OSSERVAZIONE DI EMISSIONE DI SINGOLO FOTONE DA GIUNZIONI PLANARI

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Introduction

Sharing information between individuals through secure channels has always been an extremely important issue that led to the implementation of in-creasingly efficient cryptographic systems. Nowadays, the most widespread secure communication protocols are based on public-key approaches whose security is guaranteed by the complexity of the algorithms needed to de-cript the message without knowing the private key. These offer the so-called computational security: computer available nowadays are not pow-erful enough to break the cyphertext in any reasonable amount of time. Anyway, computationally-secure approaches cannot be considered inher-ently safe for at least two reasons: if an efficient algorithm is found, the security of these protocols immediately fails; given the ever-growing com-puting power, ciphers that are impossible to break now, will – most likely – became insecure in a few years.

The research field of Quantum Cryptography offers a different approach to secure information sharing: bits are encoded in the quantum state of a single particle, typically the phase or the polarisation state of single photons, and the security is guaranteed by the fact that an eavesdropping attempt – a measurement on the state in the quantum-mechanics language – would alter the state and leave a detectable trace.

Since its first proposal, in 1984 by Bennet and Brassard, many quantum-cryptography protocols have been proposed and some of them also imple-mented in commercial systems. The widespread diffusion of these is anyhow hindered by the lack of real, practical single-photon sources.

My thesis work took place in this context. It was focused on the reali-sation of planar light-emitting junctions and their study in terms of single-photon emission and was carried out at the NEST laboratory of Scuola Nor-male Superiore, where researchers have been involved in several EU-level projects focused on the realisation of single-photon sources for quantum-cryptography applications. The activity in this field led to the recent demon-stration of a novel kind of p-n planar junction based on semiconductor het-erostructures and characterised by a very high operation bandwidth and electron-to-photon conversion efficiency. My project was motivated by the observation that these sources, when driven close to their conduction thresh-old, emits from selected spots along their perimeter. The sub-micron size of

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these spots suggested that single-photon emission could be achieved due to Coulomb-blockade effects from each spot. During my work I have verified this hypotesis by fabricating devices with a reduced junction area, in order to limit the number of emission spots. The analysis of the emitted-photon statistic performed by means of a dedicated Hanbury-Brown-Twiss setup that I have realised during my thesis allowed me to observe for the first time emission from these devices in the antibunching regime, demonstrating that they can be operated as single-photon sources and are suitable for the implementation of QKD systems.

This work is organised in four chapters:

Chapter 1 is a brief introduction to Quantum Cryptography and reports on the state-of-art of the main components of a QC setup. At the end of this chapter I shall describe the experimental setup needed to characterise a source in terms of single-photon emission.

Chapter 2 introduces the semiconductor heterostructures and the litho-graphic processes used during my work to realise my planar p-n junctions. In Ch.3 I shall describe the experimental setup and report the preliminary characterisation of the sources in terms of electrical and optical properties. Chapter 4 reports the observation of light emission with sub-Poissonian statistics from my devices, which constitutes the main result of my work.

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Chapter 1

Quantum Criptography

Cryptography is the art of hiding a message in a gibberish using an encryp-tion algorithm, named cryptosystem1_{. The true contents in the message} can be read only knowing the decryption code. Although someone refers to cryptography and cryptology interchangeably, cryptography is the ensam-ble of the cryptographic techniques while cryptology is a broader field that includes cryptography and cryptoanalysis, the science that studies how to break the code and decrypt the true message within the text.

The cryptosystem algorithm combines the input message (plaintext ) with additional information (known as the key) to produce a safe ciphertext, i. e. a message that is extremely hard to decrypt without knowing the key.

1.1 Cryptosystems

Since the Roman age cryptography has been employed in military and diplo-matic affairs, and its use was restricted to only few people.

In this century everybody, even unknowingly, makes use of cryptosystems. Connecting to the internet or simply making a telephone call involves some cryptographic techniques. Depending on how the cryptosystem algorithms resolve the problem of keys distribution we can group them in two families: symmetrical and asymmetrical cryptosystems.

Conventionally, the sender, the receiver and the eavesdropper are named Alice, Bob, and Eve and in the following I shall employ this notation. In symmetrical cryptosystems Bob and Alice share the same key. To keep this cryptosystem safe, they have to find a trusted way to exchange the key like, for example, meeting themselves.

If the sender needs to exchange information with many people, the repeated

1

Actually, the cryptosystem is compound by two alphabets, a collection of letters, two numbers that define the lenght of the words in input and in output and an injective function that encrypts the message [38].

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use of the same key increases the chances that Eve finds it. On the other hand, using different keys for each receiver leads to more difficult key man-agement and key generation. Asymmetrical cryptosystems partially resolve these problems, but also these methods are far from being perfect.

Other symmetrical cryptosystems (like DES2, IDEA3.), that use a shorter key, are currently used and preferred to their asymmetrical counterparts (see below) since more secure. Nevertheless, like the asymmetrical counterparts, also this kind of cryptosystems offer only a computational security (see asym-metrical cryptosystems for further details).

Asymmetrical cryptosystems are suitable to exchange information in crowdedsystems.

The receiver calculates a public key, which is then distributed publicly, start-ing from a private key known only by himself/herself. By usstart-ing the public key it is only possible to encrypt a message, while to decrypt a ciphertext the corresponding private key is needed. Although this kind of algorithms can manage many users (Bob does not send his private key to anyone), it needs some form of authentication: Eve could pretend to be Bob without Alice suspecting it.

It should be remarked, however, that the security of asymmetrical cryp-tosystems (but also of many symmetrical ones) is based on unproved com-putational hypoteses. For example, RSA is based on the yet-to-be-proved assumption that it is impossible to decompose a large number in prime factors in polynomial time making it impossible to break the code in a rea-sonable time. This poses two severe limits to the reliability of computational cryptography:

if an efficient algorithm is found, the security of these protocols imme-diatly fails;

given the ever-growing computing power, ciphers that are impossible to break now, will – most likely – became insecure in a few years, when faster computers will allow finding the keys even by brute force approaches. This limits in time the security of computational cryp-tosystems making them a less-than-ideal choice when important infor-mation must be stored for long times.

