d + w) − ln(d)id+w d , (3.8)

So the magnetic flux through the meander of the other nanowire is,

φ = Lµ0I 2π

h

ln(d + w) − ln(d)id+w

d . (3.9)

4

Methods

The previous chapter presented the theory of possible causes for crosstalk. The current chapter explains the methodology used to find the main causes of crosstalk between the pixels of a multi-pixel SNSPD. First, the experimental setup and the geometry of the multi-pixel SNSPD are presented. Then, the method to acquisition is explained. Finally, a Comsol model used to determine the capacitance and the method to calculated the S-parameters in Matlab will be discussed.

4.1

Experimental setup

A schematic diagram of the measuring setup is shown in figure 4.1. A diode that emits light at a wavelength of 781 nm and a power output of 10 nW is connected to an attenuator. The attenuator reduces the number of photons that arrive at the detectors. To reduce the signal to single photon level, the signal is attenuated with 30 to 50 dB depending on the efficiency of the detector. Although the signal is attenuated significantly, the detector is still subjected to ∼400k photons per second. The photons reach the detectors in the cryostat via a single mode fiber. This fiber is coupled directly to the detector and has a diameter of ∼9 µm. The dimensions of the detector are ∼14 to ∼28 µm, so, all the photons that are leaving the fiber can potentially be absorbed. The coax cables are connecting the cryostat to a SNSPD driver in which the electric signals are amplified and count. The signal as a result of photon absorption is in the order of a few microvolts, so a power amplification of about 1,000,000 times is needed to reach a reasonable range of amplitudes in the order of ∼1 V on the oscilloscope.

Figure 4.1: In this figure a schematic diagram of the measurement setup is shown. The laser diode sends photons with a wavelength of 781 nm via a single mode fiber to an attenuator. The attenuator is setup such that ∼400k photons per second are subjecting the detector in the cryostat. The signal is amplified by ∼60 dB in the driver before the signal is led to the oscilloscope by four coax cables [16].

To perform a measurement the detectors need to be in a superconducting state which occurs below the critical temperature. Reaching temperatures close to the absolute zero is a challenge because of the thermal radiation caused by the environment at room temperature. The entire system is pumped vacuum, the vacuum is then used as an insulation. Two heat shields are used to protect the detectors from thermal radiation, as is shown in figure 4.2. In between the heat shields the temperature drops to 40 K. Helium gas is pumped inside the metal bar on which the detectors are mounted and cools the detector to a temperature of 2.5 K.

Figure 4.2: This figure is a simplified illustration of the cryostat. The outer en inner heat shield protect the system from the thermal radiation caused by the environment at room temperature. The vacuum inside the system is used as an insulation. Helium gas is pumped inside the metal bar on which the detectors are mounted to reach a temperature of 2.5 K.

The multi-pixel detector is mounted onto a chip using a special glue which can withstand the cryogenic temperatures without breaking, as shown in figure 4.3 a). The detector is bonded to the PCB and mounted inside the cryostat where it is connected to coax cables with SMP connectors, labelled 5 to 8 as seen in figure 4.3 b). A single mode fiber is directly coupled on top of the detectors and kept in place by a zirconia sleeve.

Figure 4.3: The SNSPD and inside of the cryostat. a) The detector is glued with a special glue which can withstand the cryogenic temperatures. b) The detector is bonded to the PCB and mounted inside the cryostat. The single mode fiber is directly coupled to the detector and kept in place by a zirconia sleeve. The coax cables lead from the PCB to the driver. c) The inside of the cryostat is shown over which two heat shields will be placed.

