Detecting Single Photons with Graphene-Based Josephson Junctions

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Citation Walsh, Evan Daniel. 2020. Detecting Single Photons with Graphene- Based Josephson Junctions. Doctoral dissertation, Harvard University Graduate School of Arts and Sciences.

Citable link https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37368983

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DISSERTATION ACCEPTANCE CERTIFICATE

The undersigned, appointed by the

Harvard John A. Paulson School of Engineering and Applied Sciences have examined a dissertation entitled:

“Detecting Single Photons with Graphene-Based Josephson Junctions” presented by: Evan Daniel Walsh

Signature ______Typed name: Professor P. Kim

Signature ______Typed name: Dr. K. C. Fong

Signature ______Typed name: Professor D. Ham

Signature ______Typed name: Professor D. Englund

October 2, 2020 Detecting Single Photons with Graphene-Based Josephson Junctions

a dissertation presented by Evan Daniel Walsh to The School of Engineering and Applied Sciences

in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Applied Physics

Detecting Single Photons with Graphene-Based Josephson Junctions

Abstract

In this work I present theory, modeling, and experimentation demonstrating that the graphene-based Joseph- son junction (GJJ) is a capable system for the detection of single photons across the electromagnetic spectrum, from the microwave to the infrared. Two different detection mechanisms are exposed: 1) heating of the graphene weak link in the GJJ and 2) quasiparticle generation in the GJJ superconducting contacts. The first relies on graphene’s exceptionally low heat capacity and its decoupled electron and phonon systems. I show in modeling that these thermal properties can lead to a GJJ photon detector for very low energy microwave photons. Ex- perimentally, I show that a GJJ bolometer with its graphene weak link coupled to a microwave resonator can achieve an energy resolution equivalent to a single 32-GHz photon. The second detection mechanism reveals itself in the illumination of a GJJ with near-infrared light. I experimentally demonstrate single-photon detection in this system and show that the data is well fit by a model where photon-induced quasiparticles in the superconducting contacts cause the GJJ phase particle to escape. I give an overview of the single-photon, graphene, and

Josephson-junction physics required to arrive at these results before presenting the experimental evidence for single-microwave-photon sensitivity and single-infrared-photon detection with the GJJ.

iii Contents

Title Page i

Abstract iii

Table of Contents iv

Acknowledgements vi

1 Introduction 1

2 Single Photon Detection 4 2.1 Introduction ...... 4 2.2 Quantum Description of Light ...... 5 2.3 Photon Counting Statistics ...... 9 2.4 Detection Mechanisms ...... 12

3 Graphene 16 3.1 Introduction ...... 16 3.2 Thermal Properties ...... 21 3.3 Optical Properties ...... 24 3.4 Coupling Photons to Graphene ...... 27

4 Josephson Junctions 32 4.1 Introduction ...... 32 4.2 The Josephson Tunneling Junction ...... 33 4.3 Macroscopic Description: RCSJ Model and Phase Particle Dynamics ...... 35 4.4 Microscopic Description: Andreev States ...... 40 4.5 Graphene-Based Josephson Junctions ...... 42 4.6 Absorption in Superconducting Contacts ...... 45

5 Modeling of a GJJ SPD 49 5.1 Introduction ...... 50 5.2 Device Concept and Input Coupling ...... 51 5.3 Graphene Thermal Response ...... 54 5.4 Graphene-Based Josephson Junction ...... 60 5.5 Photon Detection Performance ...... 65 5.6 Conclusion ...... 69

iv 6 Another Approach: Johnson Noise Thermometry 70 6.1 Introduction ...... 70 6.2 Johnson Noise Thermometry Basics ...... 71 6.3 Single Photon Detection ...... 72 6.4 Photon-Number Resolving ...... 73

7 GJJ Microwave Bolometry with Single-Photon Sensitivity 76 7.1 Abstract ...... 77 7.2 Introduction ...... 78 7.3 Experimental Setup ...... 78 7.4 GJJ Microwave Response ...... 82 7.5 Bolometer Performance ...... 85 7.6 Methods ...... 86

8 Detecting Single Infrared Photons with a GJJ 93 8.1 Abstract ...... 94 8.2 Introduction ...... 95 8.3 Evidence of Single-Photon Detection ...... 98 8.4 Switching Mechanism ...... 100 8.5 Absorption ...... 101 8.6 Physical Model ...... 103 8.7 Tables of measured devices ...... 104 R 8.8 Calculation of photon from laser power ...... 105 8.9 Single-Photon Detection in Pulsed Measurements ...... 106 8.10 Proportionality of Γmeas in laser power ...... 108 8.11 Sweeping Experiments versus Counting Experiments and their correspondence ...... 109 8.12 Experimental determination of polarization orientation ...... 110 8.13 HFSS simulation ...... 111 8.14 Γmeas of JJ using tetra-layer graphene ...... 112 8.15 Modeling of Γmeas in Fig. 8.3B ...... 113 8.16 Gate Dependence of the GJJ Single-Photon Response ...... 116

9 Conclusion 119

Appendix A Experimental Setup 121 A.1 Device Mounting ...... 121 A.2 Optical Setup ...... 122 A.3 Electronic Setup ...... 125

References 128

v Acknowledgments

God has abundantly blessed me in the 9 years since I started my Ph.D. with a path I would have never expected.

The family and friends who have supported me during that time have been nothing short of astounding and I would not be at this milestone in my life without them. First and foremost, I acknowledge my wife, my forever friend, my love, Beth. Who knew the adventure we would be embarking on when we went to play frisbee 7 years ago? She has been there with me every step of the way, through multiple new career opportunities, through dat- ing, engagement, and marriage, through starting a new church, through four children! Her love, encouragement, and prayers for me have provided me with the strength to continue on in my Ph.D. work. A loving wife, a loving mother, dedicated and strong, I could go on and on! So to Beth — thank you. To my four children, Aubree, Na- talee, Daniel, and my youngest son who I can’t wait to meet — I love you and love watching you grow up. Thank you for knowing how to put a smile on Daddy’s face and being the loving, silly, amazing kids that you are. To my parents, Dan and Joy — your loving support throughout my life has set the foundation for me to succeed.

Your advice of “half work, half life” when I started graduate school stuck with me throughout my Ph.D. studies. I could not ask for more loving parents. To my brother Brady — your love and support of me, and now my family, have been great blessings to me throughout my life.

Of course the work in this dissertation was not just “my” work but the work of a large team of people. First, I would like to thank KC Fong who was the driving force behind the project. Thank you for your hard work and dedication throughout this project and for teaching me all the interesting physics and experimental techniques I needed to be successful. Next, thank you to my advisor Dirk Englund, for the advice and support along the way and for making me feel a part of your lab even though I took a non-traditional path to get there. To Philip Kim — thank you for always having an open door for me. One person who is truly an indispensable member of the team

vi is Gil-Ho Lee. Gil-Ho — thank you for all the devices, all the teaching on Josephson junction physics, and all the overall hard work you put into this project. To Dima Efetov — thank you for showing me the ropes and all the advice you gave me during our time together. Almost all of my work was carried out at BBN Technologies and I want to thank everyone in the Quantum Group there for welcoming me in. The conversations, the help on my experiments, and the general camaraderie made for a great graduate experience.

Beyond those directly involved in the work, so many other friends and family helped me get to this point. To my Aunt Sylvia, Grandpa Armenak (how I wish you could see this day!), Grandma Winnie, Uncle Yervant, Uncle

Peter, and my cousin Ben — I treasure your love and support. Steve, Jordan, Chen, Demetrios, Paul, Nishant, and Alexei — your friendship throughout the years has obviously been a huge blessing to me and I’ve learned actually that good friends like you can become family. Michael Rabenberg — my Ph.D. would not have been possible without all the deep philosophical discussions we shared over burgers. To my old lab mates, Tony, Jesse,

Rui, Tommi, Daniel, Danny, and Xi — it was great working together and supporting one another. To Yi Lin — thank you for being a great advisor and making my time at NASA so enjoyable. To my new colleagues at STR — thank you for the support while I finished this degree and for creating such a great team atmosphere to work in.

And to my church family — your prayers were invaluable to me and my family throughout my graduate career.

vii And God said, “Let there be light,” and there was light.

And God saw that the light was good. And God separated the light from the darkness.

Genesis 1:3-4 ESV 1 Introduction

Light in its many forms is an integral part of daily life, from the visible sunlight that illuminates our world to the radio waves that are the basis for cell phones, WiFi, and even microwave ovens. The smallest constituent of light, a single photon, has such low energy that a photon in the visible regime is barely perceptible (if at all) to the human eye.1 The limitation of the human eye to seeing these photons is two-fold: 1) the efficiency of the receptors in our eyes is not high enough for the photon to reliably trigger a response and 2) there are other background noise processes that trigger false detections, or “dark counts,” which prevent us from perceiving true single-photon events above this background. Efficiency and dark count rate are the two most fundamental parameters that will determine if any device is able to detect single photons or not. These parameters become even

1 more important for radio-frequency photons, whose energy is 1000 times less than that of the photons visible to humans.

Why might someone want to detect a single photon in the first place? Near-infrared photons are the basis of many quantum cryptography schemes, with the promise of secure communications over thousands of kilome- ters.2–4 In the far-infrared/terahertz regime, small numbers of photons from astrophysical sources are continuously arriving at Earth — being able to detect them could reveal further information about our universe.5–8

Some theorize that elusive dark matter could interact with matter to produce terahertz photons that could be detected with single-photon detectors.9,10 Radio-frequency detectors can be used in quantum computing as an alternate readout scheme for quantum states.11,12 Amazingly, the device investigated in the current work shows potential for detection of light across the electromagnetic spectrum, with demonstrated energy resolution on the scale of single radio-frequency photons and verified detection of single infrared photons. The device isa graphene-based Josephson junction (GJJ).

Graphene is a two-dimensional material comprised of a single layer of carbon atoms in a honeycomb hexagonal lattice while a Josephson junction is a device in which two superconductors are separated by a non-superconducting material (in this case, graphene) but that still allows a supercurrent to flow through it. We initially chose the

GJJ for the task of single-photon detection due to the unique thermal properties accompanying graphene’s two- dimensional character, properties that allow graphene-based detectors to have high efficiency and low dark count rates. Modeling and experimental results in the radio-frequency regime confirmed this to be true. The JJ was originally viewed merely as a useful tool for capitalizing on graphene’s exceptional properties. However, as experiments to detect infrared photons with the GJJ began, it soon became evident that the JJ itself had its own set of properties desirable for the detection of photons. In these experiments, the graphene played the supporting role while the interesting single-photon physics belonged to the JJ. In the end, the GJJ proved to be a fascinating device with the combined properties of the graphene and JJ being enablers for single-photon detection.

2 This dissertation is laid out as follows. In Chapter 2, I lay out what is meant by a single photon (a more controversial term than one may think) and briefly discuss detection in general. In Chapters 3 and 4, I explain the physical properties of graphene and Josephson junctions, respectively, and how those properties are desirable for detecting single photons. In Chapters 5 and 6, I present the modeling for graphene-based detection of single photons using two different detection readout schemes: 1) the JJ and 2) Johnson noise thermometry, respectively.

The discussion of Chapter 5 is reproduced from Ref. [13] and is the method I ultimately pursued experimentally.

Chapter 7 presents the experimental demonstration of the GJJ’s ability to resolve energy down to the level of a single radio-frequency photon as originally presented in Ref. [14] and Chapter 8 presents the experimental direct detection of single infrared photons. Finally, the Conclusion is presented in Chapter 9.

3 [...] it is the nature of all bodies, as they have been formed by the coming together of particles, that those same bodies also give off particles. These particles are emitted inthe shape of the objects from which they come, and thus we call them “images.” These images are far too fine to be perceptible in themselves, but the evidence supports the conclusion that these images exist, and that as they move they preserve, in some degree, their respective positions that they held in the solid bodies from which they came. [...] these images move with unsurpassable speed. Epicurus, in Letter to Herodotus, c. 300 BC 2 Single Photon Detection

2.1 Introduction

Before discussing how a graphene-based Josephson junction can detect single photons, it must be discussed what exactly constitutes a single photon. The most basic definition of a photon is a quanta of light, i.e. a quantized excitation of the electromagnetic field. That light has a particle nature was hypothesized as far back as theancient

Greek philosophers but its wave nature, as famously captured by Maxwell’s equations in the mid to late 19th century, was more heavily emphasized (with the notable exception of Isaac Newton’s corpuscular theory of light based on the work of Pierre Gassendi) until the groundbreaking work of Max Planck in 1900 and Albert Ein-

4 stein in 1905. With that said, describing light as a particle is somewhat controversial — Lamb, for instance, gives a thorough history of the term photon and then proceeds to argue it should be abandoned altogether (or at least require a license)15 while Loudon acknowledges the term’s usefulness, its pitfalls, and cautiously gives a definition to start his book on Quantum Electrodynamics.16 In this chapter we discuss more thoroughly the quantum description of light, how one can count single photons, and mechanisms of detection.

2.2 Quantum Description of Light

The following discussion follows closely that of Schiff who gives a thorough exposition starting from the principle of least action, classical Lagrangian field theory, and classical Hamiltonian field theory leading to quantized versions of these theories.17 To show what is meant by a single photon, I start with Maxwell’s equations, show how to arrive at a Hamiltonian from Maxwell’s equations, and then show how a quantized version of the Hamil- tonian leads to a quantum harmonic oscillator solution corresponding to a photon picture of electromagnetism.

The vacuum (i.e. sourceless) Maxwell’s equations are:

1 ∂⃗B ∇ × ⃗E + = 0 (2.1a) c ∂t

∇ · ⃗E = 0 (2.1b)

1 ∂⃗E ∇ × ⃗B − = 0 (2.1c) c ∂t

∇ · ⃗B = 0 (2.1d)

5 where ⃗E is the electric field, ⃗B is the magnetic field, and c is the speed of light. Recall that ⃗E and ⃗B can be written in terms of a vector potential A⃗ and scalar potential φ as:

1 ∂A⃗ ⃗E = − − ∇φ (2.2a) c ∂t

⃗B = ∇ × A⃗ (2.2b)

Now Maxwell’s equations must be consistent with the Lagrange (field) equations given by:

" # ∂L X ∂ ∂L − = 0 (2.3) ∂ψ ∂x ∂ ∂ /∂ i j j ψi xj

where L is the Lagrangian density, ⃗ψ = A⃗ , φ are the field components, and⃗x is a four-vector consisting of the three spatial dimensions,⃗r, and one time dimension, t. It can be shown that the appropriate choice of L for consistency with Equation 2.1 is:

! 2 1 1 ∂A⃗ 1 2 L = + ∇φ − ∇ × A⃗ (2.4) 8π c ∂t 8π

To go from this Lagrangian description to a Hamiltonian description, we introduce the canonical conjugate momentum density, ∂L ⃗P = (2.5) ∂ ∂⃗ψ/∂t which for 2.4 is: ! 1 1 ∂A⃗ ⃗P = + ∇φ (2.6) 4πc c ∂t

6 (Note I’ve been lax in notation here — in Equation 2.5, ⃗P is a four-vector while in Equation 2.6 it is only a three- vector, this decision made because the φ component of Equation 2.5 is zero. A similar note applies to Equations

2.7 and 2.8). Now the Hamiltonian density in general is given by:

∂⃗ψ H = ⃗P · − L (2.7) ∂t which for electromagnetism is:

1 H = 2πc2|⃗P|2 + |∇ × A⃗ |2 − c⃗P · ∇φ (2.8) 8π

Taking the integral of Equation 2.8 to get the full Hamiltonian and noting that the third term when integrated is zero: Z 1 H = d⃗r 2πc2|⃗P|2 + |∇ × A⃗ |2 (2.9) 8π

So far the discussion has been completely classical. Here we shift to a quantized picture: we replace the vectors ˆ ˆ ⃗P and A⃗ with the vector operators ⃗P and A⃗ . We can expand these operators in terms of an orthonormal basis.

Choose plane waves of the form:

1 i⃗k·⃗r ⃗u⃗ (⃗r) = √ ε⃗ e (2.10) kξ V kξ

⃗ where ε⃗k1 and ε⃗k2 are two unit polarization vectors perpendicular to each other and to the propagation vector k and V is the volume of a large cubical box used to set boundary conditions for⃗k (V will not affect our calculations ˆ ˆ and disappears when limits to infinity are taken). Now expanding ⃗P and A⃗ :

X h i ˆ † ∗ ⃗P(⃗r, t) = ˆp⃗ (t)⃗u⃗ (⃗r) + ˆp (t)⃗u (⃗r) (2.11a) kξ kξ ⃗kξ ⃗kξ ⃗kξ

7 X h i ⃗ˆ † ∗ A(⃗r, t) = ˆq⃗ (t)⃗u⃗ (⃗r) + ˆq (t)⃗u (⃗r) (2.11b) kξ kξ ⃗kξ ⃗kξ ⃗kξ

Here, ˆp and ˆq are expansion coefficient operators, † denotes Hermitian adjoint, and the sum is over the half k plane. Now inserting Equation 2.11 into Equation 2.9, making use of the commutation relations:

h i h i † † ~ ′ ′ ˆq⃗ (t), ˆp ′ ′ (t) = ˆq (t), ˆp⃗′ ′ (t) = i δkk δ (2.12) kξ k⃗ ξ ⃗kξ k ξ ξξ

(with all other commutations zero), and noting the integrals will be trivial due to the presence of delta functions, we have for the Hamiltonian operator:

X 2 2 † k † Hˆ = 4πc ˆp⃗ (t)ˆp (t) + ˆq⃗ (t)ˆq (t) (2.13) kξ ⃗kξ 4π kξ ⃗kξ ⃗kξ

Other than normalization, this is the Hamiltonian of a quantum harmonic oscillator. The usual steps can be taken to rewrite Equation 2.13 in terms of creation and annihilation operations. Briefly, in this case the annihila-

ˆ ( ) ˆ ( ) tion operator (which can be derived through calculating the time evolution of p⃗kξ t and q⃗kξ t in the Heisenberg

~ dF = [ , ] picture of quantum mechanics via i dt F H ) is given by:

2 1 4πc iωt ˆa⃗ (t) = ˆq⃗ (t) + i ˆp⃗ (t) e (2.14) kξ 2 kξ ω kξ

† where we’ve used ω = ck. The corresponding creation operator is ˆa (t). These operators correspond to annihi- ⃗kξ ⃗ lating or creating, respectively, a plane-wave mode with propagation vector k and polarization ε⃗kξ. Defining the number operator, which has non-negative-integer eigenvalues, as:

ω † Nˆ⃗ (t) = ˆa (t)ˆa⃗ (t) (2.15) kξ 2π~c2 ⃗kξ kξ

8 we can rewrite Equation 2.13 as: X 1 Hˆ = ~ω Nˆ⃗ (t) + (2.16) kξ 2 ⃗kξ

| ⟩ It can be shown that the eigenvectors of this Hamiltonian are the number states, n⃗kξ , which are also the eigenvectors of the number operator, with

ˆ | ⟩ = | ⟩, = , , , ... N⃗kξ n⃗kξ n⃗kξ n⃗kξ n⃗kξ 0 1 2 (2.17)

and X 1 Hˆ|n⃗ ⟩ = ~ω n⃗ + |n⃗ ⟩, n⃗ = 0, 1, 2, ... (2.18) kξ kξ 2 kξ kξ ⃗kξ

This equation says that the energy measured in an electromagnetic field must be an integer number of ~ω plus some zero-point energy. To review, we started with Maxwell’s equations, calculated the corresponding Hamil- tonian through the use of the Langrange field equations, and expanded the field components in terms ofplane waves to show the Hamiltonian could be cast as that of a quantum harmonic oscillator with energy quantized as integer multiples of ~ω. Equation 2.18 captures exactly what is meant by a single photon: it is a single quantum excitation of an electromagnetic field mode. Although the current discussion focused on plane wave modes inan infinite vacuum, the ideas extend to modes in a cavity or other systems aswell.

2.3 Photon Counting Statistics

In practice, number states, including single-photon states, are hard to prepare. Instead, coherent states as prepared by a laser are often used. A coherent state, |α⟩, (sometimes referred to as a Glauber state) is an eigenstate of the annihilation operator and can be written in terms of the number states, |n⟩ (dropping subscripts for simplic-

9 ity), as:18 ∞ X n − 1 |α|2 α |α⟩ = e 2 √ |n⟩ (2.19) ! n=0 n where α is the eigenvalue of the annihilation operator acting on |α⟩. This state |α⟩ itself is not an eigenstate of the

Hamiltonian but instead is a superposition of the energy eigenstates (number states). This fact means there is not a well-defined number of photons in a coherent state but instead an expected (or average) number ofphotons.

Taking the expectation value of the number operator in a coherent state gives:

⟨α|Nˆ |α⟩ = |α|2 (2.20)

meaning |α|2 is the average number of photons in the state |α⟩. Furthermore, if we project a coherent state into the number state eigenbasis, we can see the average occupation of each number state is given by:

2n |α| −| |2 |⟨n|α⟩|2 = e α (2.21) n!

This is the equation of a Poisson distribution with the parameter of the distribution being the average number of photons in the state, |α|2. In the experiments of Chapter 8, we used a laser as our photon source so we when we say we’ve detected a single photon, we really mean we’ve projected a coherent state into an energy eigenbasis where the resulting measurement collapsed the wavefunction into a single-photon state (but the measurement could have instead yielded zero, two, three, etc. photons with probability given by a Poisson distribution).

It is easiest to think of the Poisson statistics of a coherent source in terms of a finite laser pulse. A finite (in space and time) coherent pulse when measured will yield a finite number of photons. If many such identical pulses were measured, the measured number of photons in the pulses would follow a Poisson distribution. Some experiments in Chapter 8 were pulsed. Most however, were performed using a continuous wave (CW) laser. Fur-

10 Figure 2.1: Poisson distribution of a coherent light source (A) Whether using an ensemble of pulses from a pulsed laser, dividing a CW laser into many time bins, or dividing either into many spatial bins, the number of photons per pulse, time bin, or spatial bin will follow a Poisson distribution. (B) The Poisson distribution for an average of 1 photon (blue) or 0.13 photons (red) per pulse, time bin, or spatial bin. Note that a coherent state with on average 1 photon still has a relatively high probability (∼26%) of yielding a measurement of more than 1 photon. For the experiments of Chapter 8, an average number of photons per pulse of 0.13 was chosen to ensure a very low chance of measuring multiple photons in a given pulse (∼0.8%) . thermore, all the experiments used a laser spot size much larger than the GJJ detector itself. In the CW, large spatial extent case, one can think of the number of photons being detected by the GJJ as follows. Divide the incoming laser into many pulses in both time and space (see Figure 2.1A). The number of photons measured in these pulses will also follow a Poisson distribution with the caveat that the distribution will be weighted by the intensity profile of the laser. For the single-mode-fiber-coupled laser of Chapter 8, the intensity is constantin time and has a Gaussian spatial profile. Spatially, we only sample the laser in one location (the location of theGJJ) so the Poisson statistics in this dimension are not observable. In time, however, we continuously monitor the detector for trigger events. If the detector is truly detecting single photons, then we expect that if we divide our measurement into many time bins, that the number of trigger events per bin should follow a Poisson distribution.

In Chapter 8 we show this is indeed the case. I note that caution must be taken when interpreting the observed

Poisson distribution — any independent, random events will follow Poisson statistics. If the GJJ were only sensitive to 2-photon states, for example, the detections of these 2-photon states would still be Poisson distributed in time. However, in Chapter 8 we show that the number of detection events is also proportional to the num-

11 ber of photons thus demonstrating sensitivity to 1-photon states (sensitivity to 2-photon states would result in a number of detection events proportional to the square of the the number of photons). A helpful discussion of photon-counting statistics with a slightly different derivation is given in Chapter 5 of Ref.[19].

Intuitively, one might expect that to execute a single-photon experiment with a laser one should set the average number of photons in the coherent states to be 1. However, as illustrated by the blue line in Figure 2.1B, with an average of 1 photon in a coherent state, there is still a fairly large chance of measuring multiple photons, ∼26%.

Instead, single-photon experiments with coherent states should choose average photon numbers well below 1.

For the pulsed measurements of Chapter 8, we chose an average of 0.13 photons per pulse (Poisson distribution shown in red in Figure 2.1B). For this average number of photons, the chances of a measurement yielding multiple photons is only ∼0.8%, 15 times smaller than the chance of the measurement yielding 1 and only 1 photon.

In CW experiments, the choice of photon rate (i.e. average number of photons per time bin) takes the place of average number of photons per pulse. The photon rate should be chosen such that the average number of photons reaching the detector during its response time is much less than 1. In Chapter 8, we estimate the response time of the GJJ infrared photon detector to be on the order of 1 ns, suggesting the photon rate should be kept well below

1 GHz to guarantee single photon events. This condition was easily met, with typical photon rates less than 100

Hz. In practice, the limiting factor to the photon rate was instead heating of the GJJ.

2.4 Detection Mechanisms

Now that a single photon has been defined, I discuss how one could detect these low energy excitations. InChap- ters 5 and 6 I will discuss two specific detection mechanisms: threshold detection with a Josephson junction

(JJ) and thermal detection with Johnson-noise thermometry (JNT), respectively. Threshold detection is fairly straightforward: the photon causes some parameter to change and once that parameter surpasses a threshold (determined by the detector properties), a photon count is registered. In a JJ, the photon causes the voltage to jump

12 from zero to nonzero (i.e. the JJ to switch from superconducting to resistive). These jumps are fairly large (Figure

2.2A) so the choice of a threshold voltage is somewhat trivial — there is a large range of voltages sufficiently above the noise floor and sufficiently below the peak voltage upon photon absorption to limit the number offalseposi- tives (i.e. measurement dark count rate) and false negatives (i.e. measurement detection inefficiency), respectively.

Note the inclusion of the word “measurement” to emphasize this dark count rate and inefficiency are due to the measurement setup and not the intrinsic properties of the detector itself. The threshold detection mechanism is similar to that used in superconducting nanowire single-photon detectors (SNSPDs).20,21 JNT could also be used in a threshold detector although its strength lies more in acting as a calorimeter, being able to determine how much energy was deposited as opposed to simply whether or not energy was deposited (see Chapter 6). In the

JNT graphene single-photon detector setup, a photon raises the temperature of the graphene which in turn raises its voltage noise. Measuring the power of the voltage noise, one could set a threshold between the base voltage noise power and the elevated voltage noise power upon absorption to register photon detections. However, the measured voltage noise is proportional to the temperature, unlike in a JJ where the measured voltage is a constant determined solely by the bias current with no relationship to the temperature. To take full advantage of JNT, one can then use the measured value of voltage noise to estimate the temperature as a function of time and from that deduce how much energy was deposited into the detector. This could provide much more information than a simple threshold detection, such as the frequency of the photon being detected or how many photons were being detected. Photon-number resolving with JNT is discussed in Section 6.4.