A notable exception to this is the famous symmetrical algorithm known

2

invented on 1972 by IBM and chose as standard by Federal Information Process-ing Standard for US government. It is considered unsecure and in 1999 the DES has been cracked in about two days, see http : //csrc.nist.gov/publications/f ips/f ips46 − 3/f ips46 − 3.pdf .

3

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as the one-time pad4: Alice encrypts her message, a string, m1, of binary

numbers, with a randomly generated key k having the same length of the message. The encrypting process is simple: she adds each bit of the message to the corrisponding bit of the key obtaining a scrambled text (s = m1⊕ k,

where ⊕ denotes the binary addition modulo 2 without carry). Bob decrypts the message subtracting the key from the ciphertext (m1 = s k).

The one-time pad is the only known perfectly safe classical cryptosystem5 but it also has two severe drawbacks: the key needs to be as long as the plaintext and it can not be used more than once. If k is reused, Eve’s in-formation about key and plaintexts increases (for example if s1 and s2 are

two texts encrypted with the same key, then s1⊕ s2 = m1⊕ k ⊕ m2⊕ k =

m1⊕ m2⊕ k ⊕ k = m1⊕ m2). For this reason, the one-time-pad technique

is used only for critical applications.

Additionally, even the one-time pad has a serious practical drawback: it assumes that Alice is able to send the key secretly to Bob. In this sense, the one-time pad shifts the problem from encripting the message safely to that of sharing the key. As I shall discuss in the following sections, Quantum Physics makes it possible to solve this issue.

1.2 Quantum Cryptography

According to the Heisemberg’s Uncertainty Principle it is impossible to know with arbitrary precision information about certain pairs of physical properties of a particle, e. g. momentum and position.

With other words, this principle states that when we measure one of these properties (we increase our knowledge about it) the uncertainty of the other property increases.

This property can be turned to advantage: when Alice sends her key to Bob (information in quantum cryptography could be shared through particles), any eavesdropping attempt, which in the language of Quantum mechanics consists in a measurement performed on the system, will leave some de-tectable clues.

Since its origin, many protocols have been proposed and some of them implemented in real, commercial, systems.

Quantum Cryptography protocols can be clustered in three main fami-lies [31]: discrete-variable coding, continous-variable coding and distributed-phase reference coding. Analysing the differences between these approaches is beyond the aims of this Thesis. Here, I mention them to emphasize that intense research efforts have been devoted toward faster, more efficient

pro-4

Gilbert Vernam, 1926 AT & T.

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tocols that are able to avoid the so called photon-number splitting (PNS) attack, which I shall discuss in Sec. 1.3. All the protocols follow a similar procedure, the main differences being in the quantum states that are chosen to transmit the key. In the following I shall describe in details the BB84 protocol, which was the first quantum protocol proposed and realised. Its security relies on the no cloning theorem (see [38]).

In our treatment bits are coded using the polarisation of single photons, but it is possible – in principle – to use any quantum system with two eigen-states. For example, many fibre-based-protocols use frequencies, phases or polarisation of single photons to send the key along optical fibres.

1.3 A brief History

The first protocol of Quantum Cryptography, called BB84, was proposed in 1984 by C.H. Bennett of IBM and G. Brassard [34] of the University of Montreal, based on a Wiesner’s idea [27]. Although the Brassard and Bennett’s work had been published in conference proceedings it remained virtually unknonwn to the scientific community until Ekert [32] published in 1991 a new protocol, the E91, on quantum key distribution.

One year later, Bennett, Brassard and Mermin [26] (1992) proved that E91 and BB84 are the same algorithm, and a new protocol, the B92 [19], was presented. In the same year the first experimental proof of BB84 [20] was carried out. In 1995, Bennett published the first definition of privacy ampli-fication [23], a post-processing technique needed to reduce Eve’s information about the final key.

From the experimental side, Muller and co-workers at the University of Geneva, using a Pockels cell to choose the polarisation, performed the first QC experiment with optical fibres. They used photons at 800 nm reaching 1100 m [24]. Then, the same group performed, using photons at 1300 nm, the first experiment outside the laboratory, consisting in sending a message from Nyon to Geneva [25].

With these experiments Muller realised that using the polarisation to en-code the key is highly unstable and requires frequent active alignments to compensate for the continuous depolarisation effect due to drifts of the op-tical properties of the fibre.

At the same time Townsend tested a protocol with phase coding [22] sharing a key at a distance of 10 km.

The group of Geneva [21] proposed and implemented a new technique called Plug&Play that allowed compensating automatically in a complete quantum cryptography system all optical and mechanical fluctuations, also through a commercial fibre.

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In parallel, also another channel was being exploited: free space. It is possi-ble, using satellites that orbit around the Earth, to share information. The first experiment of this kind was performed by Jacobs and Franson in 1996 [18]: they exchanged a key over 75 m in bright daylight without excessive noise. In 2001 Rarity [17] and co-workers exchanged a key over a distance of 1.9 km in nocturnal conditions.

From a theoretical point of view, B92 was analyzed more in detail [16] and a more rigorous security proof was derived [14,15].

In 2000, the Photon-number-splitting (PNS) attack [30] was described for the first time. The security of QC protocol is guaranteed only if single pho-ton sources are used, otherwise to mantain the same security level the key rate and the transmission distance must be lowered [13, 12,11]. Some pro-tocols were invented to circumvent the limitations of the light sources used in applications (typically strongly-attenuated lasers) like SARG04 [7]. In parallel to this development, the field of practical QC grew in breadth and maturity. New families of protocols were proposed, notably continuos-variable protocols [8,9] and, more recently, distributed-phase-reference pro-tocols [10].

Recently, QC reached the commercial market: MagiQ6_{, Idquantique}7 _and BBN technologies8 sell products that implement QC in optical fibre chan-nels. Furthermore, the European Community funded many research pro-grams on this topic, including SECOQC9 _{that demonstrated the first} imple-mentation of QC protocols in a large, multi-point network.

1.4 The BB84 Protocol

The BB84 is the first protocol proposed for quantum cryptography. In this algorithm two non-orthogonal bases (also known as alphabet ) are used, in this way Bob and Alice can share 4 quantum states. In QC systems a convenient quantum state is the polarisation of light: bits can be coded using horizontal-vertical polarisation, +, or using diagonal polarisation, ×, that is the same base rotated of ±45◦ respect to the vertical direction, see Tab.1.1.