4.2

The Detector

Multi-pixel SNSPD’s can be designed in many different ways. During this project, two designs will be used. The first design, is a 2x2 pixel detector as shown in figure 4.4 a), the second design is a 4x1 pixel detector as shown in figure 4.4 b). In total four detectors with 2x2 pixels and two detectors with 4x1 pixels will be used for the measurements, these are presented in table 4.1. Of interest during this project is the total length of adjacent nanowires from different pixels, because the adjacent part of the detectors is most sensitive to crosstalk. The total length of the adjacent nanowires in the 2x2 SNSPD is 7µm. The total length of the adjacent nanowires in the 4x1 SNSPD is 14µm. Every pixel is labelled according to figure 4.4 to be able to compare the electrical output signals with the location of the pixel.

Figure 4.4: Two meander shaped designs for a multi-pixel SNSPD is shown. The signal trace is indicated by an S and the ground trace is indicated by GND. Each pixel is labelled to be able to compare the output signals. a) A 2x2 pixel SNSPD is shown, the length of the adjacent nanowires is 7µm. b) A 4x1 pixel SNSPD is shown, the length of the adjacent nanowires is 14µm.

Table 4.1: The table presents all the detectors that are used during the measurement. The first column indicates the detector name, the second column presents the detector design and column 3 and 4 indicate the average rise time and dead time respectively.

Detector

Name Design

Avg. Rise Time (ps)

Avg. Dead time (ns) 10 4x1 364 5.1 11 4x1 354 3.9 G 2x2 330 5.6 M 2x2 516 4.1 4 2x2 462 4.5 6 2x2 536 4.0

A schematic cross-section of adjacent nanowires of two pixels is shown in figure 4.5 and is equal for both designs. The nanowires made from niobium titanium nitride are fabricated on top of a dielectric substrate made of silicon dioxide. The nanowires are 7 nm thick and separated by a distance of 200 nm. The thickness of the substrate is defined by the wavelength for which the detector needs to be sensitive. A golden layer used as a mirror is placed underneath the dielectric. Therefore, photons that are not absorbed at their first passage through the nanowire are reflected back and have a second chance to be absorbed.

Figure 4.5: A schematic cross-section of two adjacent nanowires from different pixels shown. The nanowires are placed on a dielectric substrate which has a electrical permittivity of 3.9. A golden ground layer used as a mirror is located underneath the substrate.

Electrically, each pixel has its own read-out system and can therefore operate independently. Still, the pixels do have a shared ground connection within the meander structure. For each detector, its signal and ground line are individually traced out, as seen in figure 4.6. The square in figure 4.6 a) represents the 4x1 SNSPD including the signal and ground traces. Figure 4.6 a) is a close up of b), in which the traces become more visible. In figure 4.6 c), the small lines leading to the four bonding pads are the signal traces. The detector is connected via aluminum bonding wires from the detector chip to the PCB. From there the signal is connected to an amplifier.

Figure 4.6: The gold signal and ground traces are shown in this figure. a) The square in the middle of figure a) represents the nanowires of the 4x1 pixel SNSPD. The golden traces are the signal and ground. b) A zoomed out version of a, in which the structure of the signal traces become more visible. c) The signal pads which connects the detector via aluminum bonding wires to the PCB is shown.

Figure 4.7 is a schematic view of the connection from the bonding pads to the PCB. The connection of the signal traces is indicated in green, the connection of the ground traces is indicated in red. In all measurements, the signal traces are connected to the PCB by one bonding wire. The ground is connected in two different ways: figure 4.7 a) shows that every ground trace is connected by one bonding wire. On the other hand, figure 4.7 b) shows that the side ground traces are each connected by two bonding wires, meaning that the other three ground traces are left floating.

Figure 4.7: This figure shows the connection of the traces from the detector’s bonding pads to the PCB. The connection of the signal and ground traces are shown in green and red, respectively. a) Every ground trace is bonded to the PCB. b) The ground traces on the sides are connected each by two bonding wires to the PCB.