So far in this section, it was assumed that the signal-to-noise ratio (SNR) was sufficient to either set a threshold that would limit dark counts without limiting efficiency or in the case of JNT, determine a temperature with enough resolution to draw conclusions about the amount of energy the detector absorbs. However, high SNR in the raw voltage or voltage noise signal is not a necessary condition for single photon detection. It was pointed out in a helpful review of calorimeters by McCammon that filtering the signal can drastically improve SNR.22 Specif-

13 Figure 2.2: Detection of high SNR versus low SNR signals (A) A voltage versus time trace from the GJJ of the experiments in Chapter 8. The voltage signal from single-photon events in this case was high above the noise floor simplifying the measurement scheme (B) In cases where the SNR is low, matched filtering can be used to extract a signal out of the noise floor. Shown is a simulated measurement withSNR of 1 (blue), the noiseless signal (red), and the output of a matched filter acting on the measured signal using the noiseless signal asthe template (black). The matched filter changes a barely discernible signal into one well above the noise floor. Such a technique couldprove especially useful in JNT.

ically, for a known signal s0(t) (often called a template) with spectrum˜s0(f) and known noise spectral density

SN(f), an optimal linear filter (a “matched filter”) will be of theform:

∗ ˜s (f) h(f) = 0 (2.22) SN(f)

∗ where the denotes complex conjugate. If we have a measured signal smeas(t), then the filtered version, sfilt(t), will be: Z ∞ ∗ ˜s0(f)˜smeas(f) sfilt(t) = (2.23) −∞ SN(f)

This filtered version of the measured signal will have peaks at times where the original signal resembled thetem- plate. I show an example in Figure 2.2B. A noisy signal (blue) is formed by adding white Gaussian noise to a noiseless signal (red). The peak signal value is 1 as is the root-mean-square (RMS) noise value, yielding an SNR of 1. As is clear, a threshold detector would not work on this raw signal. However, using a template of the signal to match filter the noisy signal, a peak rises out of the noise at the time where the signal appears (blackline).The

14 filtered signal is normalized so the RMS noise is still 1 and it can be seen that theSNRisnow ∼7. A threshold detector could now operate effectively on this filtered signal. The level of the peak will also be proportional to the energy in the original signal so JNT could be used as well. As clearly illustrated, the advantage of this method is that it can drastically increase the SNR compared to the raw measurement. However, the disadvantage is that both the shape of the signal and the noise must be known to form the optimal filter. Typically, the noise can be measured so determining SN(f) is straightforward. However, determining the proper template can be more difficult. Even so, a matched filter with errors in the template will still often increase the SNR. Alternatively,a bank of matched filters could be used on the measured signal to increase the probability of picking outlow-SNR signals if a range of templates is expected prior to the measurement. In the experiments of Chapter 8, the SNR was high enough that matched filters were unnecessary. However, future experiments, especially in JNT, could benefit from such a technique.

15 If N classical particles in two dimensions interacting through a pair potential Φ(⃗r) are in equilibrium in a parallelogram box, it is proved that every⃗k ≠ 0 Fourier component of the density must vanish in the thermodynamic limit [...]. This results excludes conventional crystalline long-range order in two dimensions [...].

N. David Mermin, 1968 3 Graphene

3.1 Introduction

Graphene, a single layer of carbon atoms held together by sp3 hybridized bonds in a honeycomb lattice and originally thought to be thermodynamically unstable,23–25 was first isolated and observed in 2004.26 Since its experimental realization, graphene has proven its worth in a myriad of applications, from sensors of light,27–30 heat,31–33 and pressure34 to electrochemical energy storage35–37 to mechanical lubrication.38 The reason for graphene’s such widespread use is the unique properties it possesses, including being the strongest material by weight (in terms of Young’s modulus), having the largest specific surface area, and having a relativistic band struc-

16 Figure 3.1: Graphene’s band structure ture. It is this last property that produces the exceptional electronic, optical, and thermal properties in graphene that render the material an ideal candidate for single photon detection and that will be the focus of the current chapter.

Graphene’s band structure was first calculated more than half a century before the material itself wasiso- lated.39 Graphene’s honeycomb structure can be thought of as a triangular lattice with two atoms per unit cell √ with the lattice spacing being 3 times the carbon-carbon spacing of a ∼1.42 Å (i.e. the Bravais lattice is formed by carbon atoms that are second-nearest neighbors instead of nearest neighbors). The reciprocal lattice is also triangular, yielding a hexagonal Brillouin zone. Using a tight-binding model only including nearest-neighbor interactions, it can be shown that the dispersion relation is given by:39–41

s √ √ 3 3 E± (k) = ±t 3 + 2 cos 3k a + 4 cos k a cos k a (3.1) y 2 y 2 x

− with the + and referring to the conduction and valence band, respectively, k = kx, ky being the electron momentum vector, and t ∼2.8 eV being the nearest-neighbor hopping energy. From Figure 3.1 it can be observed

17 that the two bands come together at the corners of the Brillouin zone which can be described by the two momen- ′ − tum vectors K = 2π 1, √1 and K = 2π 1, √1 . Taylor expanding around these so-called Dirac points in 3a 3 3a 3 terms of the momentum vector q = k − K satisfying |q| ≪ |K| gives:

≈ ±~ | | E± (q) vF q (3.2)

≈ 6 ~ where vF = 3ta/2 10 m/s is the Fermi velocity and = h/2π with h being Planck’s constant. Unlike in typical materials with a quadratic dispersion relation and a gap between the conduction and valence bands, graphene’s dispersion is linear near the Dirac points and there is no gap. This band structure is often referred to as “relativistic” because, similar to a photon, the energy is proportional to the momentum instead of its square and, also like a photon, there is a constant velocity (or in this case Fermi velocity) as a function of momentum.

This unusual band structure makes graphene a semimetal — when the graphene is intrinsic with Fermi level

42–44 EF at the Dirac point, the density of states (and carrier density) is zero and graphene is almost (but not quite ) insulating. If graphene is somehow electron- or hole-doped away from the Dirac point, the density of states and carrier density are much larger and graphene acts more like a metallic conductor. The most typical way to change the charge carrier concentration in graphene is through an electrostatic gate, a technique that was used even in graphene’s first inception.26 In this scheme, graphene is separated from another conductor by an insulator of thickness d to form a parallel plate capacitor and a gate voltage VG is applied between the two plates to change the charge density. Typically, this task is accomplished by placing the graphene on a silicon chip with a ∼300-nm silicon oxide layer, the silicon acting as the second plate in the capacitor and the oxide as the separating insulator.

Three simple relationships can be used to calculate the charge carrier density as a function of VG: 1) a capacitor’s charge Q is related to its voltage V by Q = CV where C is the capacitance, 2) the charge on graphene can be written as Q = enA where e is the electron charge, n is the surface charge-carrier density (i.e. carriers per unit area),

18 Figure 3.2: The measured RN of a GJJ versus VG (blue data points) and the best fit line using Equation 3.4 (red line). The fitting parameters for the fit line are given in the table at right.

= Aε and A is the graphene area, and 3) the capacitance of a parallel plate is given by C d where ε is the dielectric constant of the insulator between the plates. Combining these three equations gives:

ε n = V (3.3) ed G where n > 0 means electron doping and n < 0 means hole doping. In the case of 300-nm silicon oxide, this

− translates to dn = 7.2 × 1010 cm 2/V. dVG

An easily observable result of n being a function of VG is that the normal-state resistance of the graphene, RN, is a function of VG as well (here “normal” means “non-superconducting” which will be important for the discussion on Josephson junctions in the next chapter). In the ideal case (no impurities, scatterers, contact resistance, etc.), the resistivity, ρ = RNW/L, where W and L are the device width and length respectively, is inversely pro-

| | −1 portional to the carrier density: ρ = ( n eμ) with μ being the mobility. In practice, when measuring RN (and therefore ρ) of graphene, it is important to also include the effects of excess charge that allow for non-infinite ρ

(i.e. non-zero conductivity) at the Dirac point and other non-ideal sources of resistivity. The equation for ρ then

19 becomes:

| | −1 ρ = ( n eμ + σ0) + ρ0 (3.4)

where σ0 is the residual conductivity at the Dirac point and ρ0 is the background resistivity due to scatterers and contact resistance. I plot the measured RN as a function of VG (blue circles) for a GJJ (Device A of Chapter 8) in Figure 3.2 with graphene width W=2.8 µm and graphene length L = 160 nm. Note that the Dirac point (i.e. the largest value of RN) does not align with VG = 0 V as one would expect in the ideal case. Instead it occurs at

− VG = 6.3 V meaning the graphene is inherently p-doped (hole doped). This is common in GJJs with NbN contacts. I fit the data using Equation 3.4 and 3.3 (with VG replaced by VG+6.3 V) using μ, σ0, and ρ0 as fitting

2 parameters. I find μ=5600 cm /Vs, σ0 = 70 µS, and ρ0 = 8.4 Ω. From σ0 it is also useful to calculate the minimum

10 −2 achievable charge carrier density, nden = σ0/eμ, which is 7.8x10 cm in this case.

Changes in carrier doping also change the Fermi level as available states in momentum space are filled or emp- tied. Close to the Dirac points, the total area in momentum space for a given q is simply πq2. The area of a single momentum state is (2π)2/A (where A is still the graphene area in real space) so the total number of charge carriers N is:

2 2 N = 4(πkF)A/(2π) (3.5)

where kF is the q vector at the Fermi level and the degeneracy factor of 4 is included to account for two atoms per p | | | | unit cell and two spin states per electron. Solving for kF in terms of n = N/A yields kF = π n . Combining this result with equation 3.2 gives for the Fermi level relative to the Dirac point at a given charge carrier density:

p ~ | | EF = sgn (n) vF π n (3.6)

∼ ∼ × 12 −2 Pushing the gate to the limits of the breakdown voltage of the oxide layer (VG 100 V) gives n 7 10 cm

20 ∼ and EF 310 meV. For Pauli blocking of 1550 nm light (see section 3.3) gating to Fermi levels of at least 400 meV is required. To achieve such large doping levels, electrolytic gates have been developed that can increase the

× 14 −2 45 carrier density up to 4 10 cm corresponding to EF = 2.3 eV.

Closely related to the charge carrier density and the Fermi level is the density of states per unit energy, D(E) = dN = dN dk dE dk dE . Using Equations 3.2 and 3.5, the density of states for graphene is then given by:

2A |E| ( ) = D E ~2 2 (3.7) π vF where E is measured in reference to the Dirac point. It is important to note that because graphene has a gapless band structure, E, and therefore D(E), can be tuned arbitrarily close to zero. Having such a small density of states will have a profound impact on graphene’s thermal properties as will be discussed in the following section.

3.2 Thermal Properties

Two thermal properties of graphene make it well suited for single photon detection: 1) its low electronic heat capacity and 2) its fast electron-electron scattering time in comparison to its electron-phonon scattering time.

The former is a direct result of the band structure discussed in the previous section. Heat capacity, a measure of the amount of energy required to raise the temperature of a material, can be calculated for any metal within the free electron gas approximation as: π2 C (T) = k2T D (E ) (3.8) e 3 B e F

where kB is Boltzmann’s constant, Te is the electron temperature, and D (EF) is the density of states evaluated at the Fermi level EF. This expression is valid for graphene in the degeneracy regime, i.e. when the charge carrier

& density is large compared to thermally excited carriers, or equivalently when EF kBTe. Using Equation 3.7, the

21 Figure 3.3: Heat capacity per square micron of a graphene sheet as a function of electron density and temperature. electronic heat capacity in degenerate graphene is then given by:

5/2 2 4Aπ kB 1/2 Ce (T, n) = Tn (3.9) 3hvF

In the nondegenerate regime, thermal excitations lead to the number of mobile electrons and holes being on par with one another so that Coulomb interactions become important and the free electron gas approximation is no longer valid. Such systems have recently been observed experimentally46 and it is has previously been shown theoretically that the heat capacity should be approximately quadratic in temperature with a logarithmic sup-

47 pression at lower temperatures. In the present work, Te is in the range of millikelvin to a few Kelvin which

− would require n . 5 × 108 cm 2 to be in the nondegenerate regime, far below doping levels currently achievable experimentally so that Equation 3.9 will be used throughout. The importance of Equation 3.9 consists of its dependence on n. As mentioned in Section 3.1, the carrier density in graphene can be tuned very close to the

Dirac point with values much smaller than those in typical metals. Figure 3.3 shows Ce (T, n0). For reference,

22 × 12 −2 n0 = 1.67 10 cm is a conservative value of the carrier density that will be used in the modeling of Chap- ter 5 and much of the experiments of Chapter 8. For reference, some of the most sensitive bolometers, based on

4 6 superconducting nanowires, have heat capacities on the order of 10 kB, three orders of magnitude larger than a

1 µm2 sheet of graphene at the same temperature.

Beyond its extremely small heat capacity, graphene’s thermodynamic properties also aid in the material’s ability to detect single photons. The details of the thermodynamic processes in graphene are discussed thoroughly in

Section 5.3. Briefly, upon absorbing a photon (see next section) an electron is excited high into graphene’s conductance band. This high-energy electron can lose energy either to other electrons or to phonons. It has been shown experimentally that the electron-electron scattering time is much faster (femtoseconds) than the electron- phonon scattering time (picoseconds) and that up to 80% of absorbed energy is used to create a hot-electron cloud before energy is lost to phonons.48,49 For single-photon detection, these scattering properties mean the graphene electrons stay in a heated state for an extended period of time allowing more opportunity for detecting the temperature rise. Specifically, in the case of a GJJ single-photon detector, the extended elevated temperature increases the chance that the phase particle escapes its potential well to cause a non-zero voltage to develop over the junction (see Chapter 5). The duration of the elevated electron temperature is determined by thermal pathways to the electrical contacts to the graphene (a pathway largely blocked for superconducting contacts), by radiation of photons into the environment (only significant at temperatures below ∼10 mK), and by scattering with phonons. Electron-phonon scattering has a highly-nonlinear temperature dependence resulting in a highly- nonlinear temperature-transient in the graphene electrons (see Equation 5.7 and Figure 5.4 of Section 5.3). Using this temporal temperature profile, the photon-detection efficiency of a GJJ can be estimated.

23 3.3 Optical Properties

To add to the list of graphene’s remarkable properties, a free standing sheet of the material, even though it is merely one atom thick, will absorb 2.3% of the light impinging on it at normal incidence.50 This value can be derived from the Fresnel equations for transmission through a thin film, i.e. a film with thickness d satisfying d ≪ λ where λ is the wavelength of light being transmitted. For such a film on a semi-infinite substrate ofre- fractive index ns, the total transmission through both film and substrate, Tfs, is given in relation to the substrate

51,52 transmission alone, Ts, by:

−2 Tfs Z0σ ≈ 1 + (3.10) Ts 1 + ns p ≈ Here Z0 = μ0/ε0 377Ω is the impedance of free space and σ is the (generally) frequency-dependent sheet conductivity (or optical conductivity) of the film (note for a three-dimensional material, σ would be replaced by

σd). Further corrections can be included by not assuming the substrate is semi-infinite and instead including back side reflections.52 In graphene, it was predicted theoretically in 200253 that the optical conductivity would be a constant as a function of frequency for visible to near-infrared (NIR) light given by:

e2 σ = (3.11) 4~ where e is the electron charge. As was first noted in Ref. [54] and has since been shown to hold true numerous times experimentally,50,55,56 substituting equation 3.11 into equation 3.10, assuming air or vacuum on either side of the graphene, yields a transmission through graphene (Tgr) completely independent of any material properties and only dependent on fundamental constants of nature:

−2 Z e2 − T = 1 + 0 = (1 + πα) 2 ≈ 1 − πα (3.12) gr 4~

24 − ≈ ≈ 53 where α is the fine structure constant. The absorption is then 1 Tgr πα 2.3%. The original derivation of graphene’s constant conductivity made use of the material’s unique band structure but more recently57 it has been suggested that any direct-gap two-dimensional material would exhibit such behavior, regardless of dispersion relation, with experimental evidence being observed in InAs systems.58

Equations 3.11 and 3.12 hold true for photon energies that are much higher than kBTe and EF where interband absorption dominates. As the photon energy decreases, however, two more effects come into play: 1) the intraband component of optical conductivity and 2) Pauli blocking.55,56 The intraband conductivity is usually

σDC σDC taken to have a Drude form, σ (ω) = with real part Re [σ (ω)] = σ1 (ω) = 2 2 and imaginary part 1+iωτsc 1+ω τsc

− σDCωτsc Im [σ (ω)] = σ2 (ω) = 2 2 where ω is the photon frequency, σDC is the DC conductivity, and τsc is the 1+ω τsc scattering time. This simple form has been shown to fit well to transmittance and reflectance measurements of

55 the intraband optical sheet conductivity of graphene using measured values of σDC and τsc. However, a more exact form (that is still Drude-like) can be obtained making use of the Kubo formula:56,59–62

2 E e kBTe EF − F ( ) = − + ln kBTe + σgr,intra ω i ~2 − 2 e 1 (3.13) π (ω i2Γsc) kBTe

~ ≫ where Γsc = 1/τsc is the carrier scattering rate and i is the imaginary unit. It can be seen here that for ω

~ Γ, kBTe this term is negligible. Practically speaking, the intraband contribution to the sheet conductivity starts becoming appreciable in the mid-infrared (MIR) region of the spectrum.

The second effect that starts to contribute to graphene’s optical conductivity at low photon energy isPauli blocking, which takes its name from the Pauli exclusion principle. As graphene is doped further away from the

| | ≥ Dirac point, more carrier states become filled. At zero temperature, if the graphene is doped suchthat EF

~ω/2 then there will be no available carrier states for an interband transition (a transition from −~ω/2 in the valence band to ~ω/2 in the conduction band) to take place. To account for this effect, the DC conductivity of

25 Figure 3.4: The magnitude of the real and imaginary parts of the intraband (blue and red) and interband (green and orange) contributions to graphene’s optical conductivity as a function of wavelength. In the visible and near-infrared portion of the spectrum, interband transitions dominate. Once the wavelength is long enough that the photon energy is less than twice the Fermi level, interband transitions are forbidden by Pauli blocking. Intraband transitions do not become significant until the mid-infrared portion of the spectrum. The charge − carrier density was chosen as n = 1.67 × 1012 cm 2 to match parameters used in Chapter 5. This value of n corresponds to a Fermi level of 150 meV so that interband transitions are blocked for light with wavelength &4 µm.

Equation 3.11 can be modified to be:61

2 | | − ~ − − ie 2 EF (ω i2Γsc) σgr,inter (ω) = ~ ln | | ~ − (3.14) 4π 2 EF + (ω i2Γsc)

~ ≪ Equation 3.14 falls off rapidly for ω/2 EF (the Pauli blocking regime) and returns to the universal con-

~ ≫ ductivity of equation 3.11 for ω/2 EF. Such behavior has been observed experimentally by an increased

transmission (decreased absorption) for photon energies in the MIR when the graphene was gated far from the

Dirac point.55,56,63,64

I plot the magnitude of the real and imaginary parts of Equations 3.13 and 3.14 in Figure 3.4. The charge

× 12 −2 carrier density was chosen as n = 1.67 10 cm to match parameters used in Chapter 5 and τsc was taken

26 − to be 5×10 13.62 As can be seen, for short wavelength (high energy) light, the interband conductivity dominates and is equal to that given in Equation 3.11. At longer wavelengths approaching the MIR, Pauli blocking sets in. The value of n chosen corresponds to a Fermi level of 150 meV so that interband transitions are blocked for light with wavelength &4 µm which is where the transition is observed in the figure. It can also be observed inthe figure that the intraband contribution does not become significant until the MIR region of the spectrum.

3.4 Coupling Photons to Graphene

Although 2.3% absorption is impressive for just a single layer of atoms, this performance would be rather disap- pointing as a single-photon detection efficiency. However, various schemes have been demonstrated to increase graphene’s interaction with light.65 The earliest attempts to increase graphene absorption, for the purpose of making an optical modulator, involved placing graphene atop a silicon waveguide.66–75 In this scheme, evanescent waves extending out of the waveguide mode couple into the graphene. Instead of only interacting with the graphene for a length on the order of the graphene thickness (. 1 nm), photons are absorbed along the length of the structure with most devices having lengths of 10s of microns. Typical maximum absorption per length of these graphene-coated waveguides is on the order of 0.1 dB/µm where 0.1 dB loss is approximately 2.3%. This means only about one micron of graphene coupled to a waveguide is needed to get the same absorption as free- standing graphene that is free-space coupled to light at normal incidence. The highest reported absorption in such a device has been 16 dB (97.5%) for a 300 µm long graphene-waveguide structure.74 It should be noted that these systems suffer from large insertion losses> ( 3 dB≃50%) but the absorption of the graphene can be dis- tilled from the total loss by making use of Pauli blocking. As mentioned, the application being sought after with graphene-coupled waveguides was optical modulators. To modulate the absorption, EF can be tuned through an electrostatic gate to beyond ±~ω/2, where interband transitions are Pauli blocked, allowing light to be transmitted through the waveguide mostly unimpeded by the graphene. Any change in transmission as the gate voltage is

27 changed must be caused by a changing absorption in the graphene because the gate has no effect on the transmission of the bare waveguide itself. The absorption directly caused by the graphene then is the maximum transmission as a function of gate voltage minus the minimum transmission as a function of gate voltage and these are the numbers that are quoted above. A closely related concept for a graphene modulator involves laying the graphene over a microring resonator coupled to a straight waveguide.76 Here, when the graphene is in the Pauli blocking regime the resonator is low loss and well coupled to the waveguide causing transmission through the waveguide to decrease. When the graphene is absorbative, the resonator is lossy and poorly coupled to the waveguide allowing light to be transmitted.

Another method for increasing the coupling strength of light to graphene, and the one which we propose to use in Chapter 5, couples the graphene to a two-dimensional planar photonic crystal77 (PhC) cavity.78–82 PhCs are periodic arrays of materials with different dielectric constants analogous to the periodic potential formedby arrays of atoms in solid state (electronic) crystals. Similar to the band gap in a solid which blocks the propagation of electrons with certain energies, a photonic band gap can be opened up in a photonic crystal that blocks the propagation of certain frequencies of light. By disrupting the periodic structure, propagation can be allowed in certain directions, allowing for light to be guided in the form of a waveguide or confined in the form of a cavity.

Placing graphene over one of these cavities allows for longer interaction times and enhanced absorption with recent experiments showing >50% absorption in two sheets of graphene coupled to a cavity.81 That experiment involved free-space coupling light to the cavity but performance can be enhanced even further by “critically coupling” light from a PhC waveguide to the PhC cavity. (We note that others have proposed enhanced graphene absorption using free space critical coupling to a PhC with no cavity instead of waveguide coupling to a cavity.)83

Critical coupling means that in the waveguide-cavity system there is no scattering (γs = 0 where γs is the scattering rate) and the absorption rate γa is equal to the coupling rate γw between the waveguide and cavity. If these conditions are met, 100% absorption is possible. This method has the advantage of limiting the footprint of the

28 graphene needed for total absorption (on the order of 100s of nanometers instead of 100s of microns in the case of waveguides) but limits the bandwidth of light being absorbed to the resonance of the cavity which could be a disadvantage depending on the application.

Our proposed use of a PhC waveguide-cavity system is depicted in Chapter 5 (Figure 5.2) with further details discussed in the current section. Infrared radiation passes from a ridge waveguide into a PhC nanocavity via a short section of PhC waveguide. The evanescent cavity field couples to the graphene monolayer, positioned over the cavity, while the JJ is located at the other end. Our PhC cavity design uses a thin air slot to concentrate the

EM field into the graphene sheet; this air slot (and the graphene absorber) can be deeply sub-wavelength. Other designs have used nanobeam cavities instead of an air slot but such systems lose the advantage of a planarized surface.84 Using a finite-difference time-domain (FDTD) simulation tool (Lumerical), we calculate the reflected, absorbed, and scattered power for a broadband optical input pulse from the ridge waveguide. For a graphene- cavity quality factor of 800, the calculated power spectrum indicates a peak graphene absorption of 93% for 1550- nm wavelength photons. Since the optical losses in silicon are comparatively negligible, the remaining losses are due to optical scattering. To estimate the scattering, we can make use of temporal coupled mode theory.84,85 This

| |2 | |2 theory relates the optical energy in the cavity ( Acav ), to the optical power input to the system ( Sin ), reflected

| |2 | |2 | |2 ( Sr ), absorbed ( Sa ), and scattered ( Ss ) by the equations:

dA p cav = −iω − γ A + 2γ S (3.15a) dt r t cav w in

p − Sr = Sin 2γwAcav (3.15b)

p Sa = 2γaAcav (3.15c)

29 p Ss = 2γsAcav (3.15d)

where ωr is the resonant frequency of the cavity and γt = γw + γa + γs is the total decay rate. Solving Equation

3.15 for the reflection (Rr), absorption (Ta), and scattering (Ts) gives

| |2 − 2 Sr γa + γs γw Rr = = (3.16a) |S |2 2 − 2 in γt + (ω ωr)

| |2 Sa 4γaγw Ta = = (3.16b) |S |2 2 − 2 in γt + (ω ωr)

| |2 Ss 4γsγw Ts = = (3.16c) |S |2 2 − 2 in γt + (ω ωr)

× 11 −1 × 11 −1 × 10 −1 Fitting the FDTD results to Equation 3.16 yields γa = 3.2 10 s , γw = 4.4 10 s , and γs = 3.0 10 s .

Although the scattering rate is an order of magnitude smaller than both the absorption and coupling rates, it is nonzero which contributes to the peak absorption being less than 100%. This effect could likely be eliminated by further numerical optimization.

In the discussion so far it has been assumed that the photons being detected by graphene are in the visible to infrared regime. However, as will be shown in Chapters 5-7, graphene could also be used to detect photons down into the GHz regime so graphene’s microwave coupling must also be analyzed. At low frequencies, graphene is essentially a lump resistor with impedance depending on its dimensions and gate-tunable charge carrier density.

Quarter or half-wave resonators, stub or LC matching networks,31 taper transformers, and log periodic anten- nas86 can be employed to achieve an input coupling approaching unity. Broadband 50 Ω matching is achievable with highly doped monolayer graphene. Detailed microwave circuitry for coupling low-energy photons to

30 graphene will be discussed further in Chapters 5 and 7 with the experiments of Chapter 7 showing greater than

99% coupling efficiency for 7.9 GHz radiation.

31 Josephson’s theoretical discoveries showed how one can influence supercurrents by applying electric and magnetic fields and thereby control, study and exploit quantum phenomena on a macroscopic scale. His discoveries have led to the development of an entirely new method called quantum interferometry. This method has led to the development of a rich variety of instruments of extraordinary sensitivity and precision with application in wide areas of science and technology. Nobel Award Ceremony Speech, 1973 4 Josephson Junctions

4.1 Introduction

Superconductivity was first discovered in 1911 by Heike Kamerlingh Onnes when he observed a sudden drop

87 to zero in the resistance of mercury below a critical temperature, Tc, of 4.2 K. In the over 100 years since that discovery, much has been learned about superconductors although many mysteries remain. One of the most important advances came in 1957 when Bardeen, Cooper, and Schrieffer (BCS) wrote down the theory describing the microscopic picture of superconductivity.88 A year prior, Cooper had argued that any attractive force between electrons, regardless how weak, causing them to form a bound state (now known as a Cooper pair) would

32 cause the Fermi sea to be unstable and Cooper pairs (which are bosonic) would condense until an equilibrium were reached.89 Such an attractive force is supplied by phonon interactions in which one electron polarizes the lattice which attracts the second electron. Superconductors can then be thought of as materials containing a

Bose-Einstein condensate of Cooper pairs with macroscopic wavefunction, or order parameter,

ψ = |ψ|eiφ (4.1)

where φ is the macroscopic phase of the superconductor and ρ = |ψ|2 is the Cooper pair density. When two superconductors are separated by an insulator or by some other non-superconducting “weak link” the macroscopic phases of the two superconductors can lock to each other and remarkably still pass a zero-resistance current in what is called the Josephson effect, first predicted for the insulating case by Josephson in 196290 and first observed by Anderson and Rowell in 1963.91 The details of the Josephson effect are the focus of the current chapter in which will be discussed the macroscopic picture of Josephson coupling, the microscopic picture of

Josephson coupling, Josephson coupling with a graphene weak link, and absorption in the superconducting contacts of a Josephson junction.