Typically, maximally-conjugated bases are chosen, i. e. any pair of vec-tors, one for each basis, have the same overlap: | < ×|+ > |2 = 1₂. Some quantum algorithms were proposed where the overlap is not perfectly the same. In this manner the key shared between Alice and Bob is less sensitive

6 www.magiqtech.com 7_{www.idquantique.com} 8 www.bbn.com 9_{www.secoqc.net}

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Alphabet Bit 1 Bit 0

+ vertical polarisation horizontal polarisation

× +45◦ polarisation -45◦ polarisation

Table 1.1: Alphabets

Figure 1.1: Example of a simple QC setup: Alice selects photons bases with Pockels’ cell sending them to Bob through a quantum channel. To read them Bob uses a 0.5/0.5 beam splitter and a polarising beam splitter. The beam splitter splits the photon flux randomly while the polarising beam splitter uses birifrangent materials to separate photons with different polarisations.

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to noise, but it is also more difficult to detect the presence of Eve.

Table 1.2shows the steps required to share a secret key once the bases are chosen:

1. Alice chooses, in a completely random way, a sequence of bits. Note that, at the end of the process, only few of these bits will become the key.

2. Alice chooses, again completely randomly, a base of the alphabet for each bit (in our case she chooses between × or +). A Perfectly random choice is a critical point for QC implementations; computers are deter-ministic device that cannot create truly random numbers. Moreover, to make sure that the process does not merely appear random while having some hidden deterministic pattern, the process needs to be completely under control. A natural solution is to send single photons to a beamsplitter and detect the output direction.

3. Bob chooses randomly, for each incoming photon, the receiving base. Due to both Alice’s and Bob’s random choices, they will not always choose the same base. If he chooses the wrong one he can read likewise a bit 0or 1. At the end of this step, Bob obtains a string, called raw key, with 25% of error rate.

4. Using a public channel Bob announces, for each detected particle, which base he chose.

5. Alice reveals publicly when Bob used the same base as hers. Note that both of them do not reveal the value of the bits (in Tab. 1.2

ycorrisponds to the cases when the same bases were used).

6. Alice and Bob discard the wrong bits and use the remaining sequence of bits to encrypt and decrypt the message (sifted key).

Steps 1)key 0 1 1 0 1 1 0 0 1 0 1 1 0 0 1 2) × + + × × + × + + × + + × × + 3)RK + + × × + + × × + × + × × + × 4) y y y y y y 5)SK 1 0 1 1 0 0 Table 1.2: BB84 procedure.

Let us now analyse the security of the above ideal protocol (let me remark that, up to now, the errors produced by technical imperfections were not

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taken into account). If Eve intercepts a particle propagating from Alice to Bob, the latter will not receive it and will tell Alice to disregard the cor-responding bit. To this end it is crucial that Bob knows when a photon is supposed to arrive. To avoid that, Eve has to send another particle to Bob. Ideally she would like to send this particle in its orginal state, keeping a copy to herself. Owing to the non-cloning theorem Eve cannot do this without introducing errors.

The simpler strategy that she can adopt is called intercept-resend : Eve measures each particle in one of the two bases and then she sends to Bob a particle in the state that she has measured. In about one half of the cases, Eve is lucky: she chooses the same base as Alice and the particle sent to Bob is in the correct state. In the other cases, since her measurement has an overlap of 1₂ with the correct states, Alice and Bob experience an unex-pected increase of the error rate that shows the eavesdropper’s attempts. In summary, Eve gets 50% of the information about the sifted key and Alice and Bob have an additional 25% of error rate on this key. To estimate the error rate they can, for example, divide the key in subsets and communicate publicly the parity of each set.

If, however, Eve applies this strategy to only a fraction of the particles shared, maybe 10%, then the error rate is only 2.5%. Such a low value can be covered by errors due to technical imperfections, while Eve’s information will be a non-negligible 5%. Alice and Bob can reduce the amount of infor-mation detained by Eve by using Privacy-amplification techniques.

1.5 Photon-Number Splitting attack

The practical implementation of Quantum Cryptography algorithms is by no means a simple task. Many factors may lead to transmission errors and mask Eve’s attacks, therefore a real system has to be designed very carefully. In particular, in 2000, G. Brassard described [30] a new kind of eavesdrop-ping attack that exploited the non-ideality of the photon source. While QC theoretical protocols assume a source that emits exactly one photon at a time, in real devices multiphoton emission can occur. Eve can take advan-tage from multiphoton-emission events to obtain information about the key, while remaining perfectly hidden.

Let us consider, for example, that Alice uses a photon source that emits, with 90% of probability, a single photon while the other times a multipho-tons pulse occurs. Naturally, Alice does not know when a multiphomultipho-tons pulse will be generated.

I assume also that the channel losses amounts to 90% so that Bob is ex-pected to receive only 10% of the photons sent by Alice.

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measures the number of photons of each pulse and if it is one, she just blocks it. Instead, if the bit is carried by a multiphoton pulse she can split the pulse and preserve one photon for herself while sending the other to Bob via an ideal lossless channel10. When Bob measures the losses of his raw key, he obtains what he expects so he cannot have any clue about Eve’s presence. Then, when Alice and Bob reveal publicly the bases used, Eve knows how to measure her photons to obtain all the information shared between Alice and Bob.

Brassard concludes in his article that if the probability of multiphoton gener-ation is higher than the yield, i. e. the probability of a photon to be detected by Bob, the protocol is totally insecure.

Actually the severity of this kind of attack can be reduced, even without a single-photon source, by using decoy pulses. These are high intensity pulses sent at random intervals. Since Eve does not know when they are generated, she has to apply the PNS attack also to them and the resulting higher losses can be detected by Bob and Alice, revealing Eve’s presence.

1.6 Technological Challenges

In this Section I shall analyse the main components that enter a complete Quantum Cryptography setup highlighting the impact that they have on the system performance. In particular, I shall focus on the single-photon sources and the detectors.

1.6.1 Sources

Until the PNS attack was first described, lasers were the obvious photon source of choice for the implementations of the BB84 protocol. Afterwards, many research projects were started aiming at new kind of sources, with lower multiphoton-emission statistics, called sub-poissonian sources.