4.3

Data acquisition

The goal is to measure crosstalk between adjacent pixels. Therefore, just one pixel is needed to absorb a photon and generate an electric pulse. That pixel is biased close to the critical current. The other three pixels only need to be able to transport electrical signals due to crosstalk and exclude signals due to photon absorption. These pixels are still in a superconducting state but are not biased and, therefore, nonsensitive to photon absorption. In figure 4.8, the electric schema represents the detectors and the electric read-out. The kinetic inductance Lkrepresents the nanowire of a pixel. The bonding wire is indicated with Rb. One

of the pixels is biased with a bias current IB, the other pixels are not biased. When a photon is absorbed in

the biased pixel, the biased current is shunted and an electric pulse will travel to the driver to be amplified for read-out.

Figure 4.8: The electric circuit represents the multi-pixel SNSPD mounted in the cryostat and connected to the driver. A crosstalk measurement is performed using this setup. One of the four pixels is biased with a bias current IB, the other three pixels are not biased and therefore nonsensitive to photon absorption.

A typical example of the electric pulse generated due to absorption of a photon is shown in figure 4.9. The small peaks are due to reflections caused by the connections of the coax cables. An oscilloscope is used to read out the electric signals of the pixels. The oscilloscope measures over 10,000 sweeps to average out electric noise.

Figure 4.9: A typical example of an electric pulse after absorbing a photon is shown. The small peaks at ∼10, 20 and 30 ns are caused by reflections at the coaxial cable connections.

4.4

Modelling the Capacitance

The amount of capacitive coupling depends on the capacitance between conductors. Hence, a finite element method is used in the software package ’Comsol’. Comsol is used to calculate the capacitance between two parallel nanowires and the case of only one nanowire with a golden mirror, geometry A and geometry B respectively, illustrated in figure 4.10. Geometry A represents two parallel nanowires of adjacent pixels fabricated on silicon dioxide, which has a relative electric permittivity of 3.9. The red trace defines the signal trace and blue trace defines the ground. Geometry B contains one conducting trace only, which defines the signal. The golden mirror is defined as ground. Both geometries are placed within the simulation environment in a big vacuum box which mimics the vacuum of the measuring setup as is discussed in section 4.1. The dimensions of the multi-pixel detectors are described in section 4.2. The smallest element of the mesh used to calculate the capacitance needs to be at least half the size of the smallest part of the model, which is the 7 nm thickness of the nanowire. Therefore, the smallest element of the mesh has a thickness of 3.5 nm.

Figure 4.10: Two geometries are modelled in Comsol to calculate the capacitance between conducting traces. a) Geometry A illustrates two conducting traces placed on a dielectric substrate. The red trace defines the signal, the blue trace defines the ground. b) Geometry B illustrates one conducting trace is placed on a dielectric substrate. Under the substrate, a ground layer is modelled which represents the golden mirror of the SNSPD.

4.5

Modelling Crosstalk

The amount of crosstalk is calculated using the transfer function of an electric system. It compares the output signal with the input signal. The detector setup is transformed into a simplified model-circuit comprising only of capacitors and resistors to calculate the crosstalk due to capacitive coupling, as shown in figure 4.11. The voltage source (V) simulates the electric pulse in the signal trace generated by the absorption of a photon in one pixel. The capacitors represent the capacitance C1 between the two parallel nanowires. C2 and C3 represent the capacitance between the nanowires and the ground layer. These are determined and described in section 4.4. R1 represents the impedance of the coax cable which connects the not biased nanowire to the read-out system [17].

Figure 4.11: An electric schema which represents the capacitive coupling between the nanowires and the golden mirror.