4.2 The Josephson Tunneling Junction

Although graphene falls into the weak link category of Josephson junction, or “proximity Josephson junction”, where a supercurrent is passed through a material that would normally conduct but not necessarily superconduct, the tunneling Josephson junction will first be explored (based on Feynman’s argument instead of Josephson’s

92 ~ ∂Ψ = ˆ original paper). Starting from the time-dependent Schrödinger equation i ∂t HΨ, where Ψ is the system

33 wavefunction and Hˆ is the Hamiltonian, gives for the coupled superconductor system:

            ∂ ψ1 U1 K  ψ1 i~   =     (4.2) ∂t ψ2 KU2 ψ2

th th with ψn being the order parameter of the n superconductor, Un being the energy of the n superconductor, and K being the coupling between the two superconductors. If a voltage V is applied between the two supercon-

− ductors, then U2 U1 = 2eV where the 2 comes from dealing with pairs of electrons instead of single electrons.

− Taking U1 = eV and U2 = eV together with Equation 4.1 gives four coupled differential equations:

∂ρ 2K√ 1 = ρ ρ sin φ − φ (4.3a) ∂t ~ 1 2 2 1

∂ρ 2K√ 2 = − ρ ρ sin φ − φ (4.3b) ∂t ~ 1 2 2 1

r ∂ψ K ρ eV 1 = − 2 cos − − ~ φ2 φ1 ~ (4.3c) ∂t ρ1

r ∂ψ K ρ eV 2 = − 1 cos − + ~ φ2 φ1 ~ (4.3d) ∂t ρ2

From Equations 4.3a and 4.3b we see that the change of Cooper pair density in one superconductor is equal in magnitude but opposite in sign to that in the second superconductor — these equations describe the flow of

Cooper pairs from one superconductor to the other and therefore are proportional to the current through the

34 Josephson junction. Not knowing K, the current can be written as:

IJ = Ic sin φ (4.4)

− where IJ is the supercurrent, Ic is the critical current, and φ = φ2 φ1. This implies that the maximum supercurrent that can run through the junction is the critical current. This result is known as the DC Josephson effect.

Now subtracting 4.3c from 4.3d gives: dφ 2eV = (4.5) dt ~

This illustrates what was meant in the Introduction to this section by the macroscopic phases being “locked” to-

dφ = gether: when the difference in macroscopic phases does not change in time, i.e. when dt 0, there is no voltage drop between the two superconductors and a supercurrent can flow. Another conclusion drawn from Equation

4.5 together with Equation 4.4 is that a constant voltage held across the junction results in an alternating current

2eV with frequency h . This result is known as the AC Josephson effect with the experimental evidence of its exis- tence first being presented by Shapiro in 1963.93 Although the relations 4.4 and 4.5 were derived originally for tunnel junctions, it has been proven that they are more general and apply to other weak links as well.94

4.3 Macroscopic Description: RCSJ Model and Phase Particle Dynamics

The typical way to describe a physical Josephson junction is as a set of three lump circuit elements in what is called the resistively and capacitively shunted junction (RCSJ) model.95,96 As the name suggests, this model treats the Josephson junction as an ideal junction with supercurrent given by Equation 4.4 shunted by a resistor of resistance RN (with current V/RN and where the N stands for normal metal resistance) and by a capacitor

dV of capacitance CJJ (with current CJJ dt ). Making use of Equation 4.5 to replace V, the total current Ib through the

35 Figure 4.1: (A) The washboard potential of a GJJ (Device A in Chapter 8) for two different bias currents. The phase particle can escape over the barrier ΔU by thermal activation (TA, red) or through the barrier by macroscopic quantum tunneling (MQT, blue). (B) The escape rate for TA (red) and MQT (blue) as a function of temperature for the same GJJ in (A) with = 0.9. MQT dominates at lower temperature γJJ while TA dominates at higher temperatures. TA varies significantly with temperature while MQT’s temperature dependence comes implicitly through the temperature dependence of Ic (inset). junction can then be written as:

~ dφ ~C d2φ = sin + + JJ Ib Ic φ 2 (4.6) 2eRN dt 2e dt

= ~ 2 ( ) = − cos + Defining an effective mass mφ CJJ 2e and a potential energy profile U φ EJ0 φ γJJφ with

~ E = Ic being the Josephson coupling energy and γ = Ib being the normalized bias current, Equation 4.6 can J0 2e JJ Ic be cast in the form: d2φ 1 dφ dU + = − mφ 2 mφ (4.7) dt RNCJJ dt dφ

Equation 4.7 is exactly the equation of a particle with mass mφ traveling along the coordinate φ in a potential

U while experiencing a damping force with characteristic time scale RNCJJ. This so-called phase particle picture of a Josephson junction is useful for determining how often the junction will switch into the resistive state, an important quantity for determining single-photon detection performance. The potential U is known as a tilted-

36 washboard potential because of its shape, as displayed in Fig. 4.1A for the parameters of Device A in Chapter 8.

When a phase particle is trapped in one of the wells of the washboard potential, the JJ is in the superconducting state because the phase is constant in time. When the phase particle is not trapped (i.e. when it is in the “running state”) the phase evolves in time so that the JJ is no longer superconducting and develops a voltage across itself.

For the phase particle to escape a potential well, it must overcome a barrier, ΔU, given by:

q − 2 − −1 ΔU = 2EJ0 1 γJJ γJJ cos γJJ (4.8)

To calculate the rate, Γesc, at which the phase particle can escape the well, an Arrhenius equation is employed of the form:97

− ΔU kBTeff Γesc = Ae (4.9)

where kB is Boltzmann’s constant. The prefactor A and effective temperature Teff will have different forms depending on the mechanism by which the phase particle overcomes the barrier ΔU. Either thermal fluctuations can cause the particle to be excited over the barrier, in what is called thermal activation (TA), or quantum fluctuations can cause the particle to tunnel through the barrier, in what is called macroscopic quantum tunneling

(MQT). Practically this means the escape process is probabilistic (based on these thermal and quantum fluctuta- tions). There is always some chance that the junction switches into the nonzero-voltage state even for Ib far below

98,99 Ic. For the two escape mechanisms, A and Teff in Equation 4.9 are given by:

   q  ωp + 1 − 1 2π 1 4Q2 2Q (TA) A = (4.10)  q  12ω 3ΔU (MQT) p 2π~ωp

37    T (TA) = Teff  h i (4.11)  ~ / . + 0.87 ωp 7 2kB 1 Q (MQT)

Here, Q is the quality factor of the junction given by Q = ωpRNCJJ, ωp is the plasma frequency of the JJ given

− 2 1/4 ~ 1/2 by ωp = ωp0(1 γJJ) , and ωp0 is the zero-bias plasma frequency of the JJ given by ωp0 = (2eIc/( CJJ)) .

Although both mechanisms will always take place, typically one will dominate over the other depending on the characteristics of the JJ and its temperature, with MQT dominating at low temperature and TA at high temperature. The crossover temperature between MQT and TA is given by:

∗ ~ TMQT = ATA (4.12) kB

where ATA is the prefactor for TA given in Equation 4.3. Often the Q dependence is ignored as a small correction

∗ ∼ ~ and the expression is simplified to TMQT ωp/2πkB which shows the intuitive result that the crossover occurs when the thermal energy surpasses (half) the quantum energy. I show the calculated MQT and TA escape rates as a function of temperature in Fig. 4.1B for the parameters of Device A in Chapter 8. As can be seen, ΓTA increases rapidly with temperature while ΓMQT remains relatively constant, with a slight increase at higher temperature due to Ic decreasing (inset). For the SPD mechanisms of Chapters 5 and 8, the GJJ tends to start with an MQT- dominated escape process in the dark and transition to a TA-dominated escape process upon photon absorption and subsequent temperature rise.

Once a phase particle escapes, it can either immediately retrap into the next potential well if the JJ is overdamped, returning the JJ to the superconducting state almost immediately (on a timescale commensurate with the inverse plasma frequency), or continue running along the potential if the JJ is underdamped, keeping the JJ in the nonzero-voltage state. Most of the JJs studied in this work fall into the underdamped case so that will remain

38 the focus of the current section. For an underdamped JJ, because the phase particle continues rolling after escaping, when sweeping the bias current Ib through the JJ, the resulting IV curve (where the V is the JJ voltage over the junction, VJJ) is hysteretic. When sweeping from zero current to higher current, the junction will switch to a non-zero voltage state and stay in that state. Because the escape process is probabilistic, the current at which the junction switches, Is, will be below Ic, its value depending on the Ib sweep rate. When sweeping back down from high current to zero current while in the nonzero-voltage state, the voltage will not drop back to zero at Is but instead at a lower retrapping current Ir. In the underdamped case, the phase particle’s momentum keeps it from retrapping when the potential barrier is at the same level it was when the particle escaped. Instead, the potential must be made even deeper by further lowering Ib to Ir before the particle is trapped again (in the overdamped case, the particle loses enough energy to dissipation that it retraps at the same potential barrier and therefore Is).

Even in an underdamped junction, however, when the temperature is high enough the phase particle can expe-

100,101 rience enough dissipation, likely due to temperature-dependent impedance mismatch, to retrap at Is. The

∗ temperature, TPD, at which this so-called underdamped phase diffusion (UPD) regime begins to dominate the

TA process is given by:102 3/2 ∗ ∼ EJ0 − 4 TPD 1 (4.13) 30kB πQ

∗ ∼ For high Q, this reduces to TPD EJ0/30 so that this crossover temperature depends only on Ic. It should be

∗ noted that TMQT is linear with ωp and therefore also proportional to Ic. In a GJJ, the critical current, and there-

∗ ∗ 103 fore TMQT and TPD, can be tuned with a gate voltage as was demonstrated experimentally by Lee and observed in the gate-dependent data of Chapter 8. In the UPD regime, the rate at which the phase particle retraps is given by:104,105 r | − | − ΔUr Ib Ir EJ0 k T Γr = ωp0 ] e B (4.14) Ic 2πkBT

39 with: 2 − 2 EJ0Q Ib Ir ΔUr = (4.15) 2 Ic

Accounting for this additional retrapping rate proved to be essential in explaining the data presented in Chapter

Converting from phase-particle escape rate to the probability of the particle escaping during a single-photon event is discussed extensively in Chapters 5 and 8. Briefly, the escape rate will be a function of time based onthe effective temperature response of the JJ to a single photon. This temperature response will either be determined by heating of the graphene electron gas (Chapters 5 and 7) or by non-equilibrium photon generation in the superconducting contacts (Chapter 8). In either case, the escape rate becomes time dependent, i.e. Γesc(t). Using a typical decay equation, the probability of the phase particle escaping is given by:

∫ − t1 P − 0 dtΓesc(t) esc = 1 e , (4.16)

i.e. unity probability minus the probability of the phase particle still occupying its potential well after some time t1. This equation drives the modeling of Chapters 5 and 8.

4.4 Microscopic Description: Andreev States

So far we have treated JJs in a macroscopic sense: the phase particle is a macroscopic quantum phenomenon used to reveal system characteristics of a JJ. Underlying this macroscopic picture is a microscopic description encom- passing the energy states for individual Cooper pairs and quasiparticles. A supercurrent is a flow of Cooper pairs.

However, no true Cooper pair states exist in the weak link of a proximity junction. Instead, the flow of Cooper pairs from one superconducting contact to the other through the weak link is mediated by what are called “An- dreev reflections.”106 We start by considering an electron quasiparticle in the weak link, although similar rea-

40 soning could be applied to a hole quasiparticle as well. The Fermi level of the weak link will be centered in the superconducting gap of the superconducting contacts. Electrons can live at the Fermi level in the weak link but are precluded from entering the Fermi level in the superconductors due to the gap. Instead, when a weak-link electron encounters the superconducting contact, it can only enter by pairing with another electron to form a

Cooper pair. This second electron leaves behind a hole with opposite momentum, i.e. a hole is “retroreflected.”

Assuming ballistic transport for now, this hole will travel across the weak link back to the other superconducting contact where it will pair with another hole to form a hole Cooper pair, or, to make the thought of Cooper pair flow more intuitive, to cancel an electron Cooper pair traveling into the weak link. The second hole leavesbehind a retroreflected electron that transports back to the original superconducting to start the process again. Inthis way, a Cooper pair can be dissociated at the superconducting contact (by the hole pair), travel as a correlated electron pair through the weak link, and reassociate as a Cooper pair at the other superconducting contact (by the electron pair). The electron and hole reflections at the superconducting contacts are the Andreev reflections and the bound energy states formed by these reflections are called Andreev bound states (ABS).107 In materials where diffusive, instead of ballistic, transport dominates, instead of discrete ABS, a continuum of supercurrent-carrying states is the appropriate description.108–112

To understand how the ABS or supercurrent-carrying states play into single-photon detection, note that these states account for supercurrents traveling in both directions through the junction. Negative energy states (compared to the Fermi level) carry supercurrent in the direction of the bias current while positive energy states carry

± 107,113 supercurrent in the opposite direction. For ABS, there are two discrete sets of energy states, En , given by:

± ~v 1 E (φ) = F 2π n + ∓ φ , n ≪ Δ L/~v (4.17) n 2L 2 0 F

where vF is the Fermi velocity, L is the junction length (approximately the length between the superconducting

41 contacts), Δ0 is the superconducting gap, and φ is the same phase difference between the superconducting contacts used in Equations 4.4 and 4.5. Using Equation 4.4, when the supercurrent is zero, φ is zero and these two energy levels are degenerate. Both levels have equal probability of being filled so supercurrents flow in both directions canceling one another out. As the bias current is increased, φ increases thus separating the levels and allowing for supercurrent to flow. However, while holding the bias current constant, if the upper statesagain become filled, either thermally or through some other means, the supercurrents in the negative states willagain be canceled out, effectively lowering the critical current of the junction. Similar arguments hold for a continuum of supercurrent-carrying states. It has been predicted and demonstrated experimentally that by controlling the occupation of these states via voltages perpendicular to the supercurrent direction, one can control the super-

108–114 current flowing through the junction, lowering the Ic or even reversing the supercurrent direction. For single-photon detection, an incoming photon either excites quasiparticles directly in the weak link or in the superconducting contact that can then diffuse into the weak link. These high-energy quasiparticles fill the positive supercurrent states thus effectively lowering the critical current of the junction. The drop in critical current can cause the junction to switch depending on the number of quasiparticles produced, the number of states available, and the bias current.

4.5 Graphene-Based Josephson Junctions

Shortly after graphene’s experimental realization, the material was considered for the weak link in a Josephson junction115 and soon the first graphene-based Josephson junction (GJJ) was demonstrated.116 Using graphene in a JJ provides the advantage of being able to probe multiple sets of parameters in a single device through the tuning of graphene’s charge carrier density with a back gate (see Section 3.1). In the first GJJ it was already shown that Ic could be tuned by tuning VG with later results showing that the phase particle dynamics (MQT, TA, or PD),103 the type of Andreev reflection (retro or specular),117 and the distribution of the Andreev bound

42 Figure 4.2: Gate-dependent parameters of a GJJ. (A) The measured critical current of Device A of Chapter 8. The critical current of the

GJJ is tunable through its electrostatic back gate. (B) The Josephson coupling energy, EJ0 (green, right axis), is directly proportional to Ic while the characteristic voltage, eIcRN (purple, left axis) is nearly constant as a function of gate. eIcRN typically has weak dependence on the weak link in a proximity JJ which is why it changes little and coupled with graphene’s tunable RN, explains the tunability of Ic. The Thouless energy, ETh (dark red, left axis) is also tunable based on its dependence on the charge carrier density in graphene. (C) The gate dependence of the phase particle escape rate for MQT (blue) and TA (red) with = 0.9. MQT, which dominates the dark count rate γJJ in single-photon experiments, decreases much more rapidly than TA, which dominates the single-photon response of the GJJ, as VG increases which makes higher gate voltages more favorable for single-photon detection. The TA curve is plotted for a temperature of 2 K and is scaled by a photon rate of 53 Hz and an “on time” of 1 ns per photon to better represent the actual observed switching rate of the GJJ under illumination (see Chapter 8). states118 can be tuned as well. For a thorough review of GJJs see Ref. [119].

Here I present typical parameters of the GJJs used in the experiments of Chapter 8. These GJJs use niobium nitride (NbN) as the superconducting contacts in a lateral configuration with current flowing in-plane with the graphene lattice (as opposed to a one-atom-thick vertical configuration with current flowing transverse tothe graphene plane).120 The lateral configuration is particularly useful for photon-detection experiments because it allows direct access to the weak link and junction contacts so photons, whether RF or optical, can be coupled directly to the junction. This characteristic is in contrast to typical stacked JJs where the junction itself is sand- wiched between layers of superconducting material rendering it not easily accessible. The junctions of Chapter 8 had length (distance between NbN contacts) of ∼160 nm and width 1.5 or 2.8 µm.

When discussing proximity JJs in general (regardless if it’s a GJJ or not), three energy scales are typically considered: 1) the Josephson coupling energy, EJ0, defined in the previous section, 2) the Thouless energy, ETh, and

3) the characteristic voltage IcRN (which can be converted to an energy by multiplying with a factor of e). EJ0

43 is proportional to Ic and can be thought of as a measure of the strength of the Josephson coupling through the

~ weak link. ETh = /τt is the characteristic energy associated with quasiparticles traveling through the weak link, where τt is the characteristic transport time, and therefore takes on different values if the junction is in a ballistic regime (i.e. junction length less than the quasiparticle mean free path) or a diffusive regime (i.e junction length

~ greater than quasiparticle mean free path). For the ballistic regime it is given by ETh = vF/L and for the dif-

~ 2 fusive regime by ETh = D/L where L is the junction length, vF is the Fermi velocity, and D is the diffusion constant (given by D = vFlMFP/2 for two-dimensional graphene with lMFP being the mean free path). Finally,

IcRN measures the junction coupling while removing most of the dependence on the weak link itself. Specifically, for tunnel junctions and weak links in the short diffusive regime (short meaning L is less than the superconducting coherence length), IcRN depends only on the superconducting gap, Δ0, of the superconducting contacts and the temperature, with no dependence on the material between the contacts.94 More general expressions have been derived for other regimes of weak links94 and specifically for GJJs showing a weak dependence on the Fermi level of the graphene weak link.115

I plot EJ0, ETh, and eIcRN as a function of gate voltage in Figure 4.2B for Device A of Chapter 8. I use mea-

⟨ ⟩ sured values of Is to estimate Ic combined with measured values of RN to find EJ0 and eIcRN directly. For ETh,

I fit the measured value of RN versus VG using the method in Section 3.1 to find the mobility μ, use the mobil- √ ~ ity to find the mean free path, lMFP = μ πn/e, and the mean free path to find the diffusion constant and

ETh. As seen in the figure, the level of IcRN remains fairly constant as expected based on the discussion of the previous paragraph. This fact coupled with graphene’s RN being strongly dependent on gate explains the strong

∝ dependence of EJ0 ( Ic) on gate. This dependence is one reason that the GJJ is an intriguing device for studying the capabilities of Josephson junctions. Having the ability to tune the critical current enables the study of different operating regimes in a single device. For the single-photon detection of Chapter 8, for instance, wewere specifically interested in the ratio of ΓMQT, which dominated the dark count rate, to ΓTA, which dominated the

44 photon-induced switching. Using Equations 4.9-4.11, I plot ΓMQT and ΓTA versus VG for γJJ=0.9 in Figure 4.2C.

Note that ΓTA uses T=2 K and is then scaled by 53 photons per second and 1 ns of “on” time per photon to better represent the observable switching rate of the GJJ under illumination (details in Chapter 8). Both ΓMQT and

ΓTA decrease as VG increases. However, ΓMQT has a stronger dependence on Ic than does ΓTA so as Ic decreases with increasing VG, ΓMQT decreases much more rapidly than does ΓTA. With the dark count rate depending on

ΓMQT, it is favorable to operate at higher gate voltage to drastically reduce the dark count rate while only slightly decreasing the single-photon response. Such studies are only possible because of the tunability of the GJJ.

4.6 Absorption in Superconducting Contacts

So far it has been assumed that the incoming light to our single-photon detector would be directly coupled to the graphene weak link in the Josephson junction (see Section 3.4). However, in the experiments of Chapter

8 the entirety of the GJJ is illuminated. In this configuration, we found most photon absorption occurs inthe

NbN superconducting contacts rather than in the graphene. The optical properties of NbN have been well studied in the past 17 years121–123 because this material is one of those preferred for fabricating superconducting- nanowire single-photon detectors (SNSPDs)20,21 with some pre-SNSPD studies of NbN’s absorption existing as well.124–127

NbxNy can appear in various crystal structures and with various stoichiometric ratios of niobium to nitrogen

(for a review see Ref. [128]). For most superconducting applications face-centered cubic (fcc) δ-NbN crystals are used and will be what NbN is referring to in the current section. In general a metal’s permittivity for radiation of a given angular frequency ω can be expressed in the form:

2 X 2 ωp,sc ωs,k = ∞ − + εr ε 2 2 2 (4.18) ω + i2πω/τsc ω − ω − iωω , k 0,k d k

45 Figure 4.3: Optical properties of NbN (A) The real (blue) and imaginary (red) parts of NbN’s permittivity given by Equation 4.18. (B) The absorption of NbN thin films as a function of thickness for 500 nm (blue), 1550 nm (red), and 5000 nm (yellow) incident light.(C)The absorption of a 50-nm thick film of NbN, comparable to the thickness used in Chapter 8, as a function of wavelength. Althoughtheab- sorption appears to level off at ∼15%, the structures in Chapter 8 have dimensions shorter than or comparable to the wavelength so plasmonic modes enhance the absorption as revealed by numerical models. where the second term is the Drude term for intraband transitions, the third term is the Lorentz term for interband transitions, and the first term is an offset to account for higher energy transitions not included inthesum of Lorentz oscillators. Here ωp,sc is the plasma frequency of the NbN (not to be confused with the JJ plasma frequency), τsc is the elastic electron-scattering time, and ωs,k, ω0,k, and ωd,k are the oscillator strength, natural fre-

th quency, and damping rate of the k transition, respectively. Various groups have measured εr for thin NbN films through ellipsometry and have found fits that agree well with a Drude model.121,123,125 Improvements to the fits can be made by adding the Lorentz term with one or two transitions included in the sum for higher energy (lower

123 × 15 123 wavelength) radiation. Using ωp,sc = 13.4 10 rad/s and τsc = 3.3 fs, I plot the real and imaginary parts of Equation 4.18 in Figure 4.3A for wavelengths up to 5 µm.

From the permittivity, an impedance-matching model (similar to the Fresnel equations) can be used to calculate the expected absorption of a thin metal film where thin means the thickness of thefilm d satisfies d ≪ λ.

The optical impedance Zi of material i can be defined as

Z0 Zi = (4.19) ni

46 ≈ where Z0 377Ω is the vacuum impedance and ni is the index of refraction of the material. For light incident from material 1 on a thin film material 2 laying on a substrate material 3, the reflection Rfilm and transmission

122,126 Tfilm are given by:

− 2 Zload Z1 Rfilm = (4.20) Zload + Z1

2 Z1 2Zload Tfilm = (4.21) Z3 Zload + Z1

where Zload is the combined load impedance of the thin film on the substrate and can be approximated as:

≈ RZ3 Zload (4.22) R + Z3 with:

Z0 R ≈ Imε (4.23) kd 2 being the square resistance of the thin film and k being the radiation wave vector (in free space). The absorption

− − of the thin film is just the fraction of light not transmitted or reflected, Afilm = 1 Tfilm Rfilm, which, from equations 4.19 to 4.23, is: kdImε = 2 Afilm 4n1 2 (4.24) (n1 + n3 + kdImε2)

In the experiments of Chapter 8, light is incident on the NbN from vacuum (n1 = 1) and the NbN is on a SiO2 substrate (n3 = 3.9). Thin adhesion layers of Ti and Nb are neglected. Equation 4.24, making use of the NbN permittivity of equation 4.18, as a function of film thickness for 500 nm, 1550 nm, and 5000 nm light is plotted in Figure 4.3B. For the experiments of Chapter 8, 1550 nm was used which has a maximum in absorption at a thickness of ∼10 nm while the films used in Chapter 8 are 50-nm thick. Figure 4.3C plots the absorption for

47 50-nm thickness as a function of wavelength. The absorption levels off at ∼15%. However, in the experiments of Chapter 8, the NbN has dimensions smaller than or comparable to the wavelength of light so that plasmonic modes begin to play a role. These modes were accounted for with an HFSS model (others have used coupled- wave analysis122 or finite-element analysis129) and shown to enhance the absorption far above 15%. Other techniques involving different geometries,130 photonic crystal cavities,131 and distributed Bragg reflectors132 have shown to increase NbN’s absorption as well (in SNSPDs) and such techniques could translate to the GJJ SPD.

48 5 Modeling of a GJJ SPD

Here, the physics of graphene and Josephson junctions discussed in the previous chapters come together to form the basis for a single-photon detector. The focus of this detector is exploiting the extreme thermal properties of graphene to break superconductivity in the Josephson junction. The experiments of Chapter 7 were based off of the theory of the current chapter and demonstrated noise equivalent power that corresponds to energy resolution on the scale of a single microwave photon. The experiments of Chapter 8 demonstrating single-photon detection in the near-infrared regime ended up being dominated by a different mechanism because the photons were not directly coupled into the graphene weak link in those setups. However, with a suitable coupling mechanism, such as the photonic crystal cavity described in Section 5.2, the detector theory of the current chapter could be

49 extended to the infrared as well.