Lasers

The emission of a pulsed laser in a given mode is described by a coherent state of the field:

|√µeiθi ≡| αi = e−µ/2

∞ X n=0 αn √ n! | ni, (1.1) 10

When an attack is formalized theoretically Eve is assumed to be with unlimited re-sources, so she can have also a lossless channel. Her only limits are imposed by the Physics laws

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where µ =| α2|. The factor eiθ_{is a reference for the phase and it is useful for}

both continuous-variabile or distributed-phase-reference protocols. Control over the phase is available in systems such as mode-locked lasers [31]. When the reference control is not needed:

and ρ the emitted state. Therefore, laser light contains µ photons per pulse on average with a Poissonian distribution.

The probability of having a multiphoton pulse, assuming that we know that at least one photon is present, can be written as:

P (n > 1 | n > 0, µ) = 1 − P (0 | µ) − P (1 | µ) 1 − P (0 | µ) = 1 − e−µ(1 + µ) 1 − e−µ ' µ 2, (1.4) while the probability of empty states can be derived by

P (0 | µ) = 1 − µ. (1.5)

According to Brassard [30], to avoid PNS attack the probability of multi-photon emission has to be lower than the yield. Equation 1.4suggests that it is possible to reach this condition by attenuating the laser. This kind of single-photon source, which is called faint-pulse laser, however, has the drawback that the probability of having no photons at all in a given pulse increases as µ is decreased (see Eq. 1.5). To avoid the resulting reduction of the key-generation rate, the pulse rate and the detection bandwidth need to be increased. Increasing the bandwidth, anyhow, adversely affects the dark-count rate of some detectors (especially InGaAs-based ones).

Furthermore PNS attacks can still be performed if the losses of the system are larger or equal to the probability of non-multiphoton pulses, in other words if transmission is lower than the multiphoton-emission probability [30] given by Eq.1.4:

t < µ

2, (1.6)

where t is transmission probability of the quantum channel.

It has also to be taken into account that, if Alice lowers too much the average number of photons per pulse, the numbers of errors affecting Bob’s key will increase due to the dark counts of Bob’s detector. To allow Bob and Alice to construct a sifted key, the probability of a dark count in a unity of time (pdark) has to be lower than the rate of “good” counts:

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Figure 1.2: Comparison of theoretical (curves) and experimental (symbols) quantum communication rates as a function of transimission losses for a faint-pulse laser source (dashed lines and star symbols) and for a single-photon source (solid line and “x” symbols) [41].

where η is the detector efficiency.

Brassard showed that the transmission efficiency for faint-pulse laser should be larger than:

pdark

ηµ +

µ

2η. (1.8)

Considering that the optimal choice for Alice’s rate of emission is µ = √2pdark

[30], the limit for the transmission probability of the quantum channel is given by:

t > √

2pdark

η . (1.9)

In the case of true single-photon sources (µ = 1), Eq.1.7 yields a transmis-sion limit of:

t > pdark

η . (1.10)

Comparing Eq. 1.8 and 1.10 it is obvious that true single-photon sources permit longer transmission distances than faint-laser sources. Figure 1.2

shows a comparison between quantum communications rates with a true single-photon source and with faint-pulse laser as a function of tramission probability.

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Sub-Poissonian sources

The limits imposed by faint-pulse lasers are, until now, the main obstacle to QC-protocol implementations. To overcome them new kinds of sources are needed.

Single photon sources have been proposed and realised based on many dif-ferent approaches. In the following I shall briefly describe the main classes: Atoms and ions are perfectly reproducible systems with very well characterized narrow optical transitions. With the appropriate choice of atom or ion all the spectrum from visible to near IR regions can be covered. Unlike solid-state devices, atoms do not have non-radiative recombination channels: the fluorescence quantum yield, defined as the ratio between number of photons emitted over the number of photons absorbed, is unity.

Simultaneous emission of two photons is negligible if a single atom or ion is isolated. The main disadvantage of this kind of single-photon sources is the isolation, manipulation and trapping of the single atoms or ions.

In 2000 Pinkse et al. [42] succeded in trapping a single atom in a cavity, and seven years later the same group published a detailed work [43] about a single photon source based on a Rubidium-85 atom confined in an optical cavity.

Organic molecules can be used to produce light from visible to near-IR. Molecular optical transitions are characterized by strong an-tibunching at all temperatures (from low to room temperatures), but molecules are also characterised by low photostability due to bleaching phenomena. The fluorescence yield decreases with increasing temper-ature but it remains higher than 90% for many fluorescent dyes. Self-assembled quantum dots are the main alternative to atom

sources. They are structures based on semiconductor materials which confine electrons in a tridimensional potential that leads to the quan-tization of the energy levels similar to what happens in atoms. The growth process used to produce self-assembled quantum dots does not guarantee a precise control over the dimensions of the dot or over their positioning on the substrate. This makes them not particularly useful for widespread applications: several devices need to be fabri-cated out of a wafer and then individually tested to identify those that actually contain just one quantum dot that emits at the desired frequency.

Contrary to atoms, quantum dots present also non-radiative recombi-nation channels that broaden the spectral-width of the emission and reduce the efficiency of the device. Additionally, they tipically display

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several closely-spaced emission lines and require filtering of the output photons by means of a monochromator. At increasing temperatures, the emission linewidth broadens increasing the probability of multi-photon emissions. To avoid that, the quantum dots must be operated at cryogenic temperature.

The dot excitation can be performed with electric and optical signals. In 2000 Mitchler et al. demonstrated sub-Poissonian emission from a quantum dot [44].

1.6.2 Quantum Channels

The most used physical channels for QC implementations are optical fibre and free space due to the high-transmissivity windows that they feature at selected frequencies.

As discussed above, the losses of the quantum channel are of paramount importance as they determine the transmission distance that can be reached with a given source.

Losses in optical fibres are due to absorption and leakage and depend on the fibre length (with exponential behaviour) and on the wavelength. For the widely used data fibers, losses are minimal at the two so-called telecom windows: around 1330 nm and 1550 nm (see Fig.1.3).

Atmosphere has many windows of transmission (see Fig. 1.4) at 800, 1300

Figure 1.3: Loss spectrum of a single mode commercial-Optical-fibre. and 1550 nm. Among all the above-mentioned spectral windows, one of the best choices is the one at 800 nm thanks to the high quality of the Si-based detectors available.