The transfer function describes the amount of crosstalk compared to the amount of voltage generated by absorption of a photon. The transfer function is described here,

Vout Vin = jωRC1 jωRC1+ jωC2+ 1 , (4.1) M agnitude (dB) = 20 log V_out Vin  . (4.2)

5

Results

The previous chapter desribed the methodology used to find the main causes of crosstalk. In this section, the results of the performed measurements of each detector design are presented and discussed. First, the voltage pulse generated by photon absorption in four pixels of one detector are shown, then the measured crosstalk is examined. Thereafter, three different sources of crosstalk are discussed of which two can be excluded as the source of the crosstalk. First of all, the capacitive coupling, the capacitance between the nanowires and between one nanowire and the golden mirror, is calculated, which then is used to determine the ratio of output to input using the transfer function. From this calculation, it will be concluded that the capacitive coupling is not the source of the crosstalk. Thereafter, the magnetic field coupling is estimated by the amount of magnetic flux through the nanowire after which it can be concluded that the magnetic coupling is also not the main source of crosstalk. Finally, the connection from the detector to ground is discussed.

5.1

Crosstalk Measurement

First, the voltage pulse due to photon absorption is shown. Then, the crosstalk measurement of a 2x2 pixel and a 4x1 pixel SNSPD is presented. Every ground trace of the detectors is connected to the PCB, as indicated in figure 4.7 a). Following, a crosstalk measurement of a 2x2 pixel SNSPD, in which only the side ground traces are bonded as indicated in figure 4.7 b).

Figure 5.1: The voltage pulse due to absorption of a photon in pixel 4 of a 2x2 pixel SNSPD is shown. The signal has a rise time of 514 ps and a dead time of 5.9 ns. The small peak that appears at ∼10 and ∼20 ns is a result of the reflection from the connection of the coax cable.

The voltage pulse generated due to absorption of a photon in pixel 4 is shown in figure 5.1. The pulse has a rise time of 514 ps and a dead time of 5.9 ns, which are measured using the oscilloscope. At ∼10 ns and ∼20 ns a small peak appears due to reflections from the connections of the coax cable. The pulse corresponds to the pulse seen in example 4.9. In figure A.1 in appendix A, the main pulses in the other pixels are shown. The rise time of the pulses generated in pixels from detector M range from 508 to 522

Subsequently, the crosstalk measurements when every ground traces of the detectors are bonded are presented and discussed. The detectors that were used to measure the crosstalk were not biased, this makes the pixels nonsensitive to photon absorption. In addition, the electric noise is averaged out by taking the average over 10k sweeps. Therefore, only signals due to crosstalk are measured. Figure 5.2 shows the measured crosstalk in pixel 1-3 when a photon is absorbed in pixel 4 of a 2x2 pixel SNSPD. The electric pulses show a similar characteristic and have a 45-50 dB attenuated and inverted signal strength compared to the signal in the biased detector.

Figure 5.2: The crosstalk measured in pixel 1-3 when a photon is absorbed in pixel 4 of a 2x2 pixel SNSPD is presented. All the signals show a similar characteristic, the strength of the pulses have 45-50 dB attenuated and inverted signal compared to the main pulse.

Figure 5.3 presents the crosstalk signals measured in a 4x1 pixel SNSPD in which all the ground traces are connected. The crosstalk measured does not differ from the crosstalk measured in the 2x2 pixel SNSPD and show the same characteristics.

Figure 5.3: This figure presents the crosstalk measured in pixel 2-4 in a 4x1 pixel SNSPD. The crosstalk signals are attenuated by 40-50 dB compared to the main pulse.

Lastly, the crosstalk measured when only the side ground traces of the detectors are bonded as shown in figure 4.7 b) of the 2x1 SNSPD are presented and discussed. Figure 5.4 shows the electric signal in the three detectors that were not biased. The signals are inverted and the signal’s strengths are 45-50 dB attenuated compared to the main pulse. A big difference between the crosstalk signals is the oscillation which happens at a frequency of ∼1 GHz. In addition, the magnitude of the voltage is larger than when

Figure 5.4: The crosstalk measured in the three pixels of a 2x2 pixel SNSPD that were not biased is presented. The signals strength is 40-50 dB attenuated and are inverted compared to the main pulse. In addition, the signals are oscillating at a frequency of ∼1 GHz, which dampens out around 10 ns.