The remainder of this chapter is reproduced from E. D. Walsh, et al., Graphene-Based Josephson-Junction

5.1 Introduction

Detecting single light quanta enables technologies across a wide electromagnetic (EM) spectrum. In the infrared regime, single photon detectors (SPD) are essential components for deep space optical communication133 and quantum key distribution via fiber networks.134 In frequencies on the order of terahertz, single photon detectors will allow the study of galaxy formation through the cosmic infrared background with an estimated photon flux <100 Hz.6 Microwave SPDs and photon number resolving counters are required in a number of proposed quantum technologies, including remote entanglement of superconducting qubits,135 high-fidelity quantum measurements,136 and microwave quantum illumination.137 However, detecting low frequency photons is challenging because of their vanishingly small energy. A prominently used detection scheme is to exploit the heating effect from single photons. For instance, transition edge sensors and superconducting nanowire single photon detectors can register infrared photons as they break Cooper pairs in the superconductors.3 High sensitivity calorimeters can detect single photons by reading out the temperature rise induced by the absorbed photon but require better heat absorbers to reach single photon sensitivity at lower frequencies.138

Graphene is a promising material for single photon calorimetry.31,139,140 With its pseudo-relativistic band structure, graphene can efficiently absorb photons from a wide EM spectrum, making it attractive for expanding the availability of SPDs to applications in a broader frequency range. Compared to metals, the electron-phonon coupling and electronic specific heat capacity of monolayer graphene are extremely small31,141 due to the shrinking density of states near its charge neutrality point (CNP). Therefore, a single photon absorbed by graphene can heat up the electrons significantly. This approach relies on thermal physics in this extraordinary material in

50 Photon

Te Superconductor Superconductor

Graphene

Figure 5.1: Device concept to detect single photon using a graphene-based Josephson junction. contrast to atomic-like systems such as quantum dots and superconducting circuits142,143 which require more complicated operation protocols such as microwave pumping before detecting photons.

Sensing the heat pulse generated from a single photon can be challenging experimentally. Although noise thermometry may have the bandwidth and sensitivity to read out the temperature rise, this rise of electron temperature also degrades its temperature resolution which can translate to poor dark count characteristics.139 Here we propose using the graphene-based superconducting-normal-superconducting (SNS) Josephson junction (JJ) as a threshold sensor to detect single photons across an extremely wide spectrum. Since the first observation of the superconducting proximity effect in graphene,116 many advances have been made in the fabrication and performance of graphene-based JJs (GJJs).103,144–149 To emphasize feasibility with existing materials and fabrication technologies, we used measured parameters from a GJJ to calculate the performance of our proposed SPD. Our modeling suggests that a low dark count probability with an intrinsic quantum efficiency approaching unity is achievable.

5.2 Device Concept and Input Coupling

Our proposed device is a hybrid of the calorimeter and the SNS JJ. SNS JJs have been recognized for their use as superconducting transistors109 and as sensitive bolometers.150–152 The concept is to achieve control of the supercurrent by perturbing the Fermi distribution of the normal constituent in the junction through joule heat-

51 ing.108 Compared to metals and semiconductors, graphene is a more favorable weak link material with its high electronic mobility, sensitive thermal response, and field-tunable chemical potential. When an absorbed photon raises the electron temperature in the graphene sheet, the calorimetric effect can trigger the JJ to switch from the zero voltage to resistive state (see Fig. 5.1). We can describe this heating with a quasi-equilibrium temperature, Te, of the graphene electrons as they thermalize quickly through electron-electron interactions.48,49 A GJJ SPD can be achieved by efficient photon absorption (discussed in the present section), appreciable temperature elevation

(Section 5.3), and sensitive transition of the GJJ (Section 5.4).

A practical challenge concerns the efficient absorption of EM radiation by the graphene sheet. Using Boltz- mann transport, the high frequency conductivity of monolayer graphene can be described in a Drude form.153

At low frequencies, graphene is essentially a lump resistor with impedance depending on its dimensions and gate-tunable charge carrier density. Quarter or half-wave resonators, stub or LC matching networks,31 taper transformers, and log periodic antennas86 can be employed to achieve an input coupling approaching unity.

Broadband 50 Ω matching is achievable with highly doped monolayer graphene. At optical frequencies, normal incidence light absorption is given by πα ≃ 2% where α is the fine structure constant, due to the universal a.c. conductivity.154 However, EM waves can be absorbed efficiently by evanescent wave coupling, with light grazing across the graphene sheet, using waveguides and photonic crystal (PhC) structures.78 Fig. 5.2a and b illustrate the proposed GJJ SPDs using impedance-matched resonators at microwave and infrared frequencies, respectively.

To couple microwave photons and apply a dc current bias simultaneously, the GJJ is embedded in a four- terminal geometry as shown in Fig. 5.2a. Supercurrent flows from the narrowly gapped (vertical direction in the figure) superconducting contacts through the monolayer graphene (orange) by the proximity effect.The inductive chokes following these JJ contacts isolate the microwave coupling and permit fast JJ switching. For microwave operation, one quarter-wave microwave resonator is in contact on each side of the graphene sheet along the direction of wider separation between the superconducting terminals (horizontal direction in the figure). To-

52 (a) dc (c) Josephson −100 −80 −60 −45 −30 −15 −5 0 junction quarter-wave input resonator

quarter-wave resonator resonator

(b) photonic Josephson cavity junction

waveguide

(d) 1001 (e) (f) R r 0.880 Ts Ts z y Ta 0.660 Ta d1 d2 0.440 superconducting 20 photonic electrodes 0.2 y x

graphene Transmission (%) 0

crystal T ransmission 0 1552 1554 1556 1558 1560 1530 1532 Wavelength1534 1536 1538 (nm) 1540 0.0 0.2 0.4 0.6 0.8 1.0

Figure 5.2: Device schematic for (a) microwave and (b) infrared single photon detection. (a) The graphene flake is located at the current antinode of a halfwave microwave resonator for maximizing input efficiency. Two stages of inductors and capacitors form a high impedance network at microwave frequency for the dc measurement of the GJJ. (b) A graphene sheet lies on top of a PhC cavity to increase its absorption through critical coupling. Light is coupled into the cavity through an in-chip waveguide. (c) Simulation results for critical coupling. Top view of the PhC structure overlaid with the mode profile, |E|2, using a dB scale normalized to a maximum of 1. (d) Spectra showing the wavelength dependence of the reflection, Rr, absorption, Ta, and scattered power, Ts. (e) Cross-sectional and (f) planar view of close-ups of the cavity mode, |E|2, using a linear scale. The parameters of the structure are: Membrane thickness, h = 250 nm, lattice period, a = 0.27λ, hole√ radius, r = 0.31a, cavity slot width, ws = 0.032λ, cavity hole shifts, d1 = 0.365a, and d2 = 0.153a, − waveguide width, ww = 2(W 3a/2 r) where W = 1.04. The holes at the termination of the PhC waveguide are shifted along the x-axis by s1 = 0.44a, s2 = 0.27a, and s3 = 0.1a.

gether, they form a half-wave resonator with the dissipative graphene sheet at the microwave current antinode.

High microwave absorption can be achieved by impedance matching the half-wave resonator while the temporal

mode of the single photon determines the optimal quality factor for single photon detection.155

For infrared photodetection, a dielectric photonic crystal cavity can provide the impedance-matching element

to reach near-unity light absorption by the graphene sheet, as illustrated in Fig. 5.2b. Infrared radiation passes

from a ridge waveguide into a PhC nanocavity, via a short section of PhC waveguide. The evanescent cavity field

couples to the graphene monolayer, positioned over the cavity, while the JJ is located at the other end. Graphene

can be critically coupled to the cavity78 so that all incident light is absorbed. Our PhC cavity design uses a thin air

slot to concentrate the EM field into the graphene sheet; this air-slot (and the graphene absorber) can bedeeply

sub-wavelength. Fig. 5.2c shows the EM field concentration into the PhC cavity air-slot. Using a finite-difference

53 time-domain simulation tool (Lumerical), we calculate the reflected, absorbed, and scattered power for a broadband optical input pulse from the ridge waveguide (plotted in Fig. 5.2d). For a graphene-cavity quality factor of 800, the calculated power spectrum indicates a peak graphene absorption of 93% for 1550 nm wavelength photons. Since the optical losses in silicon are comparatively negligible, the remaining losses are due to optical scattering, which can likely be eliminated by further numerical optimization.

5.3 Graphene Thermal Response

Upon absorbing a single photon, the thermal response of the graphene electrons can be characterized by the thermal time constant τth, heat capacity Ce, and thermal conductance Gth to the reservoir. Due to the fast electron- electron interaction time, the photon energy can quickly thermalize among the graphene electrons and establish a quasi-equilibrium in typically tens of femtoseconds.48,49 Therefore, both the heat injection from the photon and the initial temperature rise can be considered instantaneous when compared to the thermal time constant of the graphene electrons.

This initial temperature rise is determined using the electronic heat capacity of the monolayer graphene. In the degeneracy regime,156

Ce = AγT (5.1)

5/2 2 1/2 where A is the area of the graphene sheet and γ = (4π kBn )/(3hvF) is the Sommerfeld coefficient with kB and h being Boltzmann’s and Planck’s constants, respectively. This is in contrast to non-degenerate Dirac

∝ 2 47 fermions where Ce T . In graphene, the electron Fermi energy EF has a Dirac-like dispersion relation, i.e. √ ~ ~ 6 −1 EF = vFkF, where = h/2π, vF = 10 ms is the graphene Fermi velocity, and kF = πn is the Fermi momentum with n being the charge carrier density. Thus for the typical charge carrier density ranging from

54 3 1 4 (a) 10 (b)10 (c) 10 n 12 -2 operating 0 0=1.7x10 cm 3 10 10 regime

) T0=25mK

2 2 0 10 (K) 10 2 ∆ m 0

T =3K 10 T/T µ 0 / B

- T - -1 1 10 (k E/h 10 -2 ∆

E/h (GHz) E/h 10 e 1 -1 ∆

peak 10 cm T/ 10 10 ∆ C T T 11 -2 0 T/h 10 cm 10 k B 12 -2 1.7x10 cm -2 0 -2 -1 10 10 10 10 -2 -1 0 1 0 1 2 3 4 5 -2 -1 0 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 T0 (K) f p (GHz) T0 (K)

Figure 5.3: (a) The specific heat of a 1 µm2 graphene sheet as a function of base temperature. (b) The initial temperature rise vs. fre- × 12 −2 11 −2 10 −2 quency with electron density n0 = 1.7 10 cm (solid, used for modeling), 10 cm (dashed), or 10 cm (dotted) at a base temperature of 25 mK (blue) or 3 K (purple). For an infrared detector T0 = 3 K will suffice but for microwave detection a lower T0 is re- 2 quired to acquire a noticeable temperature change. (c) Energy resolution and temperature fluctuation of a 1 µm graphene sheet at n0, representing the intrinsic noise of the calorimeter.

10 12 −2 10 to 10 cm , EF is higher than 10 meV so that the Fermi temperature is about 135 K, justifying the use of

Eqn. (5.1). Ce can be tuned using a gate voltage reaching a minimum at the charge neutrality point, where it is

157 × 12 −2 limited by the residual puddle density. We plot Ce in Fig. 5.3a at a carrier density n0 = 1.7 10 cm that we will use in the modeling. Compared to that of a metallic nanowire used in photon detection at the same temperature,6 the electronic heat capacity of a graphene sheet of 1 µm2 area would be more than three orders of

| | ~ 2 magnitude smaller. This dramatic improvement is due to the shrinking density of states, D(E) = 2 E A/π( vF) with E being the energy measured from the CNP, in monolayer graphene.

We can estimate the initial temperature Tpeak of the hot electrons by equating the integrated internal energy R Tpeak ( ) 139 T0 Ce T dT to the photon energy such that:

q 2 Tpeak = (2hfp)/(γA) + T0 (5.2)

where T0 is the base temperature and fp is the photon frequency. Fig. 5.3b plots the temperature rise for various photon frequencies and charge carrier densities at 0.025 and 3 K. The temperature rise is higher for a lower charge carrier density or with a higher energy photon. Here we assume a full conversion of the photon energy to

55 internal energy in the graphene electrons. This assumption is justified by pump-probe experiments from which it was inferred that up to 80% of absorbed photon energy is cascaded down to heat electrons.48 This efficient energy conversion is due to the domination of the electron-electron scattering process over the coupling to the optical phonons. For lower energy photons at microwave frequencies, the heat leakage to optical phonons is negligible as the energy scale is further below the optical phonon energy.

The heat capacity also determines the root mean square fluctuations in energy of the graphene sheet, ΔE,

158 shown in Fig. 5.3c for n = n0. This intrinsic noise of the calorimeter is given by:

p 2 ΔE = CekBT (5.3)

and describes the thermodynamic fluctuations of the electrons in graphene as a canonical ensemble in thermal equilibrium with a reservoir. ΔE sets the SPD energy resolution for a measurement time much longer than τth. q 2 2 2 The fluctuation power spectral density at spectral frequency f rolls off as 4τth/(1 + 4π τthf ) since τth determines the time scale of the energy exchange between the ensemble and reservoir. In principle, widening the measurement bandwidth B can allow detection of the sharp temperature increase due to a single photon thus circumventing the limitations of calorimetry imposed by these intrinsic fluctuations.22 However, it can also expose the JJ to high frequency noise and increase the thermal conductance of the radiation channel. In this report, we shall focus on the small bandwidth regime in which the SPD requires photon energy larger than ΔE. The color gradient orange region in Fig. 5.3c highlights the requirement of operating temperature for a given photon frequency to avoid both the energy fluctuation and thermal noise. Related to the energy fluctuations are temperature fluctuations with root mean square ΔT given by:

p 2 ΔT = ΔE/Ce = kBT /Ce. (5.4)

56 (a) (b) graphene 5 Grad 10 ))

amp e elec. 2 3 4 e-ph

G m diff 10 T µ Gep 1 10 ph (fW/(K h -p 3 e th -1 T

G 10 Rad -3 base 10 -2 -1 0 1 10 10 10 10 T0 (K) -4 (c) 10 (d) 250 T4 e-ph 4 T e-ph + Rad 200 4 -6 T e-ph + Rad 10 T3 e-ph + Rad 150 (s)

th 3 τ

-8 T e-ph (mK) T 100 10 T3 e-ph + Rad 50 -10 10 -2 -1 0 1 0 10 10 10 10 0 1 2 3 4 5 T0 (K) t (µs)

Figure 5.4: (a) Thermal diagram depicting three heat transfer pathways from graphene electrons (e.) to the thermal reservoir (base) via graphene electron-phonon coupling (ph.), to the electrical contacts (elec.) via diffusion, or to the electrical environment, such as an amplifier (amp.), via photon emission (rad.). The thermal conductances in the linear response regime aredenotedas Gep, Gdiff, and Grad, correspondingly. (b) The thermal conductance vs. base temperature for clean graphene with a T4 e-ph coupling law (blue), for disordered 3 graphene with a T e-ph coupling law (red), and for the radiation channel (purple). (c) The thermal time constant τth = Ce/Gth in the linear response regime for clean (green (blue) including (excluding) radiation) and disordered (orange (red) including (excluding) radiation) graphene. (d) The transient thermal response of the graphene sheet upon absorbing a 26 GHz microwave photon for clean (green) and disordered (orange) graphene including radiation.

Curiously, when Ce decreases to kB, ΔT/T approaches unity (Fig. 5.3c). Possible modification of Boltzmann-

Gibb statistics to describe the fluctuations when the degrees of freedom in the system are close to one isbeyond

the scope of this report.159 However, temperature fluctuations can affect JJ transitions and will be included in

the performance calculation in Section 5.5.

Fig. 5.4a depicts the thermal pathways of a graphene sheet.31,160 At low temperatures, the absorbed photon

energy in the graphene electrons can dissipate through three major channels: electronic heat diffusion, photon

emission, and electron-phonon coupling. The electron heat diffusion is the heat transfer channel out ofthe

graphene sheet to the electrodes. However, at the superconductor-graphene contact, Andreev reflection can sup-

57 press the thermal diffusion and quench this thermal conductance channel.160,161

Photon emission from the graphene sheet to its EM environment can also be an effective cooling channel at

162,163 low temperatures. For a small measurement bandwidth B, such that B < kBT/h this radiation thermal conductance Grad is given by

≃ Grad r0kBB (5.5)

where r0 is the impedance matching factor. We can reduce this heat transfer channel by narrowing down the measurement bandwidth or deliberately mismatching the normal JJ resistance away from the amplifier input impedance. However, this may trade off the photon counting speed and JJ voltage measurement signal-to-noise

≃ × −17 ratio, respectively. For 1 MHz measurement bandwidth and r0 = 1, Grad 1.4 10 W/K (Fig. 5.4b). This cooling channel is only significant at about 0.01 K when it is numerically comparable to the thermal conductance due to the electron-phonon coupling Gep.

Internal energy can be transferred from electrons to phonons by scattering.156,164 Similar to Stefan-Boltzmann blackbody radiation, this heat transfer is a high power law in temperature, originating from the integral of bosonic and fermionic occupancies and density of states in the Fermi golden rule calculation. However, we can linearize this function to extract a thermal conductance Gep.

~ −1 The Bloch-Grüneisen temperature TBG = 2 skF/kB, where s = 26 km s is the speed of sound in graphene, marks the temperature when the Fermi momentum of the electrons is comparable to that of the graphene acoustic phonons. For the carrier density we consider here, T < TBG and the heat transfer from electrons to acoustic phonons in graphene is given by:31,156,160,164–166

δ − δ Pep = ΣA(Te T0) (5.6)

58 where Σ is the electron-phonon coupling parameter. The power δ is determined by the disorder in graphene.

Disorder effects dominate the electron-phonon coupling when the typical phonon momentum is smaller than

1/lmfp, the inverse of the electron mean free path. Thus for T higher (lower) than Tdis = hs/kBlmfp, the electron-

5/2 4 D2 1/2 ~4 2 3 phonon coupling is in the clean (disordered) limit. For clean graphene, δ = 4 and Σ = π kB n /(15ρm vFs )

3 D2 1/2 3/2 ~3 2 2 D ≃ whereas for disordered graphene, δ = 3 and Σ = 2ζ(3)kB n /(π ρm vFs lmfp), where 18 eV is the

× −19 −2 deformation potential, ρm = 7.4 10 kg µm is the mass density of graphene, and ζ is the Riemann zeta − ≪ ≃ − function. In the linear response regime, when (Te T0) T0, the Fourier law is recovered: Pep Gep(Te T0)

δ−1 with Gep = δΣAT0 as the electron-phonon thermal conductance (Fig. 5.4b). The total cooling power, includ- − ing radiation and electron-phonon coupling, is Pep + Grad(Te T0). We can equate this rate of heat transfer to

Ce(dT/dt) such that: dT ΣA(Tδ − Tδ ) + r k B(T − T ) ≃ −C e (5.7) e 0 0 B e 0 e dt

In the linear response regime, Eqn. (5.7) reduces to a simple RC circuit with a thermal time constant τth =

Ce/(Gep + Grad) (Fig. 5.4c). Effectively, the weak electron-phonon coupling in graphene helps maintaining the heat in the electrons for a longer period of time. τth determines the intrinsic dead time of the graphene SPD so that an infrared detector operating at a few Kelvin could count photons at a rate up to GHz while a microwave

≫ detector operating at 10s of millikelvin could have a count rate in the MHz range. When Gep Grad, thermal

∝ time constants increase rapidly as operating temperature decreases because Ce T while Gep has a higher tem-

δ−1 perature power law i.e. T in monolayer graphene. In this regime τth is independent of carrier density because

1/2 ≪ both Ce and Gep are proportional to n . In contrast, when Gep Grad, τth decreases with decreasing T (Fig.

5.4c, green and orange curves) because Ce decreases while Grad is constant in T for a given bandwidth. Grad also √ ∝ has no n dependence so that τth n, allowing for the possibility to quickly reset the graphene SPD through its gate voltage.

59 Modeling Parameters Graphene dimensons 5 µm x 200 nm JJ channel length L 200 nm JJ channel width W 1.5 µm 12 −2 Electron density n0 1.67x10 cm Electronic mobility μ 8000 cm2/Vs JJ normal resistance Rn 63 Ω Mean free path lmfp 120 nm Electronic heat capacity Ce(T0) 6.3 kB Disorder temperature Tdis 10.4 K Bloch-Grüneisen temp. TBG 90.5 K Ic(T0)Rn product Ic(T0)Rn 223 µeV Thouless energy ETh 990 µeV JJ coupling energy EJ0(T0) 7.25 meV Plasma Freq. ωp0(T0) 156 GHz

McCumber parameters βSC 0.2 NbN superconducting gap Δ0 1.52 meV

Table 5.1: List of device parameters to model the graphene-based Josephson junction single photon detector in this report.

We solve Eqn. (5.7) numerically to find T(t) for the absorption of a single 26 GHz photon by a 1 µm2 device operating at 25 mK and plot the result in Figure 5.4d. Because of the high-temperature power law in the electron-

− ≫ phonon heat transfer for (Te T0) T0, Te drops faster than exponential, followed by a slow decay at the time constant τth.

For this modeling, we employ lmfp = 120 nm at n0, deduced from the electrical transport measurements on the typical GJJ devices that we fabricate (See Section 5.4). This electrical transport mean-free-path would put the disorder temperature higher than the operating temperatures we consider here. Hereafter we will use the graphene thermal response in the disorder limit, i.e. δ = 3.

5.4 Graphene-Based Josephson Junction

Detection of single photons relies on the GJJ transition from the zero-voltage to resistive state. This switching is a probabilistic process described by the escape rate, Γ, of the JJ phase particle from the tilted washboard potential

60 (a) 3 600 I

2 S )

400 ( µ Ω ( A) n R 1 200

0 0 -10 -5 0 5 01 51 02 VG (V) (b) 250 200

V) NbN

µ 150 ( n

R 100 S

I Ib 50 Graphene 0 -10 -5 501 0 51 02 VG (V)

Figure 5.5: (a) Measured Rn (blue) and Is (red) as a function of VG (b) IsRn versus VG. Inset: Optical micrograph of the measured GJJ whose device performance parameters were used in this report. Blue colored region is the graphene channel encapsulated by hBN and the region emphasized by orange colored lines are the NbN electrodes. Scale bar (white) is 1.5 µm. in the resistively and capacitively shunted junction (RCSJ) model.167 The rate of switching depends intimately on the JJ critical current, Ic, which takes on different forms as a function of temperature depending on whether the JJ is short or long and whether it is diffusive or ballistic. Superconductor-graphene-superconductor JJs have

103,145–149 been studied in these different regimes. However, experimental values, such as Ic and the parameters in the long diffusive junction, have fallen short of theoretical expectations due to impurity doping. Toemphasize the feasibility of GJJ SPDs under currently realizable parameters, we model the device performance based on experimentally determined values (summarized in Table 5.1) of the GJJ shown in the inset of Fig. 5.5b instead of analyzing the GJJ from a purely theoretical standpoint.

61 The measured graphene monolayer is encapsulated between atomically flat and insulating boron nitride∼ ( 30 nm thick) using a dry-transfer technique.157 The doped silicon substrate serves as a back gate electrode. The superconducting terminals consist of 5 nm thick niobium and 60 nm thick niobium nitride (NbN) after reactive ion etch and electron beam deposition of 5 nm titanium to form the etched one-dimensional contact.168 The distance between the superconducting electrodes L is about 200 nm, forming a proximitized JJ with monolayer graphene as the weak link.

The GJJ is mounted at the mixing chamber (MC) of a dilution refrigerator with base temperature 25 mK.

The electrical transport measurement is performed through a standard four-terminal configuration with the silicon substrate as the back gate to control the carrier density in graphene. All DC measurement wires are filtered by a two-stage low pass RC filter mounted at the MC with an 8 kHz cutoff frequency. Ib is set by a DC voltage output through a 1 MΩ resistor while the voltage measurements are taken by a data acquisition board after a low noise preamplifier with a 10 kHz low pass filter.

− Fig. 5.5a shows the resistance as a function of gate voltage. CNP is observed to occur at VG = 5V. The chosen operating carrier density n0 corresponds to VG = 20 V for a 300 nm thick dielectric material composed of silicon dioxide and hexagonal boron nitride. At this gate voltage Rn is measured to be 63 Ω. The mobility is

~ 1/2 calculated as μ = L/(neRnW), the mean free path as lmfp = μ(πn) /e, and the diffusion coefficient as

2 −1 −1 2 −1 De = vFlmfp/2. At n0 this yields μ = 8000 cm V s , lmfp = 120 nm, and De = .06 m s .The slope of con-

= /( ) = 1 dσ ≃ ductance, σ L RnW , versus n gives similar results for mobility using μ e dn . lmfp 120 nm is comparable to the channel length L ≃ 200 nm so this device is in neither purely ballistic nor purely diffusive regime. When the bias current through the JJ increases, it switches from supercurrent to resistive state at a switching current Is depending on the gate voltage as shown in Fig. 5.5a. Despite Is and Rn vary for different gate voltages, their product approaches to a constant in both the highly electron- and hole-doped regime (Fig. 5.5b). Consistent to the experimental results in Ref. [145, 146, 148], the IsRn product for the niobium-based GJJ has a smaller value than

62 the calculation from the superconductor critical temperature probably due to impurity doping.

Fig. 5.6a shows typical IV characteristics for VG = 20V at 25 mK and 200 mK, which correspond to T0 and Tpeak at 26 GHz, respectively (Fig. 5.4d), measured by ramping up the bias current at a rate of 0.1 µA/s. In order to measure the escape rate, we repeat the IV measurement 100 times at each base temperature and find the

⟨ ⟩ average switching current, Is(T) (Fig. 5.6a inset), which at this sweep rate is typically about 90% of Ic. From the histogram of these 100 GJJ switching events, we obtain the probability density of the switching current P(Is) for each measured temperature. Fig. 5.6c shows P(Is) at 25 mK and 200 mK. We can derive the phase particle escape

169 rate Γ from the P(Is) data using:

Z Is dI − ′ ′ P(Is) = Γ(Is)/ 1 P(I )dI (5.8) dt 0 where dI/dt is the bias current ramping speed.

Fig. 5.5c right panel shows the extracted Γ data which can be described by:98

ΔU Γ = A exp − (5.9) kBTesc

q − 2 − −1 where ΔU = 2EJ0( 1 γJJ γJJ cos γJJ) is the energy barrier of the washboard potential and Tesc is the “escape temperature” which sets the energy scale competing with ΔU for the phase particle to escape from the washboard

~ potential. Here, EJ0 = Ic/(2e) is the Josephson coupling energy and γJJ = Ib/Ic is the normalized bias current. In the thermal activation (TA) regime, Tesc is simply the temperature of the device while in the macroscopic quantum tunneling (MQT) regime:

0.87 T = ~ω / 7.2k 1 + (5.10) esc p B Q

63 (a) (c) 4 3.15 25 mK

2 3.10 Is(T) A) µ 0 0 0.5 1 3.05 I T (K) b ( µ Current( A)

-2 Ib 3.00

(b) -4

) "0" "1" 2.95 -1 0.4 δVAmp 200 mK (µV 0.2 P 0.0 -2 0 2 2.90 -200 0 200 0 01 02 03 10 10 10 Voltage ( µV) P (µA -1 ) Γ (Hz)

Figure 5.6: (a) Measured GJJ IV characteristics for electron temperature at T0 = 25 mK (green) and Tpeak = 200 mK (orange) with car- − rier density of 1.7 × 1012 cm 2. Inset: The measured average switching current vs. temperature. (b) The JJ voltage and the expected voltage noise from an amplifier at 1 MHz bandwidth. The blue dotted line depicts, for a given bias current in a photon event, theorderof

magnitude of the JJ voltage V = IbRn. (c) The switching probability and escape rate of the phase particle as a function of the JJ current bias. The solid line is the best fit probability distribution to the data assuming the escape mechanism is MQT. Solid lines are givenbyEqn.