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Figure 1.4: Atmospheric transmittance as a function of the wavelenght.

1.6.3 Detectors

The quality of a detector for QC applications depends essentially on the ability in measuring accurately in time the arrival of a single photon with high efficiency. The main parameters that quantifies the quality of a detector are:

quantum efficiency: the probability that a impinging photon generates an electron-hole pair (ηq).

detection efficiency: the probability that the generated pair leads to a measurable current (ηd).

dark-count rate: which is the rate at which false-positive detections occur.

afterpulsing: an increase in the dark-count rate immediately after the arrival of a photon.

Noise-Equivalent Power (NEP): is defined as NEP = (hc)/(ληq)

√ 2ND,

where h is the Planck’s constant and c velocity of light, λ is the wave-lenght of the impinging light, ND is the number of dark counts. It

represents the signal power necessary to attain a unity signal-to-noise ratio in one second of integration time.

time jitter: the time between the arrival of the photon and the genera-tion of the consequent electrical signal. The value of this parameter is determined from the full width at half maximum of the peak generated through multiple measurements of this delay.

dead time: the time needed by the detector (typically APDs’ are af-fected) to return to measuring condition after a photon is detected.

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The most widely used detectors for discrete-variable protocols are Avalanche photodiodes (APDs). A diode is a junction between a semi-conductor with an excess of holes (p) and one with excess of electrons (n). The abrupt discontinuity of the two material creates an electric field that deplets a region near the junction of all the carriers (depletion region).

When a photon with appropriate energy (higher than semiconductor bandgap) impinges on the depletion region of the diode, typically held in reverse bias close to or slightly above the breakdown voltage, it generates an electron-hole pair and both particles are drifted away by the electric field generating an avalanche by means of impact ionization and therefore a mea-surable current [47]. The process is illustrated in Fig.1.5.

To stop the avalanche effect and restore the normal detection mode, the reverse voltage between n-p regions is compensated by a quenching circuit. Si-based Single Photon Avalanche Diode (SPAD) are the most widely used detectors since they work well in visible range (400-1000 nm) with good detection efficiency ('50%) but for telecommunication wavelenghts SPADs based on Ge are used. These devices, however, have higher dark counts and therefore need to be cooled.

Alternative candidates to Ge-based detectors for near-room-temperature single-photon counting at a wavelength of 1550 nm are devices based on InGaAs/InP heterostructures. However, devices based on these materials report many issues regarding aferpulsing and dark count rates and need to be further optimised.

Currently, the best single-photon detectors for telecommunication wave-lenght are based on narrow superconducting stripes [29]. They work at telecommunication wavelengths, have very low dark count and good detec-tion efficiency and can be rearranged in an array. However, their main drawback is that their operating temperature is of a few Kelvins and they are extremely expensive.

Table 1.3 presents some characteristic values for the most widely used detectors are presented.

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Figure 1.5: Impact ionization process occuring in Avalanche Photodiode: in the top is schematised a process of impact ionization occurring in an avalanche photodiode: an electron-hole pair is generated by the absorption of a photon in the depletion region of a reversed bias APD. In the lower schematic the electron causes further impact ionizations, multiplying the number of electron-hole pairs and producing a self-sustaining avalanche of carriers [29].

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APDs λ NEP Timing ηd rD Diam T

jitter

(nm) (WHz−1/2) (ps) (%) (Hz) (µm) (K)

Si 830 3.7 · 10−18 500 45 25 150 253

InGaAs 1500 9 · 10−17 55 10.9 2925 30 243

Table 1.3: Overview of typical parameters of single-photon detectors: de-tected wavelength (λ), single-photon detection efficiency (ηd), Noise

equiv-alent power (NEP), dark-counts rate (rD), timing jitter, operating

temper-ature (T ), diameter of detection area (Diam)

1.7 Hanbury-Brown-Twiss Interferometer and the

antibunching effect

In the previous sections I have emphasized that QC protocols need a source that emits single photons on demand with high repetition rate. Here I shall discuss how it is possible to quantify the probability of multiphoton emission and therefore characterise a single-photon source.

To this end, Time-Correlated-Single-Photon-Counting (TCSPC) measure-ments are performed with the setup proposed by R. Hanbury Brown and R.Q. Twiss [40] (HBT), an example of which is shown in Fig.1.6.

The light arriving from the device under test (the left side of Fig. 1.6) is

Figure 1.6: Hanbury-Brown-Twiss interferometer [41].

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SPADs, connected respectively to the START and STOP inputs of an ac-quisition device (in the following I shall refer to the SPADs connected to the START and STOP inputs as START and STOP detectors respectively). The device records the delays between each START-STOP sequence, build-ing an histogram of the results.

If the source emits in the single-photon regime it is impossible to record a delay equal to zero.

This procedure yields the second-order correlation function of the source that I shall discuss in the following in details assuming photons as indivisible particles [39]. If n(t) photons arrive at a detector per unit time, the counting rate of that detector is:

R(t) = βn(t), (1.11)

where β is the detection efficiency.

A 0.5/0.5 beam splitter is a device that splits a packet of n photons in two packets which contain k and n − k photons where k can be 0, 1, 2, 3.., n. The probability of a particular outcome (two packets of k and n − k photons) is wk,n−k: wk,n−k = 1 2 n n k . (1.12)

The number of coincidence counts measured by the HBT setup at a given time delay (τ ) is proportional to the probability of detecting a photon with the STOP detector a time τ later than the detection of a photon by the START detector.

This coincidence count rate for a delay equal to zero is: K(t; 0) ≡ β2 X

k

wk,n−kk(n − k) =

β2

4 n(t)[n(t) − 1] (1.13) where the double brackets indicate the average over the ensamble consisting in the statistical fluctuations of the emitted photons.