5.2

Capacitive Coupling

Capacitive coupling could be a source of crosstalk because of the capacitance between conducting traces or planes. To investigate the capacitive coupling, first, the capacitance between two adjacent nanowires and the capacitance between one nanowire and the golden mirror is calculated using finite element method in Comsol. Figure 5.5 shows a schematic of two nanowires modelled on top of the dielectric substrate. The red trace indicates the signal trace in which a photon is absorbed. The blue trace indicates the nanowire that is not biased, and therefore, is defined as the ground. Figure 5.5 a) shows a top view of geometry Awhich is used to calculated the capacitance in Comsol. Figure 5.5 b) represent the side view of the geometry. In figure 5.5 c), a top view of the traces fabricated on the dielectric shows the contour plot of the electric potential calculated in Comsol.

Figure 5.5: A schematic view of two nanowires fabricated on top of a dielectric substrate is shown. The signal trace is indicated by red and the blue trace represents the nanowire that is not biased and defines the ground. a) A top view of geometry A modelled in Comsol and used to calculate the capacitance. b) This figure shows a side view of the nanowires fabricated on the substrate. c) The contour plot of the electric potential is presented. The rectangle shape which follows the dimensions of the dielectric substrate is a result of the difference in electric permittivity between the dielectric and the vacuum box surrounding the

Due to the substantial difference in dimensions of the geometry, i.e. 7 nm thickness and 14 µm length of the nanowire, it costs Comsol too much computing power to calculate the capacitance of design 2. However, nanowire lengths up to 8 µm can be calculated. At this scale the capacitance between the traces is linearly dependent on the length of the trace if the rest of the dimensions are kept constant, except for the length of the dielectric which changes with the length of the trace. Therefore, the capacitance is calculated at variable lengths from 0.5 µm to 8 µm, in steps of 0.5µm. The results are plotted in figure 5.6. This graph shows that the capacitance grows linear with length. Equation 5.1 describes the trend line fit through the simulation points, C is the capacitance in [fF] and l the length in [µm] which indicates a growth of 2.76 · 10−2fFµm−1. Using equation 5.1, the capacitance for both designs can be calculated by filling in the length of the trace and extropolating the trend.

C = 2.76 · 10−2l + 2.41 · 10−3 (5.1)

Figure 5.6: The graph shows the capacitance between the traces as a function of the length of the traces calculated in Comsol. A trend line is fit through the simulation points. The capacitance for both designs can be calculated using the equation of the trend line.

The capacitance between the nanowire and the golden mirror for both designs is calculated using the same methodology. The capacitance of both designs and geometries is presented in table 5.1. To estimate the crosstalk due to capacitive coupling, the capacitance is filled in equation 4.2, as discussed in section 4.5. The output to input signal ratio of both designs are calculated and represent the attenuation, as shown in figure 5.7. In both designs the signal is attenuated by more than 75 dB.

Table 5.1: This table shows the calculated capacitance in femto Farad for geometries A and B and for the 2x2 pixel and 4x1 pixel SNSPD. The capacitance is calculated using an extrapolation by trend line equations.

C [fF] 2x2 4x1 A 0.20 0.39 B 0.36 0.72

108 109 -110 -100 -90 -80 -70 -60 -50 Magnitude (dB) 4x1 2x2 Bode Diagram Frequency (Hz)

Figure 5.7: This figure present the attenuation of the strength crosstalk signal, the red curve indicates the attenuation of the signal in the 2x2 pixel detector, the blue curve indicates the attenuation of the signal in the 4x1 pixel detector.

Figure 5.8 shows the circuit of the nanowires fabricated on the dielectric with the golden mirror. The expected crosstalk signal due to capacitive coupling should be attenuated and should be positive. However, the crosstalk signal which was measured is negative. In addition, the crosstalk signal due to capacitive coupling would be to small to measure as the main pulse would be attenuated by more than 75 dB. Therefore, it can be concluded that the crosstalk signal measured is not due to capacitive coupling.