9 with Ic = 3.565 µA (25 mK) and Ic = 3.425 µA (200 mK).

− 2 1/4 where Q = ωpRnCJJ is the JJ quality factor with ωp = ωp0(1 γJJ) being the JJ plasma frequency, ωp0 =

~ 1/2 (2eIc/( CJJ)) being the zero bias JJ plasma frequency, and CJJ being the effective junction capacitance. While

the geometrical capacitance between the superconducting terminals is estimated to be sub-femtofarad lower

bounded by the parasitic capacitance to the substrate, there is an effective capacitance due to electronic diffu-

~ 103 ~ 2 170 sion given by CJJ = /RnETh. ETh = De/L is the Thouless energy where De = vFlmfp/2 is the diffusion

∼ constant. ETh is the characteristic energy scale of a diffusive GJJ and is estimated tobe 11 K for the device char-

64 ∼ acterized for our modeling which gives CJJ 11 fF. Using the effective capacitance, we estimate ωp0 of this GJJ to

× 2 be 2π 156 GHz. The Stewart-McCumber parameter βSC = Q is about 0.2. This implies that the GJJ should be an overdamped junction with its phase particle retrapped quickly after switching to the resistive state. How- ever, the measured junction is hysteretic probably because the resistive state bias current self-heats the junction at low temperatures.161

103 Using ωp and Q above, we can estimate this GJJ to be in the MQT regime for temperatures below 470 mK.

With the prefactor A of Eqn. 5.9 given by:

   q  ωp + 1 − 1 2π 1 4Q2 2Q (TA) A = (5.11)  q  12ω 3ΔU (MQT) p 2π~ωp

we fit the extracted Γ values with Ic as the only fitting parameter (all other parameters depending on Rn, CJJ, and

Ib) and find the fitting consistent with the MQT process as expected. The solid lines in Fig. 5.6c showtheMQT

Γ with Ic equal to 3.57 µA and 3.43 µA at 25 mK and 200 mK, respectively. The experimentally determined Γ as a function of Ib and Ic(T) will help us to calculate the SPD performance in Section 5.5.

5.5 Photon Detection Performance

≃ The GJJ can be set to detect single photons by setting a bias current Ib (as an example, Ib 3.03μA in Fig. 5.6c) below the switching current, so that when photons raise the graphene electron temperature, the GJJ may switch to the resistive state as its critical current quenches. As the electrons cool, the GJJ will switch back to the superconducting state for a junction with no hysteresis. For a hysteretic junction, the detector can be reinitialized using the bias current or gate voltage, as both the switching and retrapping current depend on the gate voltage. The

GJJ can function as an SPD when either the MQT or the TA process dominate the JJ transition because its opera-

65 Dark Count Rate (Hz) (A) (B) (C) -5 -3 -1 1 3 5 10 10 10 10 10 10 3.6 9 10 1 1

dI /dT 5 Γ c 3.4 MQT dIc/dT 10 0.1 (µA/K) 0.1 A) (µA/K) -1 µ η (Hz) ( γJJ =0.85 -1 dIc/dT c -2 I -2 1 (µA/K) 3.2 10 Probability 0.01 -3 0.01 -3 -1 Red: η -2 Blue: DCP -3 3.0 -3 0.001 0.001 10 -12 -9 -6 -3 0 0.0 5.00.1 5.1 0.2 0.80 0.85 09.0 59.0 00.1 10 10 10 10 10 t ( µs) γJJ Dark Count Prob.

Figure 5.7: (a) Change of critical current and phase slipping probability due to a single microwave photon, impinging at t = 0 s and T = 0.025 K, deduced from the graphene thermal properties and measured GJJ parameters. (b) Intrinsic efficiency and dark count probability at 1 MHz count rate versus bias current. Vertical gray dashed line indicates bias current for which η >0.99 for the measured dIc/dT. (c) Trade-off between the intrinsic quantum efficiency η and the dark count rate at various bias currents from (b). Gray circle corresponds to gray line from (b). tion depends on the change of Γ as the electron temperature increases. However, the phase diffusion process103 is not desirable because the finite sub-gap resistance can diminish the signal-to-noise ratio of the GJJ voltage signal readout significantly.

The voltage drop across the GJJ upon detection is given by IbRn which is about 190 µV for the measured device (marked by the vertical dashed line in Fig. 5.6a). The inaccuracy in measuring this voltage with a short √ averaging time will be dominated by the amplifier noise because the Johnson noise spectral density, 4RnkBT, √ 1/2 F 1/2 is only about 10 pV/Hz while the shot noise, Rn 2eIb, is about 33 pV/Hz for Ib = 3.03μA and Fano factor F ≃ 0.29.171,172 Using a typical low noise voltage amplifier with power spectral density of 1 nV/Hz /2, the amplifier voltage noise at 1 MHz measurement bandwidth is 1 µV. Signal-to-noise of the GJJ readout islarge as depicted in Fig. 5.6c, where two well separated Gaussian peaks centered at Vb = 0 (“0” no photon state) and

Vb = IbRn = 190 µV (“1” photon state) with FWHM corresponding to the amplifier voltage noise. A Josephson coupling energy that gives an IcRn product of about 100 µV will be sufficient for making a GJJ SPD.

The performance of a GJJ SPD can be calculated by the probabilistic JJ transition at an elevated temperature.

66 P We choose to benchmark the GJJ SPD by its intrinsic quantum efficiency η and dark count probability dark for single photon detection at 26 GHz with improved performance expected for higher energy photons. We note that for infrared photons, the peak temperature will be higher than the superconducting gap energy of niobium nitride, resulting in heat leakage that can reduce the peak temperature and shorten the duration of the heat pulse.

Calculation of the spectral current109 will probably be required to understand the GJJ under high energy photon excitation and is beyond the scope of this work. For microwave photons, the temperature rise is much smaller than the gap energy and the detector remains in the quasi-equilibrium regime.

P We can calculate dark using Γ. The total escape probability in a measurement integration time tmeas is given R − exp(− tmeas ( ) ) by 1 0 Γ τ dτ . In the absence of incident photons, Γ is equal to the dark count rate Γdark and would be a constant Γ0(T = T0) if the temperature were fluctuation free. To include the effect of temperature

R ∞ ⟨ ⟩ = ( ) ( ( )) ( ) fluctuation, we use the averaged Γ, i.e. Γ 0 dTp T Γ Ic T where p T is a Gaussian distribution cen- ⟨ ⟩ ≥ P tered at T0 with standard deviation ΔT. Since Γ increases quickly as a function of T, Γ Γ0. dark equals

− −⟨ ⟩ − − 1 exp( Γ tmeas) and can be significantly higher than 1 exp( Γ0tmeas) depending on the size of dIc/dT and

≃ P ΔT/T0. At T0 = 25 mK and Ib =3.28 µA (γJJ 0.91), dark for tmeas = 1 µs is about 0.07 using an estimated

ΔT/T0 = 0.8 (Fig. 5.3c) and the fitted Γ from Section 5.4.

Upon photon absorption, the electron gas heating and the subsequent cooling from Tpeak result in a time- dependent Γ that can be used to calculate η. Using the Te(t) in Fig. 5.4d, Is(T) in Fig. 5.6a, and fitted Γ(Ic), we calculate both the critical current and Γ as they recover to their nominal values after the photon incidences at t = 0 (see Fig. 5.7a). We calculate the intrinsic quantum efficiency η of the GJJ SPD from:

 Z tmeas − − η = 1 exp Γ (Ic(τ)) dτ . (5.12) 0

P The detection efficiency increases with the measurement time, but sodoes dark. To balance between these com-

67 peting effects, we benchmark the GJJ SPD performance by taking the measurement integration time tobe1µs

( ( )) ≃ ⟨ ⟩ P / such that Γ Ic tmeas 2 Γ T=T0 . Fig. 5.7b plots η and dark as a function of γJJ for three different dIc dT ⟨ ⟩ values. With the dIc/dT of the measured GJJ at -1.1 µA/K (measured in the linear region of Is(T) above about

100 mK in Fig. 5.6a), we can set γJJ = 0.91 (vertical grey dashed line) to reach an intrinsic quantum efficiency P ≃ >0.99 while maintaining dark 0.07.

P We plot the trade off between η and dark in Fig. 5.7c by eliminating the common parameter γJJ of Fig. 5.7b. P Favorable SPD performance occurs in the plateau region where η approaches unity while dark remains small.

This SPD regime is feasible within the parameters of existing GJJs that we can fabricate. The operating point circled in Fig. 5.7c corresponds to the same bias current and integration time setting at the vertical dashed lines in

Fig. 5.7b.

To optimize the SPD performance or to operate at a different frequency, we argue that the design should focus on the dependence of the GJJ critical current on temperature. Although a lower operating temperature and a smaller area of monolayer graphene can enhance the temperature rise due to a smaller electronic heat capacity, the rate of improvement can quickly diminish because the heat loss from the electrons is dominated by the cou-

− pling to phonons and is faster than exponential when Te T0 > T0. The time integral of Γ in Eqn. (11) can only increase marginally to enhance the intrinsic quantum efficiency insignificantly. However, the phase particle escape rate of the JJ can increase by orders of magnitude with increased dIc/dT because of the exponential dependence of ΔU in both MQT and TA processes, as suggested by the initial numerical values of Γ in Fig. 5.7a.

This behavior contributes drastically to the SPD intrinsic efficiency, allowing a lower JJ current bias to further reduce the dark count probability as shown in Figs. 5.6b and 5.6c by the dashed and dotted lines for double and

P triple the measured dIc/dT, respectively. Merely doubling dIc/dT lowers darkcorresponding to η =0.99 by 5 orders of magnitude while an 8 order of magnitude improvement is possible by tripling dIc/dT. The operation of a GJJ SPD at a higher temperature up to 4 K is possible using a ballistic-like GJJ such as that demonstrated in

68 Ref. [149] with dIc/dT on the order of -1 µA/K.

5.6 Conclusion

In conclusion, we have introduced a device scheme for ultra-broadband single photon detection based on the extremely small electronic specific heat in graphene, which results from the vanishingly small electronic density of states near the Dirac point and its linear band structure. Our model analysis shows that single photon detection, together with very small dark count rates, is possible across a wide spectral region, from the microwave to near-infrared light. Efficient light absorption into the graphene absorber can be achieved by impedance matching structures, such as metallic or dielectric resonators considered here for microwave and optical frequencies. Using experimental parameters from a fabricated device, we model that a GJJ SPD operating at 25 mK could reach a system detection efficiency higher than 99%, together with one-tenth dark count probability for a 26 GHzphoton.

This device performance should improve for higher photon energies. Inductive readout138,150–152 can be used in the future to increase the SPD operation bandwidth by avoiding the Joule heating when the GJJ switches to resistive state. We have only explored a small set of possible parameters in this report. Further optimization of the GJJ

SPD will depend on application-specific performance trade-offs. Heat leakage to the superconducting electrodes as well as position and time dependence of the heat propagation will need to be included in the future for a more realistic prediction and a better understanding of the fundamental limits of the GJJ SPD. The rapid progress in integrating graphene and other Van der Waals materials with established electronics platforms, such as CMOS chips, provides a promising path towards single photon-resolving imaging arrays, quantum information process- ing applications of optical and microwave photons, and other applications benefiting from quantum-limited photon detection of low-energy photons.

69 Even noise has noise.

Jesse Crossno, 2017

6 Another Approach: Johnson Noise

Thermometry

6.1 Introduction

Before continuing to the experimental demonstrations of the Josephson-junction detector, it should be noted that graphene’s thermal properties discussed in Section 3.2 and its photon-induced thermal response discussed in Section 5.3 do not rely on the fact that it is in a Josephson-junction configuration except for the fact that superconducting contacts block heat transfer. This means there could be other ways to detect a photon-induced

70 temperature spike in graphene. Here I will discuss one of these methods that involves direct measurement of the electronic temperature in graphene: Johnson noise thermometry (JNT). As will be revealed in the following sec- tions, this technique is more experimentally taxing than the simple threshold detection of the Josephson-junction detector but allows for photon-number resolving capabilities not achievable in the other setup.

6.2 Johnson Noise Thermometry Basics

Any resistor, graphene included, with a nonzero temperature will exhibit some electronic voltage noise based on thermal fluctuations. This phenomenon was first observed by Johnson173,174 and soon after described mathemat- ically by Nyquist:175

SV(f) = 4RkBT (6.1)

where SV(f) is the voltage power spectral density as a function of frequency f, R is the resistance of the resistor at temperature T, and kB is Boltzmann’s constant. Of note is that SV(f) is frequency-independent (although it does have a high-frequency rolloff when a full quantum theory is employed).176 This means the voltage-noise power can simply be written as:

⟨ 2⟩ V = 4RkBTB (6.2) where B is the measurement bandwidth of the system. To measure the temperature of a known resistor then, one only needs to measure ⟨V2⟩. This measurement, of course, will not be deterministic but have some uncertainty associated with it. First, uncertainty arises because the accuracy of the measurement of the voltage noise will be limited by the duration for which it is measured. Second, the measurement of the resistor will include the noise from the measurement electronics as well. These two effects are captured in the Dicke radiometer equation177

(based on Rice’s equation):178 T + T δT = √ n (6.3) Btint

71 where δT is the temperature measurement uncertainty, Tn is the noise temperature of the measurement system

(which in general can be different from the physical temperature) and tint is the integration time. Using Equa- tions 6.2 and 6.3, JNT has indeed been verified to be applicable to graphene systems.29,46,179,180 In the next section, we discuss how such measurements could be used for single-photon detection.

6.3 Single Photon Detection

As discussed in Chapters 3 and 5, upon absorbing a photon, graphene’s temperature rises significantly due to its low heat capacity. The temperature then decays on timescales of order nanoseconds to microseconds. The

JNT technique measures the average temperature in the time tint so would not directly measure the peak temperature of the graphene without an ultra-short measurement time. However, as we will show, the temperature rise is enough to raise the average temperature for experimentally reasonable values of tint. This elevated temperature, including its uncertainty given by Equation 6.3, must be distinguishable from the measurement of the base temperature including its uncertainty in order to effectively detect single photons. We note the analysis ofthis section follows closely that of McKitterick.139

For the current discussion, we focus on microwave photons. We assume a graphene area of 10 µm2 and that

− the charge-carrier density is 1010 cm 2. We start by examining the possibility to detect 40-GHz photons with this graphene sheet at a base temperature of 10 mK, achievable with a dilution refrigerator (DR). Based on Equation

5.2, the peak temperature reached by the graphene upon absorption of a 40-GHz photon is ∼250 mK and, solving Equation 5.7, it takes ∼80 ns for the temperature to decay back to the base (Figure 6.1A). If we assume an integration time of 100 ns, then the average temperature upon photon absorption is ∼100 mK. Although this is ten times the base temperature, the uncertainty in the measurement can still be quite large. First, few amplifiers operate to such low temperatures so amplifiers often reside at higher-temperature stages within the DR. Assume for now, though, that we have a 10-mK amplifier and that it is quantum limited, thatis Tn = hfp/2kB where fp is

72 Figure 6.1: Single-photon detection with graphene JNT. (A) The thermal response of a 10-µm2 sheet of graphene with carrier density 1010 − cm 2 to a 40-GHz photon based on Equation 5.7. (B) The average temperature for no photon (blue) and the absorption of a single 40- GHz photon (red) using a quantum-limited amplifier with 100-MHz center frequency and 200-MHz bandwidth. The integration timeisset to 100 ns. Clear separation between the distributions means single-photon detection is possible with few dark counts and high efficiency.

the photon frequency, h is Planck’s constant, and kB is Boltzmann’s constant. If the center frequency of the am-

∼ plifier is 100 MHz, which is feasible, then Tn 2 mK. However, based on Equation 6.3, the measured temperature also plays into the temperature uncertainty. The average temperature of 100 mK will dominate δT in this case. To help mitigate this effect, we use the maximum bandwidth, 200 MHz, for a 100-MHz center frequency.

∼ This value together with the other discussed parameters, gives δTph 23 mK for the graphene in its elevated- temperature state after absorbing a photon. When the graphene is at the base temperature, assuming all other

∼ measurement parameters the same as for the elevated-temperature case, δT0 2.8 mK. We plot the expected temperature distributions in Figure 6.1B. There is a clear distinction between the dark and light distributions allowing for high detection efficiency and low dark count rate.

6.4 Photon-Number Resolving

In the GJJ SPD studied throughout the rest of this work, when enough energy is absorbed, the GJJ switches from zero voltage to a non-zero voltage determined by the bias current. Varying the amount of energy deposited into the GJJ may change the probability of the detector switching but once it switches, it will always switch to the same voltage value. In contrast, for JNT the measured voltage noise power is directly proportional to the

73 Figure 6.2: Photon-number resolving with graphene JNT. (A) The temperature distributions (solid lines) for 0- (blue), 1- (red), 2- (yellow), 3- (purple), and 4- (green) photon states. All parameters are the same as in Figure 6.1 except the integration has been increased to 700 ns. Shaded regions indicate a potential classification scheme for the number of photons in a state. For example, any temperature measured within the purple shaded region would be assigned a 3-photon state (even though there is some probability that measured temperature corresponds to a 2- or 4-photon state as well). (B) The probability of classifying a number state (actual number) with a given number state (measured number). The green boxes on the diagonal indicate correct classification while any off-diagonal probabilities represent incorrect classifications. Photon-number resolving is possible with increasing errors as the number of photons increases. temperature. A larger amount of energy deposited in the graphene will result in a higher temperature and a larger measured voltage noise. This proportionality can potentially enable the determination of the number of photons absorbed by a sheet of graphene.

I continue with the 40-GHz photon scenario of the previous section. To calculate the peak temperature upon √ ∝ absorbing N photons, hfp in Equation 5.2 should be replaced by Nhfp. Because Tpeak N, as N increases, the difference in the induced temperature by consecutive numbers of photons decreases, i.e. Tpeak(N = 3) is closer to Tpeak(N = 2) than Tpeak(N = 2) is to Tpeak(N = 1). Exacerbating this issue is that, based on Equation

6.3, δT increases as temperature, and thus the number of photons, increases as well. Using a 100-ns integration time, these two characteristics of JNT makes it difficult to resolve more than ∼2 photons. Integrating for longer, however, can decrease the measurement uncertainty (Equation 6.3) at the cost of decreasing the photon-induced measured average temperature (with only ∼80 ns of elevated temperature, averaging for longer adds more base temperature to the average thus reducing it). I find that tint = 700 ns best balances between decreasing measurement uncertainty while keeping sufficient separation between average temperatures. The temperature distribu-

74 tions for N= 0, 1, 2, 3, and 4 photons is shown in Figure 6.2A. The separation between 0 and 1 photons is still high but as N increases, the distributions begin to overlap. This overlap leads to errors in estimating the number of photons absorbed. I set cutoff temperatures between the 0- and 1-, 1- and 2-, 2- and 3-, and 3- and4-photon distributions to characterize this effect. I integrate each N-photon distribution between each set of cutoff temperatures to calculate the probability of assigning an N-photon event a given number of photons (Figure 6.2B).

Distinguishing 0 or 1 photons correctly has near-unity probability as expected and there is a ∼88% chance of characterizing a 2-photon state. For photon numbers greater than 2, the probability of assigning a photon state the correct number of photons decreases but is still above 50% for 3- and 4-photon states.

Performance of JNT for photon-number resolving could be improved in a number of ways. For instance, lower operating temperatures than those in a dilution refrigerator can be achieved with an adiabatic demagnetization refrigerator (ADR).181 Lower temperatures allow for longer integration times because the 0-photon state has a smaller T and δT. For non-zero photon states, δT could be improved with the development of wider band, quantum-limited amplifiers allowing for better separation between peaks. Instead of focusing on lowering

δT, one could also look to increase the separation between Tpeak for the different photon temperature distributions. One way to do this would be to further decrease graphene’s heat capacity, for instance, through making the graphene area smaller. Recent experiments have been probing the limits of light confinement in graphene structures with promising results that could drastically reduce the graphene area in a JNT setup.182,183 Although testing the limits of currently available technologies, the combination of JNT with graphene is already making progress in ultrafast bolometry29 and could be a useful tool for single-photon detection as technological advances are made in the above areas.

75 7 GJJ Microwave Bolometry with Single-Photon

Sensitivity

Based on the practical hurdles presented by Johnson noise thermometry for single-photon detection, the graphene- based Josephson junction detector of Chapter 5 was pursued. The current chapter demonstrates the sensitivity of the GJJ to microwave radiation. Instead of using the junction as a single-photon detector (a calorimeter, i.e. energy detector), it was used as a bolometer, i.e. power detector. The noise-equivalent power (NEP), that is the power that could be measured with a signal-to-noise ratio (SNR) of 1, was shown to correspond to an energy resolution of a single 32-GHz photon. The GJJ in the experiment was coupled to an 8-GHz resonant circuit and

76 irradiated with continuous-wave power so no single photon detection was observed. However, the single-photon energy resolution demonstrated means future experiments with different circuitry could prove the utility of the

GJJ as a single-photon detector in this regime.

The remainder of this chapter is reproduced from G. H. Lee, et al., Graphene-based Josephson junction microwave bolometer, Nature, published 2020. ©The Authors, under exclusive license to Springer Nature Limited

2020. I note a similar GJJ bolometer was demonstrated in Ref. [184] at the same time as that presented here.

7.1 Abstract

Sensitive microwave detectors are critical instruments in radioastronomy,5 dark matter axion searches,9 and superconducting quantum information science.143,185 The conventional strategy towards higher-sensitivity bolometry is to nanofabricate an ever-smaller device to augment the thermal response.6,138,152 However, this direction is increasingly more difficult to obtain efficient photon coupling and maintain the material properties inadevice with a large surface-to-volume ratio. Here we advance this concept to an ultimately thin bolometric sensor based on monolayer graphene. To utilize its minute electronic specific heat and thermal conductivity, we develop a superconductor-graphene-superconductor (SGS) Josephson junction103,147,148,161,186,187 bolometer embedded in a microwave resonator of resonant frequency 7.9 GHz with over 99% coupling efficiency. From the dependence of the Josephson switching current on the operating temperature, charge density, input power, and frequency,

− we demonstrate a noise equivalent power (NEP) of 7 ×10 19 W/Hz1/2, corresponding to an energy resolution of one single photon at 32 GHz188 and reaching the fundamental limit imposed by intrinsic thermal fluctuations at

0.19 K.

77 7.2 Introduction

Many attractive electrical and thermal properties in graphene make it a promising material for bolometry and calorimetry.29,31,139–141,189–191 It can absorb photons from a wide frequency bandwidth efficiently by impedance matching;13 the electron-electron scattering time is short and can quickly equilibrate the internal energy from absorbed photons to evade leakage through optical phonon emission;48 its weak electron-phonon coupling can keep the electrons thermally isolated from the lattice;31,141,156,161,164,192–194 most importantly, at the charge neutrality point (CNP), graphene has a vanishing density of states. This results in a small heat capacity and electron- to-phonon thermal conductance which are highly desirable material properties for bolometers and calorimeters, while maintaining a short thermal response time.29 Although the bolometric response of graphene has been tested in devices based on noise thermometry,29,31,139 their performance is severely hampered by the degrading thermometer sensitivity when the electron temperature rises upon photon absorption.139 Here, we overcome this challenge by adopting a fundamentally different measurement technique: we integrate monolayer graphene simultaneously into a microwave resonator and a Josephson junction, and upon absorbing microwave radiation into the resonator, the rise of the electron temperature in graphene suppresses the switching current of the SGS

Josephson junction. This mechanism can function as the bolometer readout and provide us a way to study the thermal response of this bolometer.

7.3 Experimental Setup

Inspired by the demonstration of using heating or quasiparticle injection to control the supercurrent in superconductor- normal-superconductor junctions in the DC regime,108,195 we design our microwave bolometer with an orthogonal- terminal graphene-based Josephson junction (GJJ) as shown in Fig. 7.1a and b. The monolayer graphene is encapsulated on the top and bottom by hexagonal boron-nitride (hBN). The proximitized Josephson junction

78 Figure 7.1: Graphene-based Josephson junction microwave bolometer. (a) Device concept of the superconducting-graphene- superconducting (SGS) Josephson junction (JJ) microwave bolometer. The hBN-encapsulated SGS JJ (1 µm wide and with a gap of ≃0.3 µm) is embedded simultaneously in a half-wave resonator to allow microwave coupling (blue) and DC readout (green) of the JJ. For clarity, the local gate is not shown. (b) Scanning electron microscope image of the orthogonal-terminal JJ. (c) Schematics of the detector setup. The graphene flake is located at the current antinode of the half-wave microwave resonator. Test microwave power is coupled tothede- tector through the 20 dB directional coupler and highly attenuated coaxial cables from room temperature. Two stages of inductors and capacitors form a low-pass filter network for the DC measurement. (d) False-colored optical image of the actual device.

(green color) is formed by edge-contacting NbN superconductors to the graphene such that dissipationless

Josephson current can flow along the JJ direction.147 A dissipative microwave current can flow along the direction perpendicular to that of the junction, with the graphene extended out by 0.8 µm from each side of the GJJ before connecting to quarter-wave resonators (blue color) to form a half-wave resonator using a NbN microstrip with a characteristic impedance of 86 Ω (Fig. 7.1c and d). This extension is narrow and long to prevent Joseph- son coupling to the microstrips and positions the graphene at the current antinode of the resonator. We have measured three devices with intended variations in graphene dimensions, superconductors, and input frequencies for testing the robustness of the bolometer performance. While their results are similar, we will present the one with the highest sensitivity in the current section and the rest in Section 7.6.

79 Figure 7.2: Characterizing the graphene-based Josephson junction (GJJ) switching current. (a) GJJ voltages with sweeping of bias current and (b) the averaged switching current vs. gate voltages at various temperatures, 0.19-0.9 K. (c) Averaged switching currents vs. ⟨ ⟩ temperatures at different gate voltages. (d) GJJ normal resistance and the Is Rn product as a function of gate voltage.

Microwave power is applied to the resonator through a 200 fF coupling capacitor. We can characterize our

GJJ-embedded resonator by reflectometry using a directional coupler. All test power is delivered via the heavily- attenuated microwave coaxial cables to filter the thermal noise from room temperature. To decouple the GJJDC measurement from the microwave resonator, two stages of LC low-pass filters are implemented to form a high- impedance line at high frequency. The 1 nH inductors are made of narrow meandered wires and are shunted by

530 fF capacitor plates.