The counting rate of each of the two detectors is: R0(t) = β X

k

wk,n−kk =

β

2 n(t) , (1.14)

while the coincidence count at delay τ is: K(t; τ ) = β2X k,j wk,n(t)−kwj,n(t+τ )−jk(n−j) = β2 4 n(t)[n(t+τ )] . (1.15)

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The photon numbers at time t and t + τ will be uncorrelated if τ goes to ∞ providing a coincidence count rate equal to:

K(t; ∞) = R0(t)R0(t + ∞). (1.16)

The parameter r, defined as:

r(t) = K(t; 0) − K(t; ∞)

K(t; ∞) , (1.17)

allows to quantify the occurrence of the antibunching phenomena: when r > 0 light is bunched [39]. A negative value of r implies that the source is emitting in the antibunching regime.

The parameter r can be written as: r = n(n − 1) − n 2 n 2 = ∆n2_{− n} n 2 , (1.18) where: ∆n2 = n2 − n 2, (1.19)

is the variance of the photon number.

Since it is well known that ∆n2 = n for a Poissonian distribution, sources whose emission follows this statistic, e. g. lasers, have r = 0 while antibunched sources emit with a narrower distribution and are therefore referred as sub-Poissonian sources.

As discussed in the previous sections, QC implementations based on laser sources have several limitations due to the large probability of multiphoton emission. To lower the latter, it is necessary to reduce the intensity of the laser, leading to a reduction both in the transmission speed and in the achievable distances. Sources emitting in the antibunching regime will allow overcoming these limitations: the photon-number fluctuations of the sources, i. e. ∆n2, is smaller than in the Poissonian case allowing the cryptography apparatus to work at larger intensities n of the source while keeping multiphoton events below acceptable values. The best-case scenario would be represented by a perfect single-photon source, where ∆n2 = 0 and n = 1. TCSPC measurements made on such a source would show a vanishing zero time-delay peak, K(0, 0) = 0. If the source emits in pulsed regime the possible delays of the coincidence count rate become discrete and K(t; τ ) 6= 0 only when τ = pj where p is the pulse period and j = 0, ±1, ±2.... In the following I shall replace K(t; τ ) with Kj in the case of pulsed sources.

The result of a TCSPC measurement performed with an HBT setup, K(t; τ ), depends on the detector efficiency through the parameter β. A quantity that depends only on the properties of the source is G(2)(τ ), defined as:

K(t, τ ) = β

2

4 G

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Figure 1.7: Measured correlation function G(2)(τ ) of (A) a mode-locked Ti:sapphire laser and (B) a single photon emission from a quantum-dot source under pulsed excitation (82 MHz). Figure taken from Ref. [44].

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Chapter 2

Lateral light-emitting diodes

In the previous chapter I have discussed the reasons behind the large re-search efforts devoted in recent years toward the realisation of sub-Poissonian sources as these would allow high key-generation rates without the risks as-sociated to PNS attacks.

Currently, the best single-photon sources in terms of repetition rate and sup-pression of multiphoton events are based on semiconductor quantum dots [29, 41], but these did not find application in commercial systems due to their complex and low-yield fabrication procedures.

Antibunching emission has also been demonstrated in high-efficiency light-emitting diodes (LEDs), [6,2], driven by very stable current sources, i. e. whose noise is well below the shot-noise regime. The high efficiency of these LEDs implies that the fluctuations of the emitted photons reflect closely the injected-electron statistics. Anyhow, this approach to antibunch-ing emission works only in the high-injection-current regime (the lowest in-jection current reached by an antibunched LED is ≥ 1 µA) [5]. Lowering the current leads to an increase of the noise up to the limit of the shot noise, which corresponds to a Poissonian electronic distribution.

In this thesis I shall demonstrate a novel LED geometry that can emit in the single-photon regime without relying on a low-noise injection current. My devices are based on a planar p-n junction [48,49,50,36] realized on a p-type doped GaAs/AlxGa1−xAs heterostructure. This approach also

ben-efits from the extremely small junction capacitance, characteristic of planar LEDs, which leads to large operation bandwith of the device (up to GHz fre-quencies) [48]. The optical observations of these junctions, biased very close to conduction threshold, have shown that the emission occurs from spatially localised spots placed along the junction. As an example, Fig. 2.1 reports an optical image of one of the devices realised at NEST laboratories at 15 K of temperature and driven by forward bias of 1.44V: emission from several spots is observed. This behaviour is attributed to the formation of localised

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states below the conduction-band bottom following the n-type-contact fab-rication. At low injection currents, electrons are preferentially injected into these states rather than in the conduction band, which is higher in energy, leading to well-localised emission spots.

As I shall show in the following the localisation occurs over sub-micron length scales. It can therefore be expected that at cryogenic temperatures the Coulomb repulsion between electrons plays a relevant role in determin-ing the occupation number of these states, preventdetermin-ing an additional electron being injected in the same state before the first electron has recombined and leading to antibunched emission.

Figure 2.1: Optical image of the device realised at NEST. The emissions, spatially localised in spot are highlighted by the arrows. Courtesy of Giorgio De Simoni.

In the next sections, after a brief introduction about semiconductor het-erostructures, I shall describe the fabrication details of planar junctions.

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2.1 Heterostructures

A heterostructure is a crystal whose chemical composition changes along a given direction.

One of the most used techniques to grow heterostructures is the Molecular Beam Epitaxy (MBE) [4]: semiconductor layers are deposited on a substrate in a high-vacuum chamber by means of molecular beams originating in ef-fusion chambers, where the chemical elements required for the synthesis are thermally evaporated.

The beams impinge on a rotating crystalline substrate which is mantained at high temperature (≈ 600◦C) to achieve an uniform spreading of the de-posited chemical elements on the surface. The chemical composition of the layers depends on the intensity of the beam flow, which can be controlled by tuning the effusion-chamber temperatures. Abrupt changes of the com-position can be achieved by opening or closing shutters placed in front of the effusion chambers.

The MBE technique allows an extremely fine control over the thicknesses of the epitaxial layers, down to the single atomic layer.

High-quality MBE growth is possible if the crystal structure and lattice constant of the layers are the same. Figure2.2shows a graph of the energy gap as a function of the lattice constant of some of the most widely used compounds. The darker areas highlight the materials that are compatible in this respect. If a compound with a crystal structure or lattice constant different from the substrate is evaporated, it is still possible to achieve a good crystal quality if the thickness of this layer is below a critical value, beyond which defects are generated.