Figure 5.8: The electric schema represents the capacitive coupling between the nanowires. The red squares indicate the signal and crosstalk trace. The crosstalk signal as a result of capacitive coupling is illustrated. The signal is positive and attenuated compared to the electric pulse in the signal trace.

5.3

Magnetic Field Coupling

The magnetic field coupling is the second possible source of crosstalk which is examined. The electric pulse that travels through the nanowire creates a changing magnetic flux which induces a voltage in the not biased nanowire. The magnetic flux changes to opposite directions due to the meander structure of the

of the magnetic field coupling. A simple estimation of the amount of induced voltage is made. First the induced voltage as a result of the adjacent straight part of the nanowire will be estimated, after which the meander structure of the nanowire is taken into account.

First the magnetic flux due to a straight wire is calculated using equation 3.9. The length L is the length of the adjacent nanowire of the 2x2 pixel which is 7 µm. The magnetic permeability µ0has a value

of 4π · 10−7Hm−1, the bias current Ibis 15 µA, the distance between the nanowires d is 200 nm and the

width w of total meander structure of the nanowire is 7 µm.

φB = Lµ0Ib 2π h ln(d + w) − ln(d)id+w d φB= 7 · 10−6· 4π · 10−7· 15 · 10−6 2π  ln(14.2 · 10−6) − ln(200 · 10−9) φB= 8.95 · 10−17Tm−2

The magnetic flux through the area of one pixel is 8.95 · 10−17Tm−2. The the induced voltage can be calculated using equation 5.3. The most significant change in magnetic flux is during the rise time of the pulse being 516 ps. ε = −dφ dt ε = −8.95 · 10 −17 516 · 10−12 = −1.73 · 10 −7_V

The calculated voltage needs to be amplified by 56 dB, as the signal is amplified in the driver:

56 dB = 20 log ε −1.73 · 10−17

ε = −0.11 mV.

The induced voltage due to the adjacent part of the nanowire after amplification is -0.11 mV. The same calculation is performed for the part of the nanowire that is a distance d further away from the adjacent pixel, as indicated in figure ??. The induced voltage as a result of this nanowire has a value of +0.09 mV. The resultant of only these two nanowires is an induced voltage of -0.02 mV, when the rest of the meander structure is taken into account, the induced voltage decreases even more. The measured crosstalk has the same "sign". However, the magnitude of the induced voltage is much lower than the signal peak measured. Therefore, it can be conclude that the magnetic field coupling is not the source of the measured crosstalk.

5.4

Connection to ground

As described, the capacitive coupling and magnetic field coupling can both be excluded as the source of the measured crosstalk. At last, the connection from the detector to ground, when every ground is connected properly, is examined. Figure 5.9 shows an electric schema of the detectors, which is explained in 4.3.

Figure 5.9: One pixel is biased with a bias current IB. An electric signal is generated when a photon is

absorbed. The positive pulse travels to the signal end, the negative pulse can not flow directly into the ground due to the resistance of the bonding wire. Therefore, the negative pulse travels into the direction of the not biased nanowires and is amplified and read-out in the driver.

When a photon is absorbed a rapid increase in resistance of ∼1 kΩ creates a positive voltage pulse into the direction of the amplifier of that pixel. The positive pulse ensures that a negative pulse travels into the direction of the shared ground. In general, a signal does not encounter any resistance and can leak into the ground. In this case, the ground traces of the detectors are connected via a bonding wire which has a resistance to the PCB. Therefore, the negative voltage pulse travels through the nanowires of the not biased pixels and are read-out in the driver. The amount of this crosstalk is determined by using the ratio of impedances, as described in equation 5.2. The coax cable has an impedance of Rcoax= 50 Ω, the

nanowire has an impedance of Z0 = jωLkand a kinetic inductance of 1 µH as stated in section 3.1,

[dB] = 20 log Rcoax4 Rcoax3+ 2πf Lk



(5.2)

The amount of attenuation of the crosstalk signal is plotted in figure 5.10 and matches the attenuation of the crosstalk signals measured. In addition, due to the positive pulse travelling to the signal end, a negative pulse starts travelling into the direction of the ground. Which explains the inverted signal.