We study the GJJ switching as a function of temperature and gate voltage. Fig. 7.2a shows the typical voltage drop across the junction VJJ as the DC bias current is swept from 1.5 to -1.5 µA at device temperatures between

0.19 and 0.9 K. Our GJJ shows hysteretic switching behavior: the switching current Is, at which the junction switches from the dissipationless state to the normal state, is different from the retrapping current, Ir. Such

80 Figure 7.3: Demonstration of the device’s operation as a bolometer and measuring the detector efficiency. (a) The scattering parameter of the device at 0.19 K shows a resonance near 7.9 GHz with linewidths of 861 and 599 MHz at gate voltages of 0.1 and 1.3 V respectively, obtained from the quality factor analysis. (b) Suppression of the average switching current at -112 dBm relative to the absence of input power at two different gate voltages. hysteresis is presumably due to self-Joule heating when the junction turns normal.161 The averaged switching

⟨ ⟩ ⟨ ⟩ currents Is are plotted at various gate voltages Vgate and temperatures in Figs. 7.2b and c. The drop of Is as temperature rises is an important feature that can determine the sensitivity of the GJJ as a bolometer as well as the quantum efficiency and dark count of the future microwave single photon detector.13 Fig. 7.2d plots the normal- state junction resistance Rn as a function of gate voltage, indicating that the CNP is at -0.9 V. We note that the unusual rise of Rn at around 2 to 3 V of Vgate may be due to the formation of a moiré superlattice with the hBN

⟨ ⟩ substrate (see Section 7.6). The Is Rn product is on the order of 0.16 mV, which is comparable to other GJJs of similar size in the long diffusive limit.13

The coupling efficiency can be characterized using reflectometry (see Fig. 7.3a). We design the resonatorto be critically coupled at about 7.9 GHz. The dissipation is dominated by the monolayer graphene, which can be

81 ⟨ ⟩ modeled as a resistor located at the current antinode. At -112 dBm input power, we measured the change in Is

⟨ ⟩ as a function of input frequency for two different gate voltages, 0.1 and 1.3 V, with Is of 0.94 and 1.17 µA respectively (see Fig. 7.3b). If the absorbed microwave photon caused a resonant excitation at the plasma frequency

⟨ ⟩ of the GJJ, we would expect the frequency of the resonance dip in Is to shift by approximately 1 GHz, because the GJJ plasma frequency is proportional to the square root of the critical current. In contrast, the suppression

⟨ ⟩ of Is aligns closely to the microwave resonant frequency measured by reflectometry. Therefore we cannot at-

⟨ ⟩ tribute the suppression of Is to resonant excitation of the GJJ. We note that the linewidths in Fig. 7.3b match to the one given by the loaded quality factors of 9 and 13 at Vgate = 0.1 and 1.3 V respectively, obtained from the fitting of the phase of the scattering parameter in Fig. 7.3a.

7.4 GJJ Microwave Response

Supercurrent switching statistics can reveal the basic properties of the GJJ103,186 and hence its thermal response as a bolometer. We measure the distribution of Is by recording the potential drop across the GJJ while sweeping the bias current for 6,000 times per gate voltage, input power, and temperature. The decay of the GJJ from the supercurrent state to the normal state is stochastic and the typical distribution is plotted in Fig. 7.4a at two temperatures, 0.19 and 0.45 K, at Vgate of 0.1 V without power input. The decay rate (also known as the escape rate from the tilted-washboard potential in the resistively and capacitively shunted junction model), and thus the switching probability, can be determined uniquely from the distribution using the Fulton-Dunkleberger method.103,186 When the experiment is conducted at 0.19 K with an increasing power input, the switching histogram shifts gradually to lower values. When the microwave input power reaches 126 fW when the device is at

0.19 K, the distribution overlaps well to that at 0.45 K with zero input power and therefore the GJJ has the same

⟨ ⟩ decay rate under these two conditions. This suggests the suppression of Is is due to the heating of graphene electrons from 0.19 K to 0.45 K by the microwave input power, instead of other mechanisms such as the AC

82 Figure 7.4: Sensitivity and the fundamental fluctuation limit of the bolometer. (a) Distributions of the switching current at 0.19K(blue circles) and 0.45 K (green circles) without microwave input, and at 0.19 K with microwave input power of 126 fW (red crosses) at Vgate = 0.1 V. (b) Average switching current as a function of microwave input power at 7.9 GHz at various gate voltages. (c) Interpolated graphene electron temperature using the results in Fig. 7.2c with 0.19 K offsets for clarity. The dashed lines are fits to the theory of heat transfer from electrons to phonons. (d) Measured noise equivalent power (NEP) and the fluctuation limit. The averaged NEP is0.7 ± 0.2 aW/Hz1/2, corresponding to an energy resolution of a single 32 GHz photon.

Josephson effect or an additional bias current across the GJJ.196

⟨ ⟩ ⟨ ⟩ Is decreases monotonically as we raise the microwave input power Pmw. Fig. 7.4b plots Is as a function of

⟨ ⟩ input power. Using the measured Is at various device temperatures in Fig. 7.2b, we can use an interpolation

⟨ ⟩ to calculate graphene electron temperature Te as a function of Is which is a function of Pmw. The resulting

Te(Pmw) are shown in Fig. 7.4c for four different gate voltages with offsets in multiples of 0.19 K in they-axis for clarity. The dashed lines plot the fits to the data using the electron-phonon heat transfer equation:156,164,192 δ − δ P = ΣA Te T where A is the area of the monolayer graphene, and Σ and δ are the electron-phonon coupling

− − parameter and its temperature power law, respectively. The best fitted Σ are 2.14, 2.04, 2.74, and 3.30 Wm 2K 3

83 in the ascending order of gate voltages with δ = 3. This temperature power law corresponds to the cooling of graphene electrons mediated by supercollision or disorder.164,192–194 However, using the deformation potential of 20 eV and measured electron mobility of 20000 cm2/Vs from the GJJ normal resistance versus the gate volt-

− − − − age, the same theory predicts a Σ of 0.0086 Wm 2K 3. We note that Σ ranges from 2.04 to 9.9 Wm 2K 3 in three devices and agrees quantitatively with another report.197 This large discrepancy suggests that the existing electron cooling theories of the defect-mediated electron-phonon coupling are not applicable when lmfp (340 nm in our sample) is larger or comparable to the sample dimensions (0.3 µm by 2.6 µm in our device). A recent scanning nanothermometry experiment198 has spatially imaged the cooling of electrons in high-quality graphene and demonstrated that the cooling of electrons can be dominated by the atomic defects on the edge rather than those in the bulk. Therefore, we can expect lmfp based on the bulk electrical transport may underestimate the total cooling rate of the electron-phonon coupling when the lmfp is larger than the sample size. Since these scatterings by atomic defects on edge scales with the sample perimeter whereas the defect-mediated scattering scales with sample area, more systematic experiments with consistently etched graphene flakes of different sample aspect ratios can provide more understanding of the cooling of electrons to achieve a higher-sensitivity graphene-based bolometer in the future.

The effectiveness of the thermal insulation at the graphene-superconductor contacts due to Andreev reflection can be evaluated using the Wiedemann-Franz law. If there is any heat diffusion at the contacts, the GJJ

(being wide and short) will have the largest contribution to the thermal conductance which, based on the one-

31 2 dimensional thermal model, is given by 4(πkB/e) (T/R) where kB is the Boltzmann constant, e is the electron charge, and R is the electrical resistance between the contacts. For Rn = 145 Ω at Vgate = 1.9 V, this would be

δ−1 about 387 pW/K and is 1000 times larger than the measured thermal conductance Gth = δΣAT , i.e. 230 fW/K, at 0.19 K. This suggests the NbN superconductor used in the experiment is acting as a good thermal insulator, prohibiting the heat diffusion at the graphene-superconductor interface.

84 7.5 Bolometer Performance

We now study the sensitivity of our detector. When operating as a bolometer, we can infer the input power from

the suppression of Is through the calibration in Fig. 7.4b. The standard deviation of the Is distribution, σIs , sets the uncertainty of a single Is measurement and thus the minimal detectable power

− = · | ⟨ ⟩/ | 1 δPmin σIs d Is dPmw Pmw=0 (7.1)

i.e. δPmin = 11.4 fW for σIs of 13.2 nA at 0.19 K and Vgate = 1.9 V. To estimate the NEP, we consider the time duration required to detect this δPmin by comparing three time scales: resonator input coupling rate, resonator dissipation rate, and thermal time constant τth. Analyzing the scattering parameter of the resonator shows the coupling and internal quality factors of 12.8 and 10.0, respectively (see Section 7.6). The resonator is nearly critically coupled with coupling rate and dissipation rate Δfint of 630 and 790 MHz, respectively. The thermal time

156 constant is given by the ratio of the graphene electron heat capacitance Ce to Gep. Since Ce = AγT where

5/2 2 1/2 γ = (4π kBn )/(3hvF) is the Sommerfeld coefficient with n as the electron density, h as Planck’s constant,

≃ × 12 −2 ∼ and vF as the Fermi velocity of electrons in graphene. At 0.19 K and n 2 10 cm , Ce 10 kB resulting

≃ in τth 0.6 ns. Therefore, the fastest GJJ detection time is bounded by the rate at which the resonator dissipates p energy into the graphene such that our GJJ bolometer has a NEP of δPmin/ Δfint. The result is plotted in Fig.

⟨ ⟩ 7.4d and the error bar is dominated by the accuracy in obtaining d Is /dPmw and Δfint. The NEP achieved by

± 1/2 this SGS bolometer is, on average across different Vgate, 0.7 0.2 aW/Hz .

Compared to the state of the art, our GJJ shows promise for a range of applications. The GJJ bolometer can

∼ 5 operate 10 times faster than the nanowire for its shorter τth, making GJJ bolometer an attractive component for ultrawide-IF-bandwidth hot-electron-bolometric mixing. It also has a much lower energy resolution,

188 29,31,139,141 equivalent to a single-32 GHz-photon energy, because of the small Ce. Unlike the superconducting-

85 qubit-based and nanowire SPDs, the GJJ detector does not require qubit state preparation nor does it rely on the breaking of Cooper pairs to generate a detectable signal, making it suitable for continuous photon sensing over a wide photon energy range.

Intrinsic thermal fluctuation of a canonical ensemble imposes a fundamental limit on the sensitivity ofa p 2 188 bolometer given by 4GthkBT . Comparison of the data in Fig. 7.4d suggests that the NEP of our bolometer as predicted by such fluctuation (based on the measurement of the electron-phonon coupling) is in close agreement to the NEP that we measure using the suppression of switching current resulting from microwave input power. This also suggests that 1/τth is nearly the same as the internal dissipation rate of the resonator. The same

− temperature scaling law projects a further improvement to 10 21 W/Hz1/2 at 20 mK. The same detector design

⟨ ⟩ 13 could perform calorimetry to detect single microwave photons with further optimization of d Is /dT. For a continuous power readout while keeping the GJJ non-dissipative in the supercurrent state, we can employ an RF resonance readout to detect the change of the Josephson inductance of the GJJ.

7.6 Methods

Device fabrication. The device is fabricated by first encapsulating the monolayer graphene between two layers of atomically flat and insulating boron nitride≃ ( 30 nm thick) using the dry-transfer technique. The superconducting terminals consist of 5-nm-thick niobium and 60-nm-thick niobium nitride deposited after reactive ion etch and electron beam deposition of 5-nm titanium to form the one-dimensional contact.13,147 Finally, we make the local gate to control the carrier density of the monolayer graphene by growing an aluminium oxide dielectric layer by atomic layer deposition and depositing a layer of gold electrode, before wiring it through the reactive low-pass filter to provide isolation to the microwave circuit. We note that the simultaneous functioning of the Josephson junction and microwave impedance matching of the device can limit the fabrication yield to about 14% when duplicating the process on a new set of cleanroom equipment.

86 Impact from contact resistance on detector performance. We can consider how the contact resistance may impact the device (1) on the Josephson junction measurement and (2) on the microwave resonator. Nyquist noise by the graphene/NbN contact in the junction direction (vertical direction in Fig. 7.1b-d) would be absent as the bolometer operates in the supercurrent regime where the two-probe resistance is zero. Nyquist noise would come into play only after the JJ switches to the resistive regime by microwave photon absorption and gives finite normal resistance of Rn. Therefore, we do not need to consider Nyquist noise for determining the NEP of the bolometer. The noise due to thermal and quantum fluctuation on the current-biased Josephson junction are included in determining the NEP because both of these noise processes contribute to the width of the switching current distribution in Fig. 7.4a. They are the mechanisms in the thermal activation and macroscopic quantum tunneling of the phase particle of the current-biased Josephson junction99 and can be determined using the

Fulton-Dunkleberger method.169

On the other hand, it is possible that the contact resistance between the graphene flake and microwave resonator degrade the bolometer by dissipating the photon energy at the contact instead of the graphene flake. To estimate this effect, let the graphene resistance along the resonator direction be Rn,res and the graphene/NbN contact resistance along the resonator (horizontal direction in Fig. 7.1b-d) be Rc. We have measured normal re-

≃ sistance Rn 145 Ω along the junction direction that has width W = 1 µm and length L = 270 nm, so square

′ ′ resistance Rsq = Rn/(L/W). With the graphene width W = 300 nm and length L = 2.6 µm along the res-

∗ ′ ′ ≃ onator direction, we can roughly estimate Rn,res = Rsq (L /W ) 5 kΩ. If we assume that the contact transparencies for graphene/NbN interfaces along the junction direction and along the resonator direction are similar since the NbN for both the GJJ and the resonator was deposited at the same time, Rc can be estimated by

′ ∗ 2 ≃ Rc = (π/kFW ) (h/4e )/T 0.4 kΩ, where kF is Fermi wavenumber and T = 0.8 is a contact transparency

∗ 2 estimated from the relationship of Rn = (π/kFW) (h/4e )/T for a ballistic graphene channel along the JJ direction (see the next paragraph for the discussion on ballistic nature of graphene in our experiment). Contact

87 2 0.1 V 1.3 V 1

-1 Phase(rad.)

-2

-3 6 6.5 7 7.5 8 8.5 9 9.5 Frequency (GHz)

Figure 7.5: Loaded Q factor of input resonator. Fitting of the loaded quality factor of the microwave resonator. Shown is the phase of S11 scattering parameter of the half-wave microwave resonator at two different gate voltages. Data from the same dataset as in Fig. 7.3a.

contribution to the photon energy dissipation by Rc is less than 10% of total resistance given by Rn,res + Rc. Thus, we expect that the contact resistance would not significantly degrade bolometer performance.

Electron mean-free-path. The graphene-based Josephson junction is at or nearly at the ballistic limit. This is because if we assume the graphene is in the diffusive regime, the Drude mobility and mean free path of graphene are estimated to be 20,000 cm2/Vs and 340 nm, respectively. However this mean free path exceeds the junction length of 270 nm. This is usually the case when graphene is encapsulated by atomically flat and insulating hBN flakes and protected from environmental contamination during the fabrication processes. Ref.[199] describes how the formation of a moiré superlattice with the hBN substrate can give rise to the unusual rise of Rn at around 2 to 3 V of Vgate.

Design of input resonator. We design the device for optimal impedance matching to a 2 kΩ graphene resistance, an estimated value based on its dimensions. Energy dissipation is dominated by Joule heating into the graphene in such a structure, since the typical internal Q-factor of NbN superconducting resonators without a graphene flake is on the order of 105 − 106, compared to the internal Q-factor of our device measured to be less than 30, based on the circle fitting method200 and the loaded Q-factor fitting (see Fig. 7.5). We achieve optimal impedance matching at critical coupling, where the resonator internal Q-factor due to the graphene resistance is equal to the coupling Q-factor, by adjusting the coupling gap capacitor. We simulate the device with different

88 (a) 20 (b) 0

10 -5

0 -10

-10 (nA) (nA) Device T

s -15

s I I T = 0.2 K -20 Test power = Device H -15 dBm -20 T = 0.3 K V -30 Test power = -15 dBm gate 0.0 V 3.6 V -25 -40 5.0 V 8.0 V

-50 -30 6 7 8 9 10 11 12 8.4 8.6 8.8 9 9.2 Input Freq. (GHz) Input Freq. (GHz)

Figure 7.6: GJJ bolometer input resonator. Suppression of switching current at the resonance frequency of the input resonator for Device H (a) and T (b) with test power of -15 dBm applied outside the cryostat at 0.3 and 0.2 K, respectively. See Table 7.1 for the dimensions and measured parameters of the devices. gap capacitor values using a Method of Moments electromagnetic simulator and determine a coupling capacitor value of 200 fF. In addition to the reflectometry (Fig. 7.3a), we can also measure the input resonator frequency by monitoring the Is as shown in Fig. 7.6.

Electron-phonon coupling. We can use the electron-phonon coupling theory in the supercollision or disorder regime to calculate Σ:164,192

2ζ(3) E D2k3 = F B Σ 2 3 ~4 2 (7.2) π vFρM lmfps

where ζ is the Zeta function, EF is the Fermi energy of graphene charge carriers, vF is the Fermi velocity in graphene,

D ~ ρM is the mass density of the graphene sheet, is the deformation potential, kB is the Boltzmann constant, is the reduced Planck constant, lmfp is the mean-free-path, and s is the sound velocity of graphene lattice. However, the enhanced electron-phonon cooling that we observed is more likely due to the resonant-scattering by defects located around the edge of graphene flake.197,198,201 Σ fitted values are listed in Table 7.1 based on the data in Fig.

7.7 and 7.8.

Σ is independent of charge carrier density if we assume the electron mobility μe is a constant of carrier density

89 such that, with e as the electron charge, and τ and m as the scattering time and mass of the charge carriers, respectively, μe = eτ/m = evFlmfp/EF. Measured electrical transport and Josephson junction parameters are listed in

Table 7.2.

(a)2.2 (b) 0.6 0.04 K 0.10 K 2 0.5 0.20 K 0.30 K 0.40 K

1.8 0.4 0.50 K A)

A) 0.60 K ( ( 0.70 K

s 0.3

s I I 1.6 0.2

1.4 0.1

1.2 0 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 V (V) T (K) gate

Figure 7.7: GJJ switching current. Average switching current of the Josephson junction for Device H (a) and T (b).

1 (a) (b)

0.2 K data 0.2 K data 0.3 K data 0.3 K data 0.5 0.2 K fit 0.2 K fit 0.3 K fit

0.3 K fit (K)

e 0.3 T

0.2

12 -2 Device H at ~0.72 10 12 cm -2 doping density Device T at ~3.2 10 cm doping density

0.1 10 0 10 1 10 2 10 3 10 0 10 1 10 2 10 3 10 4 Input Power (fW) Input Power (fW)

Figure 7.8: Electron cooling. Interpolated graphene electron temperature versus input power for Device H (a) and T (b) with carrier − − density ∼ 0.72 × 1012cm 2 and ∼ 3.2 × 1012cm 2, respectively. The lines are fits using electron-phonon heat transfer theory.

90 Device A Device H Device T superconductor NbN MoRe MoRe graphene area (μm2) 0.78 1.00 1.40 input frequency (GHz) 7.9 9.75 8.82 temperature (K) 0.2 0.2 0.3 0.5 0.2 Vgate (V) 0.1 1.3 1.9 3.1 ∼ 1 6 7 8 ∼ VCNP (V) -0.9 -1 3.6 − carrier density (1012 cm 2) 0.72 1.6 2.0 2.9 ∼0.72 1.7 2.5 3.2 Rn (Ω) 160 127 145 195 150 249 225 205 Ce (kB) 6.1 9.0 10 12 6.1 9.7 16 10 12 14 − − Σ (Wm 2K 3) from fitting 2.1 2.0 2.7 3.3 15 9.9 6.3 6.8 9.5 9.6

91 Gth (pW/K) 0.181 0.173 0.231 0.279 1.80 2.68 4.75 1.1 1.9 1.6 ⟨Is⟩ (μA) 0.943 1.17 0.978 0.714 1.932 1.887 1.832 0.349 0.508 0.582

σIs (nA) 15.0 23.7 13.2 9.96 25.8 26.2 21.4 3.35 4.21 4.48 6 |d⟨Is⟩/dP| (10 A/W) 1.1 1.5 1.2 0.43 0.18 0.11 0.081 0.014 0.022 0.018 Qint 18.3 28.5 10.0 7.9 ∼39 190 228 242 ∼ Qcouple 18.4 24.4 12.8 9.7 39 25 25 26 resonator internal dissipation rate (MHz) 432 277 790 1000 ∼250 46 39 36 − − thermal fluctuation limited NEP (×10 19 W/Hz 1/2) 0.60 0.59 0.68 0.75 2.0 3.6 8.1 1.59 1.87 1.84 − − estimated NEP (×10 19 W/Hz 1/2) 6.5 9.6 4.1 7.4 8.9 15 17 35 28 40 ratio of the estimated NEP to the 1.1 1.6 0.6 1.0 4.9 4.2 2.0 22 15 22 thermal fluctuation limited NEP

Table 7.1: List of parameters to estimate NEP and thermal properties of GJJ bolometer in Chapter 7. Josephson junction parameters of Device A at Vgate = 1.9 V JJ channel length 300 nm JJ channel width 1 μm Electron density 2.0 × 1012 cm−2 Electronic mobility 2 × 104 cm2/Vs normal resistance 59 Ω Mean free path 340 nm Disorder temperature 2.8 K Bloch-Grüneisen temp. 76 K Ic(T0)Rn product 142 μeV Thouless energy 1.2 meV JJ coupling energy 2.0 meV Effective capacitance 3.65 fF Plasma Freq. at zero bias current ≤ 904 GHz McCumber parameters 0.23 NbN superconducting gap 1.52 meV

Table 7.2: List of Josephson junction parameters of Device A in Chapter 7.

92 8 Detecting Single Infrared Photons with a GJJ

Here, the focus shifts from graphene physics to Josephson junction physics. Simply shining 1550-nm light on a

GJJ without optimizing the coupling, I was able to demonstrate that the GJJ was sensitive to single near-infrared photons. The response of the GJJ to the light followed Poisson statistics, was linear in power, and did not match the response expected for steady state heating. However, as I mentioned in Chapter 1, I initially thought, based on the modeling of Chapter 5, that the graphene would dominate this near-infrared single-photon response of the GJJ and was surprised when the data did not support that model. The first hint of this discrepancy was that the photon response seemed to improve as the gate voltage increased (Section 8.16). If the photon response were dominated by the thermal response in graphene, it would have instead decreased as the gate voltage, and

93 thus carrier density and heat capacity, increased. A higher heat capacity at higher gate voltage would mean less photon-induced temperature rise and therefore less chance for the GJJ phase particle to escape. The second set of data that turned me away from graphene physics toward Josephson junction physics was an experiment with tetralayer “graphene” (Section 8.14). In this experiment, the tetralayer-graphene device demonstrated nearly an identical single-photon response to that of a single-layer graphene device. As more pieces of evidence came together, it became clear that the absorption was dominantly occurring in the NbN contacts of the GJJ and not in the graphene weak link. We confirmed this conclusion with polarization-dependent measurements and modeling

(Figure 8.4). Photon absorption in the superconducting contacts close to the graphene interface breaks Cooper pairs which creates non-equilibrium quasiparticles. These quasiparticles form a diffusion current from which arises a shot noise. The shot noise can excite the phase particle causing it to escape and producing the voltage spike measured in our setup. Here I present the results that led me to this model of the infrared single-photon response of a GJJ. Further details of the experimental setup are described in Appendix A.

8.1 Abstract

Josephson junctions (JJ) are the basis of high sensitivity magnetometers and voltage amplifiers that can reach quantum-limited performance. JJ square law detectors for electromagnetic radiation have been studied since the discovery of the JJ. However, they have never been pushed to the limit of single optical photon detection with some single photon detectors even requiring the suppression of the Josephson effect. Here we demonstrate single near-infrared photon induced switching of a current-biased graphene-based JJ, by coupling light locally at the junction with surface plasmon-polaritons. From the switching statistics and polarization dependence, we reveal the role of quasiparticles generated from the absorbed photon in the switching mechanism. The photon-sensitive

JJ would enable a high-speed, low-power consumption optical interconnect for future JJ-based computing architectures.

94 8.2 Introduction

Exploiting its macroscopic quantum behavior, the Josephson junction (JJ) is arguably the most important superconducting device with its wide array of applications. It can perform as a high sensitivity magnetometer based on the Superconducting Quantum Interference Device (SQUID),202 as the voltage standard,203 and as a quantum noise limited microwave parametric amplifier.204 The idea to detect electromagnetic radiation by JJs has also been pursued since their earliest realizations.205–207 Despite achieving a high sensitivity,208 however, it is not clear whether a JJ can detect a single optical photon.209 In fact, a superconducting tunnel junction (STJ)210,211 can only be single-photon sensitive by deliberately suppressing the Josephson coupling with an external magnetic field. Similar to superconducting nanowires,20,21 transition edge sensors,212 kinetic inductance detectors,213 microbolometers,214 and quantum capacitance detectors,7 the operation of the STJ relies on the generation of quasiparticles (QPs) as the incident photon breaks Cooper pairs. This mechanism is analogous to the electron multiplication process in an avalanche photodiode where electron-hole pairs generated from single-photons are amplified to a sizable electrical current by accelerating the electrons in an electric field. Being a high sensitivity

QP sensor,215 the JJ should in principle be able to detect single-optical photons. Studying JJs under the influence of the QPs generated by a single-photon can provide insights into designing a longer lifetime qubit, as energized stray radiation, such as cosmic rays, is always present to produce QPs.216 In addition to the quantum computer, the JJ is also the basic component of the rapid single flux quantum circuitry for cryogenic high performance computing.217,218 A single-photon sensitive JJ naturally provides a high speed, low power consumption optical interconnect that is readily integratable into these computational systems. Here, we report an experiment to demonstrate the switching of current-biased JJs by individual near-infrared (NIR) photons. By coupling photons to a graphene-based proximity JJ using plasmonic resonance,219 we measure the single-photon induced switching as a function of current bias, temperature, photon rates, and polarization. Our analysis indicates the JJ switching is

95 (A) (D) 0.5 infrared niobium nitride Ib direction superconductor photon quasiparticles

0 (mV) JJ

graphene V

-0.5 -10 0 10 h-BN I ( A) b (B) graphene (C) (E) 0.2 Dark U TA 0.1 Dev. A 0 ∆U (mV) 0.2 100 pW MQT JJ V Dev. B 0.1 3 µm Increasing I 0 b φ 0 4 8 Time (s)

Figure 8.1: Detecting single-NIR-photons by Josephson junction (JJ) (A) Illustration of the switching of JJ induced by single-photon. Quasiparticles (QP) are generated as incident photons break Cooper pairs and trigger the JJ to switch. (B) Optical image of two graphene- ◦ based JJs oriented at a 90 angle. Each JJ is made of a hBN-encapsulated graphene strip of length 160 nm by width 2.8 µm contacted on each end by NbN pads of the same width and 1 µm length. (C) Current biased JJ can be described as a macroscopic quantum phase particle subjected to a tilted-washboard potential. The phase particle can be driven into the motion, representing the normal resistive state, from the stationary state, representing the supercurrent state, by thermal activation (TA) or macroscopic quantum tunneling (MQT). Ad- ditional noise from the QP diffusion can enhance the state transition and thus enhance the JJ switching probability. (D) Typical IV curve displaying hysteresis likely due to self-heating. (E) Switching events in the absence (top) and presence (bottom) of 100 pW of 1550-nm

light at Ib = 10.90μA.

caused by the QPs produced from the absorption of a single photon in the superconductor, clearly distinguished

from resonant excitation and bolometric effects.12,14,152,196

We investigate the photon-JJ interaction by illuminating the JJ at 27 mK with a 1550 nm NIR laser (schemat-

ically shown in Fig. 8.1A), brought via a single-mode optical fiber after filtering any unintended room tempera-

ture radiation, into the dilution fridge. The junction (an optical image in Fig. 8.1B) is fabricated by depositing

superconductor, i.e. 5-nm niobium (Nb) and 50-nm niobium nitride (NbN) with 5-nm titanium as an adhesive

layer, on the sides of graphene which is encapsulated between two atomically flat and insulating hexagonal boron

nitride layers. In contrast to traditional trilayer JJs, the lateral JJs expose and couple directly to the NIR photons.