Changing the composition of the crystal along the growth direction in-troduces abrupt discontinuities of the energy bands within the crystal and has a strong impact on the electron transport properties. Depending on the relative alignment of the gaps, heterojunctions are classified as shown in Fig.2.3. By confining a small-gap material, such as GaAs, within two layers of larger-gap semiconductor, e. g. AlxGa1−xAs/GaAs, quantum-well

struc-tures can be obtained. The free electrons in the larger-bandgap material are transferred and confined, due to the higher electron affinity, into the GaAs region. Under suitable condition, this confinament can be so strong that electron degrees of freedom in the growth direction are completely freezed. In this case, a true 2-dimensional electronic gas (2DEG) or, if the charge carriers are holes, a 2-dimensional holes gas (2DHG) is created.

An extremely useful feature of semiconductor materials is the possibility to tune the free-charge density by introducing, during the growth phase, impurities into the crystal that provide electrons (donors) or holes

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(accep-Figure 2.2: Semiconductor compound energy gaps as a function of the lattice constant.

Figure 2.3: Heterojunction classification: (a) 1st _{kind, when the energy gap}

of one layer is entirely within of the gap of the other; (b) 2nd kind staggered, when the two energy gaps are misaligned but partially overlapped; (c) 2nd kind misaligned, when the two gaps are completely misaligned.

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tors). Dopants, however, have a negative impact on the mobility of the material: raising the impurity concentration increases the probability of carrier scattering by Coulomb interactions. Furthermore, when the ther-mal energy become lower than the binding energy of the impurities, the electrons are trapped at the dopants (freeze-out ) reducing the free-charge density available. The modulation doping technique [3] allows solving these issues: exploiting the different electron affinity of the various heterostructure regions the free-charge carriers can be spatially separated from the dopant impurities, introducing the latter in the region with lower electron affinity leading to a reduced scattering probability. Confining the charge carriers away from the impurities also avoids the freeze-out. By means of this tech-nique, mobilities higher than 107 cm2/(Vs) were reported.

The sources realised during my thesis are based on a GaAs/Al0.33Ga0.67As

heterostructure (wafer A4149) grown by MBE. Table2.1 reports the com-position of the wafer. The top of the wafer (cap layer ) consists of a 10-nm-thick GaAs layer, required to prevent the oxidation of following AlGaAs layer, which contains, in its central section, a nominal density of 1.33 × 1018 carbon atoms/cm3. The carbon atoms act as acceptors creating a 2DHG in the QW, corresponding to the 5th layer in Tab.2.1. The undoped portion of the Al0.33Ga0.67As spacer layer (4th layer in Tab. 2.1) creates a spatial

separation between the acceptors and the QW improving the mobility of the 2DHG. Under the QW a 100-nm-thick layer of AlGaAs guarantee the confinement in the QW of both electrons and holes. The layers 7 and 8 are repeated 50 times to create a superlattice that increases the quality of the heterostructure and also avoids the formation of an additional two-dimensional carrier layer at the interface between the Al0.33Ga0.67As layer

(6th layer in Tab. 2.1) and the underlying GaAs layer.

The band profile of the heterostructure, calculated by solving self-consistently the Poisson and Schr¨odinger equations, is shown in Fig.2.4. The calculated density of holes in the QW was found to be 1.0 · 1011cm−2 at a temperature of 4 K.

Table2.1reports the energy value of the electronic and hole eigenstates.

2.2 LED fabrication

The result of the MBE growth is a wafer containing the heterostructure. The fabrication of a complete device requires further processing to add con-tacts and define the geometry of the device in the plane of the wafer.

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Layer Material Thickness Comments (nm)

1 GaAs 10 Cap Layer

2 Al0.33Ga0.67As 20 3 Al0.33Ga0.67As 20 doped with C 1.33 × 10+18 atoms/cm3 4 Al0.33Ga0.67As 40 5 GaAs 20 QW 6 Al0.33Ga0.67As 100

7 Al0.33Ga0.67As 2 layers 7,8 are repeated 50 times

8 GaAs 2

9 GaAs 1000

10 GaAs 0.5 mm S.I. substrate

Table 2.1: Nominal composition of the heterostructure A4149.

Figure 2.4: Left: energy-band diagram of the heterostructure along the growth direction (z) calculated by solving self-consistently the Poisson and Schr¨odinger equations. The black line represents the bottom of the conduc-tion band, the red line represents the top of the valence band, and the blue line represents the chemical potential. Right: enlargement of the top of the valence band in the QW region. The green line shows the first hole sub-band vertically translated to its corrisponding eigenvalue.

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Electronic subbands Hole subbands

Level Energy (meV) Level Energy (meV)

1E 1529.8 1H 0.7

2E 1562.1 2H -6.2

3E 1610.5 3H -7.5

4E 1672.1 4H -1.7

5E 1742.4 5H -2.8

Table 2.2: The calculated bottom (top) of the first five electronic (hole) subbands, obtained by solving self-consistently the Poisson and Schr¨odinger equations at a temperature of 4 K. The chemical potential is taken as the zero of the energy. In the valence band, only the first subband (1H) is occupied by holes, while no electrons are present at this temperature in the conduction band.

2.2.1 Mesa

To define the device geometry, it is necessary to limit the presence of the 2DHG to a selected portion of the wafer (the mesa). This is achieved by removing by chemical etching the first layers of the heterostructure, down to the QW, in the regions outside the mesa. The geometry chosen for the fabrication of my LEDs is shown in Fig.2.5.

Figure 2.5: Schematic view of the mesa. All the values are in µm. The realization of a given pattern on a wafer requires the deposition of a thin layer of a specific solution (resist) on the wafer surface. This is achieved by depositing a controlled amount of resist on the waver and spinning it at high speed to obtain the desired thickness and uniformity. The resist is then hardened by means of a soft-baking process that partially removes its sol-vent. Resist materials employed for optical lithography are sensitive to UV

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radiation. When exposed, the chemical composition of the resist changes making it soluble (positive resist) or insoluble (negative resist) to a specific developer. Using a quartz mask having transparent or opaque regions, the desired geometry can be transferred to the resist. The entire procedure is summarized in Fig. 2.6. The soluble parts of the resist are then removed by immersion in the developer. The remaining resist is further hardened by means of a harder bake.

Figure 2.6: Schematic view of the photolitography process. Panel a (b) shows the result in the case of a negative (positive) photoresist.