108 109 -50 -45 -40 -35 -30 -25 -20 Magnitude (dB) Bode Diagram Frequency (Hz)

Figure 5.10: This figure presents the attenuation of the strength of crosstalk signal. The attenuation of the signal is ∼40 dB, which corresponds to the attenuation of the crosstalk signal’s strength measured.

The measured crosstalk signal oscillates when not all ground traces are connected properly. The source of the oscillating signal is unknown. However, it can be excluded that it is due to reflections of the floating ground traces. The signal oscillates with a frequency of ∼1 GHz, meaning that it will travel a distance of ∼20 cm. If there were any reflections due to the floating ground it would have a frequency of ∼ 1011_{Hz, the amplifier is unable to amplify these frequencies, therefore, it can not be measured.}

6

Conclusion

In total six detectors are used during this project, of which four detectors are fabricated with 2x2 pixels and two detectors with 4x1 pixels. All the crosstalk measurements resulted in measuring a crosstalk signal. The crosstalk signal of interest is the crosstalk signal that appears when all the ground traces of the detector are bonded to the PCB. A single negative voltage pulse is the result of the crosstalk measurement. The signal strength of the pulse is attenuated by 40-50 dB compared to the main pulse. When the side ground traces are bonded only, an oscillating negative signal arises.

Potential sources of crosstalk are discussed during this project. Firstly, the crosstalk due to capacitive coupling is examined. The crosstalk signal due to capacitive coupling should have a positive value and the signal strength should be attenuated by more than 80 dB. Therefore, the measured crosstalk signal is not due to the capacitive coupling. Secondly, the crosstalk as a result of magnetic field coupling is estimated. The crosstalk signal due to magnetic field coupling should be negative. However, the estimated signal strength is too weak to measured. Therefore, the magnetic field coupling is not the source of the crosstalk signal.

At last, the connection from the multi-pixel SNSPD to the ground is examined. As a result of resistance of the bonding wire, the negative pulse, which opposes the positive pulse due to photon absorption, is unable to flow directly into the ground. Therefore, the negative pulse will flow through the nanowires of the other pixels and is detected. The attenuation of the signal strength is calculated using the ratio of impedances, which matches the attenuation of the crosstalk signal strength measured.

A

Main pulses

This appendix presents figure A.1, in which the electric signal generated as a result of photon absorption in a 2x2 pixel SNSPD are shown. In every figure, small peaks at ∼10 and ∼20 ns appear. These peaks are due to reflections of the connection of the coax cable. The difference is pulse height is due to the difference in bias current.

Figure A.1: The voltage pulse due to absorption of a photon in in pixel 4 of a 2x2 pixel SNSPD. The signal has a rise time of 514 ps and a dead time of 5.9 ns. The small peak that appears at ∼10 and ∼20 ns is a result of the reflection due to the connection of coax cable.

B

Crosstalk signals

This Appendix presents the crosstalk signals measured in a 2x2 pixel SNSPD and a 4x1 pixel SNSPD, in which all the ground traces are bonded to the PCB.

Figure B.1: This figure shows the crosstalk signals measured in a 2x2 pixel SNSPD. In every measurement an other pixel is biased. The crosstalk signals have similar characteristics and the signal strength is 40-50 dB attenuated compared to the main pulses.

Figure B.2: This figure shows the crosstalk signals measured in a 4x1 pixel SNSPD. In every measurement an other pixel is biased. The crosstalk signals have similar characteristics and the signal strength is 40-50 dB attenuated compared to the main pulses.

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