When a single NIR photon is absorbed into the superconductor, it will break Cooper pairs to generate QPs. The

diffusion of QPs across the JJ can act as a noise source to switch the current-biased JJ. We can understandthe

96 probability of QP-induced JJ switching by the resistively and capacitively shunted junction (RCSJ) model.167

Using the phase difference (Δφ) between the two superconducting electrodes, the RCSJ model describes JJs by a macroscopic quantum phase particle subjected to a washboard potential (Fig. 8.1C). When the phase particle is trapped initially in a local minima, i.e. dφ/dt = 0, the voltage drop across the JJ is zero. The bias current running through the JJ tilts the washboard potential and the phase particle could escape from the metastable minimum. When it escapes, either through thermal activation (TA)99 over or macroscopic quantum tunneling

(MQT)98 through the barrier (ΔU), the voltage drop across the JJ becomes finite and the JJ switches to the normal resistive state. To study the JJ switching induced by single-photons, we have prepared and studied seven JJs of different widths and graphene thicknesses (Section 8.7). All of them can detect single-photons; we will present results mostly from Device A and compare it against other junctions as controls to study its interaction with NIR photons and the mechanism that leads to JJ switching.

Fig. 8.1D shows a typical JJ current-voltage characteristic without photon illumination. The junction switches

∼ from the zero-voltage to the resistive state with normal resistance Rn = 44 Ω at the switching current Is 11μA

≃ as we ramp up Ib. It returns back to the supercurrent state at a retrapping current Ir 6.64μA as we ramp down

14,149 Ib. This hysteretic behavior, i.e. Is > Ir, is frequently observed in graphene-based JJs at low temperatures

220 due to self-heating by Ib in the normal state rather than to JJ underdamping. This hysteresis is useful to our investigation because we can apply a constant current bias to the JJ so it will latch into the resistive state after switching. We can then reset the bias current again and over time, measure the total switching count under different light intensities and conditions. As Fig. 8.1E shows, there were considerably more switching events with even just 100-pW of illumination, which is equivalent to 85.5 photons per second in a 1 µm2 area, calculated from the

Gaussian beam profile (Section 8.8).

97 Figure 8.2: Switching of Josephson junction by single-NIR-photon. (A) The observed switching events under illumination follow Poisson statistics showing the events are uncorrelated with one another as is expected for single-photon switching. Shown from top to bottom are switching events for 0, 50, 100, and 200 pW of illumination and the photon rate in a 1 µm2 area is 85.5 Hz/100 pW. Inset: The variance 2, i.e. the photon shot noise, equals to the mean (solid line shows the theory without fitting parameter) of distribution. (B) The σΓ μΓ Is switching rate as a function of power for bias currents of 10.75 (blue), 10.83 (green), and 10.88 µA (orange). At low power, the curves are dominated by the dark count rate, followed by a region of linearity where single-photon events are dominant. Solid lines are the best fit of the data up to square laser power dependence. Dash line shows the direct proportionality as a guide to eyes.

8.3 Evidence of Single-Photon Detection

The seemingly random switching with light illumination actually follows the statistics of photon shot noise. Af- ter taking 104-s time traces, similar to those in Fig. 8.1E, at various powers, we divide them into 1-s bins, count the number of switching events in each bin, and produce histograms in Fig. 8.2A. A Poisson distribution (solid lines in Fig. 8.2A) traces the experimental data closely (bars in Fig. 8.2A) as expected for uncorrelated single- photon events from a coherent source. The inset shows the variance, i.e. photon shot noise, is approximately equal to the mean of the Is distribution without any fitting. Furthermore, the average number of events per1-s bin, i.e. switching rate Γmeas, is linearly proportional to the number of incident photons (laser power) with an offset due to the false positive (dark) count. We study this property over a larger range of laser powers andatsev- eral values of Ib (Fig. 8.2B). A higher Ib produces a higher Γmeas, thus a higher detector efficiency. Γmeas is linear over about 2 orders of magnitude in laser power for a low bias current of 10.75 µA. At higher laser powers, Γmeas

98 Figure 8.3: Switching mechanism. (A) The distribution of switching current Is under three conditions: (i) at 27 mK in the absence and (ii) presence (yellow) of 100 pW of 1550-nm light, and (iii) at 325 mK in the absence of light. Although the 27-mK, 100-pW distribution has the same average as the 325-mK distribution in the dark, the shapes of the two distributions are very different, demonstrating that the JJ is not undergoing simple heating. (B) The switching rate as a function of bias current in dark and for various incident laser powers. The points are measured values while the black line is the best fit of MQT process to Γmeas in the dark and the color lines are modeled results from augmented Γesc due to enhancement from the absorbed single photon. increases super-linearly with laser power, indicating that the heating of the JJ by laser power is becoming non- negligible. The observed Poisson distribution and the linear dependence of Γmeas on the laser power, together, show that an individual NIR photon can switch the JJ.

It is imperative to distinguish our observation from the case of steady heating of the junction by the laser power. To rule out this possibility, we compare the distribution of the switching current at different laser powers and operating temperatures. Fig. 8.3A shows the histogram of Is from 1000 measurements as we ramp up repeatedly the bias current at 1 µA/s. The spread Is distribution reflects the probabilistic nature of the phase particle escaping from the metastable state in the tilted-washboard potential, thus determining the escape rate in the

(dark) 169 (dark) dark Γesc . Fitting Γesc to TA and MQT mechanisms will decide the JJ critical current Ic and coupling en-

~ ~ ergy EJ = Ic/2e with and e being the reduced Planck constant and electron charge, respectively. Therefore we measure and compare the Is distributions under three conditions: (i) at 27 mK without light, (ii) at 27 mK with

⟨ ⟩ ⟨ ⟩ ⟨ ⟩ light, and (iii) at 325 mK without light. 325 mK is chosen such that Is (ii) = Is (iii). While both Is (ii) and

⟨ ⟩ Is (iii) are suppressed equally, they do not have the same shape distribution. This is in sharp contrast to previous

99 experiments in which JJs under photon illumination have Is distributions matching those of JJs with elevated

14,209 temperature. The difference of Is distributions in Fig. 8.2 suggests the switching mechanism observed is not due to a steady heating of the JJ.

8.4 Switching Mechanism

To study the switching mechanism, we measure and compare Γmeas versus Ib with and without illumination (Fig.

(dark) 169 8.3B). Γmeas in the dark is equal to Γesc , which is nominally extracted from the Is distributions (Fig. 8.3A), and we verify the two measurement methods agree with one another (Section 8.11). Up until about 3 kHz, lim-

(dark) ited by the low pass filters along the DC bias circuit for reducing electrical noise to theJJ, Γmeas in the dark in- √ ≃ − creases with Ib due to a stronger escape tendency of the phase particle as the barrier height ΔU (4 2/3)EJ(1

3/2 (dark) Ib/Ic) decreases when the washboard potential tilts further. The nearly linear dependence of Γmeas in the

( ) − dark ∝ ΔU/kBTesc semilog plot agrees with an escape rate Γesc Ae with kB and Tesc being the Boltzmann constant and escape temperature, respectively. This is a characteristic exponential form in reaction-rate theory with an

221,222 activation energy of kBTesc. For TA escape, kBTesc is given by the thermal energy whereas for MQT escape,

≃ ~ by the energy of the harmonic oscillator of the tilted-washboard potential, i.e. kBTesc ωP/7.2(1 + 0.87/Q) with ωP and Q being the JJ plasma frequency under current bias and the quality factor of the harmonic potential,

(dark) respectively. Γmeas is best fit by the MQT theory (black solid line) with Ic of 11.99 µA such that ωP0/2π = 225

GHz and Q ≃ 1 (Section 8.15).

(dark) (light) (light) In contrast to Γmeas , Γmeas in the light does not directly measure Γesc , i.e. the escape rate enhanced by the

(light) ≫ (dark) absorbed single photon. For a brief moment Δt1 when Γesc Γesc , the escape probability of the phase R P(light) = − exp(− Δt1 (light) ) (light) particle is esc 1 0 Γesc dt . To understand qualitatively Γmeas , we can consider a constant (light) P(light) ≃ (light) ≪ (dark) Γesc during Δt1 and focus on the lower Ib regime where esc Γesc Δt1 1. With negligible Γmeas ,

(light) ≃ (light)R ≃ (light) R Γmeas Pesc photon Γesc Δt1 photon where Rphoton is the rate of photon absorption by the JJ. Since Δt1

100 (A)90 (B) (C) (D) 1 direction direction thin film ⊥ E k E ⊥ JJ-A ⊥ 0.5 k JJ-B (x=d)

0 180 ⊥ k 0 0.6

k 0.4 measured value 2.0 0.2 I d d |E| (a.u.) d d b

PolarizationRatio 0 270 1.0 0 0.2 0.4 0.6 0.8 1 distance d ( m)

Figure 8.4: NIR photon coupling into Josephson junction. (A) The polarization dependence of Γmeas for Device A and B (same orientation as the optical image in Fig. 1B) are orthogonal to each other, demonstrating that the single-photon switching is due to photon absorbed right by the JJs. (B-C) The simulated electric field strength for incident light perpendicular and parallel to the supercurrent direction, respectively. (D) Upper panel: averaged volumetric NIR absorption α of the two polarizations versus distance x from the in- terfaceR between the grapheneR and superconductor. Dashed line is the value of α for thin film NbN. Lower panel: the polarization ratio x x [ ( ⊥ − ∥) ]/[ ( ⊥ + ∥) ] versus . The dotted line marks the measured ratio 33% from (A), implying an effective absorption 0 α α dx 0 α α dx x length of 190 nm on either side of the graphene.

(light) (light) and Rphoton are Ib independent, the nearly linear Γmeas in the semilog plot suggests Γesc follows the reaction-

∗ rate theory as the single-photon-induced JJ switching can be described by an enhanced activation energy, kBTesc, and Δt1. More quantitatively, we fit the data using two parameters accounting for MQT, TA, and phase diffusion

105 R processes (Section 8.15) with photon = 53 Hz per 100 pW based on the photon absorption simulation. Color

∗ ≃ ≃ solid lines in Fig. 8.3B are the results with Tesc 2.1 K and Δt1 0.86 ns.

8.5 Absorption

To understand how the JJ absorbs single photons, we study the dependence of Γmeas on the polarization of incident light. Fig. 8.4A plots this angular dependence for two equally-sized JJs oriented orthogonally to one another on the same substrate (Fig. 8.1B) with arbitrarily oriented superconducting electrodes connecting the JJs to wire-

◦ bonding pads. Both devices exhibited the same photon rate but offset by 90 to each other. This dependence on junction geometry suggests that the detected single photons are absorbed at the JJs; not the electrodes. By inserting and rotating a polarizer in front of the JJs in a series of experiments (Section 8.12), we find the maximum

⊥ Γmeas coincides with the polarization perpendicular ( ) to the JJ supercurrent flow. Since the AC Josephson ef-

101 fect and photon-assisted tunneling requires the electric field in the same direction as the supercurrent flow, this result eliminates them as the JJ switching mechanism.

Using ANSYS HFSS, we study the angular dependence by computing the volumetric photo-absorption for the entire junction area with an incident plane wave approximation because the JJ is much smaller than and located at the center of the Gaussian beam waist of 2.41 mm. In Fig. 8.4B and C, we plot the field distribution within the NbN. When the real part of the permittivity of NbN and its adjacent medium have opposite signs, they form effectively a capacitor-inductor network that supports a localized surface plasmon along the interface aligned with the incident electric field, resulting in an intensified field distribution.223 Thus the graphene-NbN interface demonstrates such enhancement only when the E-field is in the ⊥ direction.

We can estimate the effective area of the JJ as a single-photon detector by comparing the measured againstthe simulated polarization ratio. By defining volumetric absorption zones (dashed line in Fig. 8.4B and C) at distance x from the graphene-NbN interface with area 50 nm by 2.8 μm, Fig. 8.4D plots the spatial dependence of photon

⊥ ∥ absorption coefficient αi(x), where i = , notates the polarization direction. α⊥ and α∥ differ considerably near the edge due to the surface plasmon, but approach the same value further away from the edge. This agrees with the expected photo-absorption of 50 nm thin film NbN, α = 0.27 (dashed line)122 and verifies our calculation.

R Photons could be absorbed directly into the graphene with an expected photon of 0.07/s and 0.27/s with 100

⊥ ∥ pW of laser power in the and directions, respectively. This is much smaller than Γmeas and orthogonal to the measured polarization dependence thus excluding the detected single photons being absorbed in the graphene. R R d d [ ( ⊥ − ∥) ]/[ ( ⊥ + ∥) ] We define the polarization ratio as 0 α α dx 0 α α dx , that sums up the photon absorption with increasing distance d measured from the graphene-NbN contact. It can be compared directly with the ratio of

⊥ ∥ P(light) Γmeas in -to- direction because it depends only on α⊥/α∥; not on the intrinsic quantum efficiency, i.e. esc .

To match the 33% polarization ratio in Fig. 8.4A, the simulation suggests the photons absorbed within 190 nm of the edge are those that trigger the JJ switching. Therefore the total effective area of the JJ as a single-photon

102 2 ⊥ R detector is about 1.06 µm with an averaged α of 0.58 in the polarization direction, resulting in the photon value used for data fitting in Fig. 8.3B.

8.6 Physical Model

Analogous to the QP-induced qubit relaxation, we argue that QPs are mediating the current-biased JJ switching.

When a single NIR photon impinges on the superconductor, it breaks Cooper pairs and generates ∼261 QPs,

~ given by η ωphoton/Δ with ωphoton/2π being the photon frequency, Δ being the superconducting gap energy, and

η = 0.57 being the down-conversion efficiency.224 Before recombining through inelastic scatterings, these QPs √ D (−x2/4Dt) D can diffuse across the JJ, resulting in a diffusion current Idiff = (e/4 π t)xe per QP at the JJ with =

2 225 0.55 cm /s being the diffusion constant in NbN. For x = 100 nm, Idiff = 11.5 nA which rises and subsides in

2 D ≃ − a characteristic time scale of x / 0.2 ns. It is much smaller than what would be needed, i.e. Ic Ib, to switch the JJ directly. However, unless scattered inelastically and trapped inside the JJ, these QPs at the gap energy can diffuse across the JJ with a mean-free-path of ∼92 nm in graphene through the 160 nm long normal JJ channel.

F The JJ in our experiment is quasi-ballistic such that the Idiff could generate a shot noise SI = 2e Idiff with a

Fano factor F of the order of 1. Similar to how the current noise relaxes the current-biased JJ qubit by coupling through its shunt resistance,226 shot noise can enhance JJ switchings by a mechanism that is equivalent to TA

∗ ~ −1 ~ 2 227 with an effective temperature T given by ωP/kB[2 coth (1 + QSI(ωP)/ ωPCJJ)] with CJJ = 20fF being

∗ the effective JJ capacitance (Section 8.15). For F = 1, T ≃ 1 K, same order of magnitude but a factor of two lower than our fitted value in Fig. 8.3B. We note that Idiff might rise above 1 µA for small x, i.e. 10 nm. However,

∗ 227 the characteristic time scale of Idiff is much faster than 1/ωP, invalidating the use of T equation. The role of graphene is inessential under this hypothetical mechanism except that it provides a shunt resistor, quasi-ballistic channel across the JJ and forms a proximity JJ that allows for the efficient coupling of photons by the dissipative surface plasmon. This is consistent with the lack of large dependence of gate voltage (Section 8.16) and graphene

103 thickness (Section 8.14) in subsequent experiments. In the future, theoretical calculations of Γesc enhancements

due to both adiabatic and non-adiabatic change of washboard potential will inform the microscopic mechanism

of single-photon induced switchings. Our single-NIR-photon sensitive JJ can perform as a high speed, low power

consumption optical switch for high performance computer and quantum computer at cryogenic temperature.

8.7 Tables of measured devices

Device A B C D Nickname Woody JJ90 Woody JJ2p8 Woody JJ1p5 Buzz Chip serial number BGB71 BGB71 BGB71 BGB38 JJ width (µm) 2.8 2.8 1.5 1.5 JJ channel length (nm) 160 160 160 160 Graphene layer 1 1 1 1 NbN stack (Ti/Nb/NbN in nm) 5/2.5/50 5/2.5/50 5/2.5/50 5/5/100 ⟨ ⟩ Is (µA) @ΔV(V) 10.91 @26.5 10.77 @26 5.07 @25 3.11 @24.5 Rn (Ω)@ΔV(V) 44 @26.5 27.3 @26 70.8 @25 62.5 @24.5 VCNP (V) -6.5 -6 -5 -4.5 Measured polarization ratio 0.33 0.33 0.18 0.54 to 0.75

Table 8.1: List of measured devices (1/2). The JJ channel lengths measured by scanning electron microscope are typically 40 nm shorter

than the design value of 200 nm in lithography. VCNP is the gate voltage of the charge neutrality point for the monolayer graphene.

Device E F G Nickname Stitch JJq NemoJJ0 NemoJJ1 Chip serial number BGB86 BGB57 BGB57 JJ width (µm) 2.8 1.5 1.5 JJ channel length (nm) 160 160 160 Graphene layer 4 1 1 NbN stack (Ti/Nb/NbN in nm) 5/5/75 5/5/50 5/5/50 ⟨ ⟩ Is (µA) @ΔV(V) 9.77 @20 1.71 @39.6 1.55 @35 Rn (Ω)@ΔV(V) 27 @20 101 @39.6 94.1 @35 VCNP (V) n.a. -9.6 -5 Measured polarization ratio 0.33 n.a. n.a.

Table 8.2: List of measured devices (2/2). The JJ channel lengths measured by scanning electron microscope are typically 40 nm shorter

than the design value of 200 nm in lithography. VCNP is the gate voltage of the charge neutrality point for the monolayer graphene.

104 Device A B C D Nickname Woody JJ90 Woody JJ2p8 Woody JJ1p5 Buzz Electron density (1012 cm−2) 1.99 1.98 1.89 1.67 Electronic mobility (1012 cm2/V s) 5588 7740 7739 8000 Mean free path (nm) 91.7 121 124 120 Ic (µA) 11.99 11.47 3.78 3.54 Rn (Ω) 44 40 71 63 IcRn (μeV) 528 459 269 223 Thouless energy (meV) 0.76 0.74 1.6 0.99 JJ coupling energy (meV) 25 24 7.8 7.25 ωP0/2π (GHz) 225 200 500 156 CJJ (fF) 18 22 5.8 11 Q0 1.12 1.22 1.67 0.46 Δ of NbN (meV) 1.52

Table 8.3: List of JJ properties. Values taken at gate voltage of 20 V.

R 8.8 Calculation of photon from laser power

To understand the measured single-photon induced JJ switching rate quantitatively, we should calculate the number of incident photons per unit time per unit area per unit incident laser power through the optical fiber, i.e.

R R photon. We can calculate photon using the Gaussian beam profile as follows: The light that illuminates the JJis guided by a single-mode fiber (SMF-28 designed for 1550-nm transmission). The radius of the laser beam, w, a distance z away from the end of the fiber, assuming Gaussian beam propagation, is:

p 2 w(z) = w0 1 + (z/zR) (8.1)

2 where w0 = 5.2 µm is the beam radius at the end of the fiber and zR = πw0/λ = 55 µm is the Rayleigh range with λ being the wavelength, i.e. 1550 nm. The device is 1 inch away from the end of the fiber resulting in w =

2.41 mm at the JJ with a spot size of 18.25 mm2. For a Gaussian beam, the intensity profile, I, as a function of z

105 and distance r from the beam center is given by:

 2 w − 2 I(r, z) = I 0 e 2(r/w(z)) (8.2) 0 w(z)

2 where I0 = 2Plaser/(πw0) with Plaser being the total power of the beam given by the normalization condition, i.e.

R ∞ ( ) = ( = , ) = / [ ( )]2 0 I r 2πrdr Plaser. Therefore, at the center of the beam, I r 0 z 2Plaser π w z . Assuming the

JJ is centered in the laser spot, 100 pW out of the fiber transduces to 10.96 pW/mm2 at the JJ. Using 0.80 eV for

R 2 1550-nm photons, we calculate photon = 85.5 photons per second per µm per 100 pW.

R Based on this photon and the effective single-photon absorption area estimated from the polarization ratio measurement and HFSS simulation, the expected single-photon counts are shown in Table 8.4.

Device A B C D Measured polariza- 0.33 0.33 0.18 0.65 ± 0.15 tion ratio Effective single- 2 × 2.8 × 0.19 2 × 2.8 × 0.19 2 × 1.5 × 0.27 < 2 × 1.5 × 0.05 photon absorption area, Aeff, (µm2) Averaged coupling 60% 60% 52% 94% efficiency within ⟨ ⟩ Aeff, α Photon rate 53 Hz 53 Hz 36 Hz < 12 Hz with 100 pW laser power = R ⟨ ⟩ photonAeff α

Table 8.4: Estimated photon rate based on the polarization ratio measurement and HFSS simulation

8.9 Single-Photon Detection in Pulsed Measurements

Using a pulsed laser as the photon source, instead of a continuous wave (CW) laser, can provide an independent method to cross-check the single-photon response of the GJJ by: 1) showing that the detector responds to pulses

106 with less than one photon on average and 2) verifying the linearity of the single-photon sensitivity with incident

laser power for various pulse conditions.20 In the pulse measurement, we can control the number of photons

reaching the JJ per pulse by the pulse laser power, Ppulse, and the pulse duration, tpulse. The number of photons per pulse, Npulse is given by Npulse = Epulse/Eph = Ppulsetpulse/Eph where Epulse is the energy per pulse and Eph is the photon energy. By tuning Ppulse and tpulse we can keep Npulse < 1 to ensure the GJJ is sensitive to single photons. Furthermore, if the JJ is acting as a single-photon detector then its switching rate must be linear with the incident photon rate and therefore power. For a pulsed measurement, if this is true then as we tune Ppulse we should be able to keep the probability of switching per pulse constant by inversely proportionally tuning tpulse.

(A) t pulse I On I Off b b

I b

0 -0.2 0 0.2 0.4 0.6 0.8 1 Time (ms) t ( s) (B) pulse 25 2.5 0.25 2.5

2.0

1.5 Mean Probability

1.0

0.5

SwitchingProbability perPulse (%) Dark Count Probability

0 0.01 0.1 1 P ( W) pulse

Figure 8.5: Single-photon-induced switching in a pulsed laser experiment (A) The timing of the pulsed experiment. The laser pulse starts 500 µs after the bias current is turned on to allow for a ∼300 µs ramp-up time. The bias current is turned off 200 µs after the laser pulse starts to allow for any photon-induced switching events to be measured. The bias current is 10.70 µA for the data of this figure. The pulse repetition rate was 1 Hz so that we can completely eliminate any switching events caused by Joule heating. (B) Switching probability per pulse of Device A. If the energy per pulse, or equivalently the number of photons per pulse, is kept constant (in this case 250 fJ per pulse), the measured probability of switching per pulse remains constant for various powers (bottom axis) and their corresponding pulse duration (top axis). The average switching probability across trials is 1.7% per pulse (blue line). The error bars are calculated as the square root of the measured number of counts divided by the number of pulses.

107 For the pulse experiment, we illuminate the JJ at a pulse repetition rate of 1 Hz. Such a low repetition rate can guarantee a sufficient cooling time between pulses to eliminate any Joule heating effect from frequent switching.

Additionally, we bias the junction for a short time, i.e. 700 µs, in each pulse cycle to reduce the time window for dark counts. As shown schematically in Fig. 8.5A, we turn the bias current on 500 µs before the laser pulse to allow for a 300 µs turn on time (limited by the low pass filters in the setup), and then turn it off 200 µs afterthe laser pulse. For these measurements, Ib=10.70 µA. The voltage across the GJJ is recorded to measure the probability of single-photon induced switching.

Measured on Device A using 6500 pulses for each data point, Fig. 8.5B plots the switching probability per pulse for eleven (Ppulse, tpulse) pairs such that Epulse is held constant at 250 fJ. Using the result from Section 8.8, the number of photons per pulse reaching the JJ is:

R ⟨ ⟩ Npulse = photonAeff α Ppulsetpulse = (53Hz/100pW)(250fJ) = 0.13 (8.3)

Assuming a Poisson distribution, the probability of zero, one, and two photons in each pulse are 0.878, 0.114, and 0.007, respectively. Therefore, most of the observed light-induced switching events are caused by single photons. The average switching probability across trials is 1.7% per pulse. The error bars are given by the square root of the number of counts (the standard deviation assuming a Poisson distribution) divided by the number of pulses. The result shows a constant switching rate over two-orders of magnitude in the pulse power up to 1 µW, i.e. 104 times larger than the CW power used in the main text.

8.10 Proportionality of Γmeas in laser power

(dark) ′ ′′ 2 To check the linear dependence of Γmeas on laser power, Plaser, we use Γmeas = Γ + Γ1Plaser + Γ2 Plaser to fit the data in Fig. 2B. Table 8.5 shows the fitted result.

108 (dark) ′ ′′ 2 Ib (µA) Γ (mHz) Γ1 (Hz/nW) Γ2 (Hz/nW ) 10.75 2.41 7.71 14.4 10.83 48.5 26.4 36.7 10.88 467 46.1 44.3

Table 8.5: Proportionality of Γmeas in laser power. The fitted values for the solid lines in Fig. 8.2B.

8.11 Sweeping Experiments versus Counting Experiments and their correspondence

99,103,149,228,229 The switching rate of a JJ, Γmeas, is conventionally obtained by bias-current sweeping experiments based on the Fulton-Dunkleberger method,169 but by switching rate counting experiments in our report. Our experiments show the two methods are equivalent both with and without light illumination (Fig. 8.6).

The counting experiment is performed as follows (Fig. 8.6A) : Ib is set to be a constant value for the duration of the experiment, in this case 100 s. VJJ is monitored via a comparator during this time. Any time VJJ switches to non-zero, the comparator will switch and shunt the bias current thus resetting VJJ to zero and the JJ to the superconducting state. The comparator then switches back allowing Ib through the junction once again. The number of times the GJJ switches to the non-zero-voltage state is counted and divided by the 100-s duration to arrive at Γmeas. This procedure is repeated for various values of Ib and incident laser power.

Eqn. (7) in Ref. [169] shows that the sweeping experiment and counting experiment are related by:

≥ P dIb/dt Σj k jΔIs Γmeas(I = kΔIs) = ln P (8.4) b ΔIs Σj≥k+1 jΔIs

P where j is the probability of Is falling between jΔIs and (j + 1)ΔIs in the sweeping experiment, i.e. the vertical axis in Fig. 3A of the main text. We use Eqn. 8.4 to convert the data of Fig. 8.3A into a switching rate and plot it next to the data of Fig. 8.3B in Fig. 8.6B. The results of the two measurement techniques agree very well with one another, albeit a slight offset removed due the critical current of the GJJs drifting over time between thetwo experiments. This result validates the techniques and the conclusions drawn from them.

109 (A)

(B)

3 Direct counting 10 From I distribution s

10 2

1 (Hz) 10

meas 100 pW 10 0 k

10 -1 dar

10 -2 10.4 10.5 10.6 10.7 10.8 10.9 11 11.1 11.2 I ( A) b

Figure 8.6: Counting method versus current-sweep method. (A) The circuit used to bias and reset the JJ in the counting method. When the JJ switched to the non-zero voltage state, VJJ would surpass Vthres causing the switch to shunt the JJ to ground thus resetting the JJ to the zero voltage state. The number of times the JJ switched is monitored versus time and converted to a rate. (B) The counting method (circles) agrees well with the typical current-sweeping method (x’s). The current-sweeping method data has been shifted by 35 nA to account for slight variations in experimental conditions between experiments. All data taken at 27 mK

8.12 Experimental determination of polarization orientation

We perform a series of experiments to determine the polarization orientation of our incident photon relative to the orientation of the GJJ. We do not use a polarization-maintaining fiber in our setup so the linear polarization of the light out of the fiber, and thus incident on the GJJ, is not known a priori. To determine the polarization orientations causing the highest and lowest Γmeas (Fig. 8.4A in the main text), we add a linear polarizer between the output of the single-mode fiber and the JJs. The orientation of the JJs compared to their packaging isknown and used to align the polarizer either parallel or perpendicular to the direction of the supercurrent flow in two separate cool-downs of the devices. To align the polarization orientation of the incoming light with the polarizer, we rotate the polarization while simultaneously measuring the switching rate of the JJs. The polarization is rotated until the switching rate of the JJs is at a maximum for a given laser intensity, corresponding to the maximum amount of light reaching the JJs and therefore the maximum alignment with the polarizer. At this polar-

110 (A) 10 3 (B) Device A Device B 10 2

10 1 (Hz)

10 0 switch f

-1 10 dark counts dark counts 10 -2 10.3 10.4 10.5 10.6 10.7 10.7 10.8 10.9 11 11.1 11.2 I ( A) I ( A) bias bias

Figure 8.7: Determination of polarization orientation. Measurements of the switching rate of two orthogonally-oriented GJJs (Devices A and B in (A) and (B), respectively) are carried out with a linear polarizer between the light source and the devices. The experiment is ◦ repeated twice with the orientation of the polarizer rotated 90 between the two. The triangular data corresponds to one experiment and the circular data to the second. The black data is the dark count rate. The red and blue data is the switching rate in the presence of light. The double arrows show the orientation of the incident polarization relative to the JJ depicted in the inset. We observe for both devices that the switching rate is enhanced when the polarization is aligned perpendicular to the supercurrent direction (blue in (A) and red in (B)). ization, the switching rate is measured as a function of bias current for both JJs. The results are displayed in Fig.

8.7 where the blue data is taken in one cool-down for both devices and the red in another. We observe that for both devices, the switching rate is highest when the light is polarized perpendicular to the supercurrent direction which is in agreement with our HFSS model.

8.13 HFSS simulation

To understand the polarization ratio and the photon coupling to the GJJ, we model the device using finite element analysis in ANSYS HFSS. The model is based on the geometry of Devices A, B, C, and D. The dimensions, thickness, and materials are listed in Table 8.1. The NbN contacts have the same width as the graphene and length 1 µm. The hBN-encapsulated graphene and NbN/Nb/Ti stack are on 285 nm of SiO2 with 500 µm of p- doped Si below that. To limit the boundary effects, periodic boundary conditions are used on the four surround- ing sides and an impedance boundary is applied at the bottom side to terminate the silicon layer. Regarding the

− 230 − constitutive parameters of the materials at 1.55 µm, we use: εNbN = 6.52 + 60.88i, εNb = 72.39 + 7.67i,

111 − and εTi = 7.67 + 33.92i for the key lossy structures above the silicon dioxide. Normally incident plane wave

◦ ◦ with polarization angles of 0 and 90 , corresponding to the electric field photon in parallel and perpendicular to the supercurrent flow respectively, are used with an amplitude 2of aW/µm . The volumetric absorption of the

NbN/Nb/Ti stacks can be calculated numerically within the simulation. We shift the distance of the integration box from the Josephson junction in the direction of supercurrent to obtain the data in Fig. 8.4D in the main text.

The white dashed lines in Fig. 8.4B and C outline the integration box which has a dimension of 0.05μm×2.8μm for Device A and B, and of 0.05 µm × 1.5 µm for Device C and D.

10 3 Dev. B, Dark Dev. B, 100 pW Dev. E, Dark 10 2 Dev. E, 100 pW

(Hz) 10 meas

10 0

10 -1

10 -2 9.9 10 10.1 10.2 10.3 10.4 10.5 10.6 I ( A) b

Figure 8.8: Γmeas of JJ using tetra-layer graphene. Switching rate of a JJ with tetra-layer graphene (Device E) compared to the one with monolayer graphene (Device B) at Vgate = 20 V and T = 36 mK. The switching rate is nearly identical between the two devices pointing to the the fact the absorption takes place in the NbN contacts instead of in the graphene itself. Device E data is offset by 0.65 µA to account for its difference of Ic when compared with Device B.

8.14 Γmeas of JJ using tetra-layer graphene

To further investigate the role of graphene in the photon absorption and observed JJ switching, we fabricate a JJ with tetra-layer graphene (Device E) with dimensions comparable to Device A and B. The measurement results are plotted in Fig. 8.8 with an offset of 650 nA in Ib for Device E to account for the difference in critical current from Device A. Γmeas for the two devices is nearly identical, in both the single-photon counts as well as its dependence on Ib. This result is consistent with the HFSS simulation that the photon absorption is dominated by the

112 plasmonic resonance of the superconducting electrode right at the JJ. It also suggests that the mechanism of the single-photon induced switching does not depend critically on the graphene thickness.

8.15 Modeling of Γmeas in Fig. 8.3B

In the main text, it is noted that the switching rate of the GJJ is modeled using a phase particle model with escape

98 −ΔU/kBTesc rate given by Γesc = Ae , where

   q  ωp + 1 − 1 2π 1 4Q2 2Q (TA) A = (8.5)  q  12ω 3ΔU (MQT) p 2π~ωp

   T (TA) = Tesc  h i (8.6)  ~ / . + 0.87 ωp 7 2kB 1 Q (MQT)

− 2 1/4 ~ 1/2 with ωp = ωp0(1 γJJ) being the JJ plasma frequency, ωp0 = (2eIc/( CJJ)) being the JJ plasma frequency at zero bias current, Ic being the critical current, γJJ = Ib/Ic being the normalized bias current, CJJ being the effective junction capacitance, Q = ωpRnCJJ being the JJ quality factor, Rn being the normal-state resistance, q − 2 − −1 ~ ΔU = 2EJ0( 1 γJJ γJJ cos γJJ) being the phase-particle energy barrier, EJ0 = Ic/(2e) being the Josephson ~ coupling energy, and T being the JJ electron temperature. Here, kB, , and e are Boltzmann’s constant, Planck’s reduced constant, and the electron charge, respectively. To fit the data of count rate versus bias, we include both

TA and MQT in in our model. We estimate an effective junction capacitance CJJ =18 fF from the Thouless

~ 2 ~ ∼ 6 energy, ETh = De/L , using CJJ = /RnETh. Here De = vFlMFP/2 is the diffusion constant, vF 10 m/s is the Fermi velocity, and lMFP is the electron mean free path in graphene. We determine Ic (which is in general

113 ∼ ⟨ ⟩ (dark) (dark) 10% higher than Is ) by fitting the dark count rate with Γmeas = Γesc (Ic, T0) = ΓMQT(Ic) + ΓTA(Ic, T0) with Ic as the only fitting parameter (here T0=27 mK is the base temperature). For Device A, we find Ic =11.96

µA which determines the other junction parameters such as EJ0, ΔU, ωp, and Q.

To model the enhanced switching probability induced by single photons, we assume the effective junction temperature remains at a constant elevated temperature Tel for some time Δt1 upon photon absorption in the

NbN contacts and subsequent quasiparticle diffusion. The elevated effective temperature is not a true temperature rise in the junction but instead is due to quasiparticle noise. This effective temperature will decay in time but we make the assumption that it is a constant during Δt1 to simplify the model. The probability of the phase R P = − exp(− Δt1 (light) ) ∼ − exp(− (light) ) particle escaping during Δt1 is 1 0 Γesc dt 1 Γesc Δt1 . For our range of oper- (light) ≪ P ∼ (light) P ation we have Γesc Δt1 1 so Γesc Δt1. With being the probability of switching with each absorbed

PR ∼ (light) R photon, the photon-induced switching rate in our setup is photon Γesc Δt1 photon. Combining this with

(light) the dark count rate when no photons are present, we have for Γmeas , i.e. the total measured switching rate of our setup under illumination:

(light) ≃ (dark) − R Γmeas Γesc (Ic, T0)(1 Δt1 photon) (8.7) (light) R + Γesc (Ic, Tel)Δt1 photon

∫ (light) − Δt1 (dark) (dark) − R − 0 Γesc dt Γmeas =Γesc photon(1 e ) ∫ − Δt1 (light) R − 0 Γesc dt + photon(1 e ) (8.8) ≃ (dark) − R Γesc (Ic, T0)(1 Δt1 photon)

(light) R + Γesc (Ic, Tel)Δt1 photon

(light) ∼ We fit Γmeas versus Ib for the two lowest light intensities that we study (100 and 200 pW which is 53 and

∼ 106 photons per second into the GJJ) with Δt1 and Tel as the fitting parameters and find good agreement with

114 the data.

We can further improve the model by including the effect of phase particle retrapping events. As Ib decreases, it gets closer to the retrapping current, Ir0, where the phase particle is retrapped and the JJ returns to the zero- voltage state. In practice, the phase particle will retrap at a current Ir < Ir0 due to thermal and quantum fluctuations (similar to the measured switching current Is being less than Ic). When retrapping becomes significant, the junction is said to be in the phase diffusion (PD) regime and shows reduced escape rate. Although there will still be a voltage spike in the case that the phase particle escapes and retraps, the spike will be transient so the JJ does not latch into the non-zero-voltage state, forbidding our setup from recording the event. The retrapping rate, Γr, can be calculated as:105

r − − ΔUr Ib Ir0 EJ0 k T Γr = ωp0 e B (8.9) Ic 2πkBT with

 2 − 2 EJ0Q Ib Ir0 ΔUr = (8.10) 2 Ic

(dark) Now modifying Equation 8.7 and noting that Γr is negligible in our measurement range:

(light) (dark) − R Γmeas =Γesc (Ic, T0)(1 Δt1 photon) (8.11) (light) R − (light) + Γesc (Ic, Tel)Δt1 photon(1 Γr (Ir0, Tel)Δt1)

We refit the data adding Ir0 as an additional fitting parameter to Δt1 and Tel. Here, we set an upper bound of

⟨ ⟩ ΓrΔt1 to be 1. The result is Δt1 = 0.86 ns, Tel = 2.1 K, and Ir0 =8.55 µA. For comparison, Ir =6.64 µA, 78% of Ir0, showing reasonable agreement. The model with these parameters agrees well with the data in Fig. 8.3B

115 (light) of the main text for all bias currents. We note that the model starts underestimating Γmeas at higher incident powers which we attribute to the heating of the substrate. We correct the fit by decreasing Ic slightly at the higher incident powers. This heating correction to Ic is small, requiring only a 0.4% reduction in Ic at the highest laser power that we used (800 pW).

Fig. 8.9 shows the data and fitting of Γmeas for Device B using the same methods. We find Tel=1.8 K and

Δt1=0.67 ns, similar to the values for Device A (see comparison in Table 8.6).

Device Tel (K) Δt1 (ns) A 2.1 0.86 B 1.8 0.67

Table 8.6: Fitting parameters for the single-photon response observed in Fig. 3B and 8.9

1000

100

10 (Hz)

meas 1 800 pW 400 pW 200 pW 0.1 100 pW Dark 0.01 10.4 10.6 10.8 I ( A) b

Figure 8.9: Γmeas of Device B The switching rate as a function of bias current in dark and for various incident laser powers for Device B, similar to Fig. 8.3B in the main text. The fitting parameters for Device B were Tel=1.80 K and Δt1=0.668 ns, similar to the values for Device A.

8.16 Gate Dependence of the GJJ Single-Photon Response

We study how the photon-induced switching depends on the gate voltage. We measure the switching-current distribution of Device A for 5 different values of Vgate (0 V to 20 V in steps of 5 V) using a 10 µA/s bias-current

116 169 sweep both in the dark and with 100-pW of illumination and convert to Γmeas(Ib). As shown in Fig. 8.10 for

˜ (light) (dark) Device A, the ratio Γ = Γmeas /Γmeas decreases as the gate voltage (Vgate) decreases for a given γJJ = Ib/Ic. This makes the observation of the single-photon induced switchings more favorable at high electron doping (i.e. Vgate) than at low doping. We attribute this behavior to two effects.

(light) The first effect is the relative change in ΓTA compared to ΓMQT as a function of Vgate, with Γmeas dominated

(dark) (light) (dark) by TA and Γmeas by MQT. In Fig. 8.10, the y-axis limits are the same in all plots of Γmeas and Γmeas at five different gate voltages. While the Ic increases at higher Vgate, we also plot the normalized Ib on the top x-axis as

γJJ = Ib/Ic, which shifts to higher values as Vgate increases. For a given value of γJJ, both ΓMQT and ΓTA decrease with increasing Vgate. However, ΓMQT decreases more quickly than does ΓTA. Therefore, the MQT-dominated dark count rate falls faster than the TA-dominated single-photon response, making high gate voltage operation favorable. Qualitatively, ΓMQT is heavily determined by ωp which depends on Ic, while ΓTA is determined more

˜ strongly by the temperature than ωp. Thus, the Γ ratio is higher at higher gate voltages for a given γJJ. We fit the data using the same method as in Section 8.15, optimizing simultaneously over four different gate voltages: 5 V,

10 V, 15 V, and 20 V. Vgate at the charge neutrality point, VCNP, of the device is -6.5 V. The best fit gives Tel=2.13

K and Δt1 = 0.92 ns, in agreement with the fit from Fig. 8.3B. We plot these fits as lines for all five gatevoltages.

˜ Retrapping is the second cause of a lower Γ ratio at lower gate voltage which can be observed in the Vgate=0

V case in Fig. 8.10. Retrapping causes the observed count rate to be lower than it would otherwise be with only

(light) (dark) TA or MQT present. In Fig. 8.10, at Vgate=0 V the data for Γmeas is essentially equal to the data for Γmeas . This result is in contrast to the best-fit line that suggests we should still have observed single-photon induced switching above the dark count rate. At lower gate voltages, the GJJ enters the phase diffusion (PD) regime upon photon absorption so that retrapping becomes more significant and the count rate is reduced, similar to the reduced switching rate in the low-bias-current case of Section 8.15.

117 = I /I JJ b c 0.8 0.85 0.9 0.86 0.9 0.85 0.9 0.88 0.9 0.92 0.94 0.9 0.92 0.94 10 4 data, dark data, 100 pW 3 fit, dark 10 fit, 100 pW V = 0 V V = 5 V V = 10 V V = 15 V V = 20 V (Hz) gate gate gate gate gate

10 2 meas

10 1 Gamma 10 0

10 -1 4.2 4.4 4.6 4.8 6.5 7 8 8.5 9.5 10 10.5 11 I ( A) I ( A) I ( A) I ( A) I ( A) b b b b b

(dark) (light) Figure 8.10: Gate-dependent switching rate of Device A Γmeas (gray circles) and Γmeas for 100 pW (orange circles) versus bias current (lower x-axis) and normalized bias current (upper x-axis) for five different gate voltages. The gray and orange lines are thebest Ib γJJ (light) (light) fits for (dark) and , respectively, using the model of Section 8.15. For a given , / (dark) increases with increasing gate Γmeas Γmeas γJJ Γmeas Γmeas voltage so that the single-photon switching events can be observed more clearly at higher gate voltages. Data was collected using the bias-current sweep method (see Section 8.11).

118 9 Conclusion

Throughout this work I have shown through modeling and experiments that the graphene-based Josephson junction (GJJ) can serve as a single-photon detector across the electromagnetic spectrum. The detection can either be driven by hot electrons in the graphene weak link, enabled by graphene’s exceptionally low heat capacity, or driven by non-equilibrium quasiparticles formed in the superconducting contacts, enabled by the sensitivity of the Josephson junction to such excitations. The GJJ can serve both as a practical detector for applications in quantum communications, radioastronomy, and quantum computing and as a testbed for studying the fundamental physics of photon-Josephson junction interactions. Efficiency in the infrared regime could bein- creased compared to that in Chapter 8 by optimizing the geometry of the superconducting contacts for enhanced

119 plasmonic-mode coupling or by incorporating the GJJ with a photonic crystal cavity. With a photonic crystal cavity, light could be concentrated either in the graphene weak link or in the superconducting contact, allowing for direct comparison of the two detection mechanisms revealed in this work. Detector reset times could be increased by changing the readout mechanism to prevent the GJJ from switching into the resistive state, for instance with a resonance-shift scheme. Such a scheme could also allow for improved time resolution of the photon-GJJ interaction providing further insight into the microscopic mechanisms of Josephson junction physics. I hope this works sparks others to consider the interesting physics contained within the GJJ and its interactions with the fundamental constiuents of light.

120 A Experimental Setup

A.1 Device Mounting

Graphene-based Josephson junctions (GJJs) were fabricated on a silicon chip with a 285-nm silicon oxide layer.

The silicon chip was mounted to a custom mounting board machined from Rogers RT/duroid 6035HTC ceramic-filled PTFE composite laminate with 35-µm thick copper cladding. The silicon chip backing (faceop- posite the GJJ) was scratched with a diamond-tipped scribe to ensure electrical conductivity for the back gate before the chip was affixed to the mounting board with silver paste. The mounting board has 5 SMP connec- tors and 1 12-pin Nano D connector soldered to its copper wire-bonding pads. The silver paste is extended from

121 Figure A.1: GJJ Mounting Setup. (A) GJJs are on a silicon chip affixed to a custom copper-clad composite board via silver paste. Thesilver paste also acts as an electrical connection from the GJJ back gate (silicon chip) to an SMP connector. Other electrical connections are made from a Nano D connector soldered to the board to the GJJs. The entire board is screwed to a custom gold-plated copper mount. NbN wire-bonding pads and GJJs are contained within the red box on the silicon chip. (B) A 10x magnified view of the red box on the silicon chip in (A). Twelve NbN wire-bond pads are clearly visible. The GJJs are contained within the orange box. (C) A 10x magnified view of the orange box in (B) for 100x magnification total from (A). Three GJJs are clearly visible. under the silicon chip to one of the SMP copper pads to facilitate the back gate connection. Aluminum wire is bonded from the Nano D wire-bond pads on the mounting board to the NbN wire-bond pads connected to the

GJJs on the silicon chip. The mounting board is mounted onto a custom gold-coated copper structure that is then fastened to the mixing chamber (MC) in the dilution refrigerator (DR) via screws. Care must be taken to ensure strong thermal contact between the structure and the MC and between the GJJ board and the structure.

The mounted GJJ and close ups of the NbN wire-bond pads and GJJs are shown in Figure A.1. A side view of the setup is also shown in Figure A.2A.

A.2 Optical Setup

A copper fiber mount can be affixed to the gold-plated mounting structure across from the GJJ board viascrews

(Figure A.2A). A polarizer can be inserted into the gold-plated structure between the fiber and the GJJ board using a custom copper collar that is held in place by a set screw. Different views of the polarizer can be seen inFig- ure A.2. Alignment of the polarizer to the GJJs was achieved by using known orientations of the GJJs to visible

122 Figure A.2: Optical Mounting Setup. (A) The fiber is mounted opposite the GJJ board on the gold-plated copper structure. A fittedcollar with set screw (not shown) can hold a polarizer in the gold-plated copper structure to set the orientation of the linearly polarized light reaching the GJJs. (B) The view from the fiber mount to the GJJ board through the polarizer. (C) The view from the GJJ board tothefiber mount through the polarizer. features on the GJJ mounting board. The orientation of the GJJs to these features was determined using magnified pictures such as those in Figure A.1. The fiber in the DR was a bare (no jacket) SMF-28 single-mode fiber designed for 1550-nm light. Connections on either end of the fiber were FC/PC. At the MC, the fiber was coiled

7 times with a bend radius of 0.75 inches to decouple any stray light of different wavelengths from the fiber. The fiber was fed to a fiber-coupled optical passthrough bulkhead on the top oftheDR.

Feeding the internal fiber was the setup depicted in Figure A.3. A 1550-nm fiber-coupled laser (Santec WSL-

100) was coupled to an acousto-optic modulator (AOM) (AA Optoelectronics MT80-IIR60-PM 0,5-J3V-S) to control the power reaching the GJJ in the DR. The AOM had a rise time of 60 ns and was driven by an 80-MHz

RF source. For continuous-wave (CW) experiments, the AOM was driven by an Agilent N5183A with tunable constant power output at 80 MHz. For pulsed experiments, the AOM was driven by an AA Optoelectronics

MODA80-B4-34 which output 80 MHz with power level and duration determined by a variable gate from a

Tektronix AFG3102 arbitrary function generator. This same function generator is used in the electronic setup

(next section) to simultaneously trigger a bias current pulse.

After power and duration of the light reaching the GJJ was controlled by the AOM, the polarization was controlled in a Thorlabs U-bench setup. Light was collimated into free space from the fiber through a FiberPort

123 Figure A.3: Optical Setup. 1550-nm light from a laser has its power controlled by an acousto-optic modulator (AOM). For continuous- wave (CW) experiments, the AOM is controlled by a constant output RF source (purple) while for pulsed experiments it is controlled by a gated RF source (dark orange). After the AOM, the polarization of the light is rotated in free space through a beam displacer (for cleaning the polarizaion) and a half-wave plate (λ/2). Polarization-maintaining (PM) fibers (light blue) are used prior to rotating the polarization while typical single-mode (SM) fibers (dark blue) are used after. After polarization control, 99% of the light is sent to a powermeterfor monitoring while 1% is further attenuated before being sent into the dilution refrigerator. At the mixing chamber (green) the fiber is coiled 7 times with a bend radius of 0.75 inches before light exits the fiber to illuminate a GJJ either with or without a polarizer between fiber tip and GJJ.

Collimator, the polarization was ensured to be linear by passing the free space light through a beam displacer

(which displaces one linear polarization while allowing the other to pass through), the polarization of the light was rotated by passing it through a half wave plate that could be rotated, and the light was coupled back into a single-mode fiber with a FiberPort Coupler. Note that polarization-maintaining fibers only allow two specific orientations of linearly polarized light to pass without being distorted. Therefore, all fibers after this point in the setup were single-mode but not polarization-maintaining in order to ensure arbitrary linearly polarized light could reach the GJJs. Special care was taken to not move any fibers during experimentation because otherwise inconsistent rotations of the polarization could be introduced from data set to data set.

The power-, duration-, and polarization-controlled light was next attenuated by a 99:1 fiber splitter (Thorlabs

124 10202A-99-FC). The 99% port was sent to a power meter (HP81536A) for power monitoring while the 1% port continued to a 20-dB attenuator before being fed into the fiber-coupled optical passthrough bulkhead on the DR.

Another input port on the splitter was terminated with an optical terminator. Prior to cooling down the DR, power was measured both at the fiber output at the MC and on the HP81536A for calibration.

A.3 Electronic Setup

The electronic setup used to supply a current bias, Ib, to the GJJ and measure its voltage, VJJ, is displayed in

Ω Figure A.4. Ib is supplied by a voltage source (Agilent 33220A) passed through a resistor (500 k ). The voltage source can be held at a constant value, for measuring switching rate at a given Ib for instance, or swept, for measuring switching-current distributions. During pulsed measurements, the voltage source was triggered by the same arbitrary function generator (Tektronix AFG3102) as the AOM to produce simultaneous laser and current pulses. Ib is passed through a switch (Mini Circuits ZYSW-2-50DR) that can be triggered to shunt the

GJJ to ground to reset the GJJ from a resistive state to a superconducting state. The switch trigger comes from a comparator (Pulse Research Lab PRL-350TTL) that compares VJJ to a threshold voltage set by another voltage supply (Rigol 832). For most experiments, the threshold voltage was set to 60 mV. VJJ across the junction is amplified 1000 times by a pre-amplifier (Ithaco 1201) before being sent to the comparator. The output ofthepre- amplifier is also record by a data acquisition system (NI USB-6341). All Ib and VJJ lines to the GJJ were passed through multiple stages of filtering to reduce noise. At the 4-K stage of the DR, a two-stage RC low-pass(LP) filter was mounted. The first stage had a500-Ω resistor with 2-nF capactior for a cutoff frequency of 160 kHz while the second stage had a 500-Ω resistor with 4.7-nF capactior for a cutoff frequency of 70 kHz. At the mixing chamber of the DR, a two-stage pi LP filter with cutoff frequency of 470 kHz was used. The back gate voltageto the GJJ was supplied by a voltage supply (Keithly 2400) through a 10 MΩ-resistor.

Special care was taken to eliminate ground loops in the setup. This required having only a single path to

125 Figure A.4: Electronic Setup. The GJJ is current-biased by a voltage source through a 500-kΩ resistor (purple background). The voltage across the GJJ is first amplified with a pre-amplifier before being measured by a data-acquisition system (yellow background). The measured voltage is also compared to a threshold voltage through a comparator. When the GJJ switches into the normal state, the GJJ voltage surpasses the threshold and the comparator sends a pulse to the bias-current switch to shunt the GJJ to ground. This switch re- sets the GJJ into the superconducting state. In pulsed experiments, the same arbitrary function generator that gates the laser (Figure A.3) also triggers the GJJ bias current so that the laser and the bias current pulse occur simultaneously (dark orange). All bias current and voltage measurement lines to the GJJ are passed through a dual-stage RC low-pass (LP) filter mounted at the 4-K stage (blue dashed box) and a dual-stage pi LP filter mounted at the mixing chamber (green dashed box). Not pictured is a voltage source with10-MΩ resistor for the GJJ back gate.

ground from any electronic equipment. All equipment was plugged into a single power strip. The power strip was connected to wall power through an isolation transformer. The ground pin on the isolation transformer was internally disconnected to disconnect the electronics from the wall ground (CAUTION: Electronics must

have another path to ground before performing this step. Otherwise dangerous — to the equipment and hu-

mans — voltages can build up on the equipment). Instead, all electronic equipment was grounded through a

typical 19 inch electronics rack which had a single point of connection to a thick copper cable that was driven

into the literal ground underneath the building. The DR was connected at a single point to this same grounding

126 cable. Much of the equipment required USB connections to a control computer. The control computer was on a separate power line and the electronics were connected to it through a USB optical isolator to prevent ground contamination.

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