The pattern resolution depends on several factors, the most important being the wavelength of the radiation used to expose the resist. Resolution of ∼ 1µm can be routinely achieved in UV lithography, while, for very-high-resolution processes (down to ∼ 10 nm) e-beam lithography is employed. The fabrication of the LEDs described in this thesis did not require such high resolutions and was carried out entirely by UV lithography.

The etching procedure can be performed by means of an acid solution (wet etching), or by means of a plasma (dry etching).

The fabrication process of the mesa is summarized as follows:

A thin film of negative resist (S1818) is deposited on the wafer, which is then spun for 60 s at 6000 rpm by a spin coater.

The resist is baked at 90◦_{C for 60 s.}

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The device is immersed into the development solution, MF319, for 30 s. Then, the development process is stopped in water.

The device is immersed in a solution of H3PO4:H2O2:H2O with relative

concentrations of 3 : 1 : 50. In order to remove the quantum well layer an etching depth of at least 110 nm is required. Knowing the etch rate, which is measured by a previous calibration procedure, allows estimating the processing time needed to achieve the desired depth. In the case here of interest, the sample was etched for 90 s, leading to a depth, measured with a profilometer, of 123 ± 1 nm.

2.2.2 Evaporation of p-type and n-type Ohmic contacts The complete LED structure requires the fabrication of a n-type region. Following Ref. [48], this step consists in the removal of the acceptor layer in a selected region and the subsequent evaporation of an n-type contact to provide donors and create the n-type material. An additional p-type contact allows electrical access to the 2DHG. Figure2.7shows a lateral view of the wafer at the end of the lithography process.

Since electrons and holes are confined in the QW, this process yields a lateral p-n junction.

Figure 2.7: Schematic view of the planar LEDs. The yellow shapes are the p-type and n-type contacts. The red area represents the doping layer while the blue area is the quantum-well layer.

The evaporation of metal layers on the wafer surface for the generation, for example, of gate or metallic contacts requires, as in the case of the mesa fabrication, the deposition of a thin film of resist that, after exposure and development, acts as a mask and allows evaporating the desired metals only on specific regions of the wafer surface.

The evaporation process is performed in a vacuum chamber with a base pressure in the range of 10−5÷ 10−7 Torr: high-purity metals, contained in crucibles, are thermally evaporated, diffuses in the chamber, and deposits on the resist-covered wafer forming a uniform, thin layer.

After removal of the remaining resist in acethone, only the areas of the metallic layer deposited directly on the wafer remains (lift-off process).

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During this process, a brief pre-development step and a careful choice of the baking and exposure parameters produce a certain amount of undercut (see Fig.2.8(a)), which guarantees that the metal deposited on the resist is not connected to the one on the wafer surface allowing the removal of the former without damaging the latter during the lift-off procedure.

Figure 2.8: Schematic view of the lift-off procedure.

The p-type contacts consist of three layers: 5 nm of gold, that helps the adhesion between the p-contact and the GaAs, 50 ÷ 60 nm of zinc, that acts as an acceptor for AlGaAs, and finally a layer of 100 nm of gold to protect the underlying zinc from oxidation and to facilitate the following bonding step.

P-type contacts are shaped as squares with 200 µm sides.

The p-type contact deposition is summarized in the following:

The device is immersed in a development solution, MF319, for 20 s. Then the development process is stopped in water.

The quartz mask is aligned over the wafer in order to obtain the pat-tern in the correct position and exposed for 20 s.

The wafer is baked at 120◦_{C for 20 s.}

The desired sequence of metals is evaporated.

The device is immersed in acethone to complete the lift-off process. An optical-microscope image of one of the devices realised during my thesis, after mesa and p-type-contact fabrication, is shown in Fig.2.9.

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Figure 2.9: Optical-microscope image of one of the LEDs, after mesa and p-type-contact fabrication. P-type contacts appear as golden squares.

the quantum well intact. To this end, a 60-to-80-nm-deep etching has to be performed.

The n-contact consists of four layers: 10 nm of nichel, that helps the ad-hesion between n-type contact and the GaAs, 180 nm of a gold-germanium alloy that provides donors in the GaAs, 10 nm of nichel to improve contact penetration in the semiconductor, and a final layer of 100 nm of gold that protects the device from oxidation to facilitate the following bonding step. The n-type contact, shown schematically in Fig. 2.10, consists of a 200 µm × 200 µm square and a thin finger 120 µm long and 20 µm wide.

In order to diffuse the dopants into the heterostructure a rapid thermal anneal (RTA) is performed in nitrogen atmosphere. The device is placed on a metallic stripe heated up to 450 ÷ 500◦C by means of a high-intensity current.

The n-type-contact deposition is summarized in the following:

The device is immersed in a development solution, MF319, for 20 s. Then the development process is stopped in water.

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Figure 2.10: Schematic view of the n-type contact. All the lenght values are intended in µm.

The quartz mask is aligned over the wafer in order to obtain the pat-tern in the correct position. The resist, covered or not by the mask is exposed for 20 s.

The wafer is baked at 120◦_{C for 20 s.}

The device is immersed in a solution of H3PO4:H2O2:H2O with relative

concentrations of 3 : 1 : 50 for 60 s. The echting depth, measured with a profilometer, is 79 ± 1 nm.

The desired sequence of metals is evaporated.

The device is immersed in acethone to complete the lift-off process. A rapid thermal annealing process is performed heating the device up

to 500◦C for 90 s.

An optical-microscope image of one of the devices realised during my thesis, after the n-type contact fabrication, is shown in Fig.2.11.

2.2.3 Resizing of the junction area

To reduce the number of emission spots along the juction, I have performed an additional etching step to reduce the junction size down to a length of

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Figure 2.11: Optical-microscope image of one of the LEDs, after the n-type contact fabrication. The image shows an enlarged view of the end of the finger where the junction area is realized. Inset: optical-microscope image of one of the LEDs, after the n-type-contact fabrication. The white squares indentifies the enlarged part.

25 ± 2 µm, performed following the same procedure used to realize the mesa (see Subsec.2.2.1). The final device is shown in Fig. 2.12.

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Figure 2.12: Enlargement of the junction region after the resizing step. The dashed red (solid blue) lines shows the mesa before (after) the resizing step. The yellow line indicates the region where the p-n junction is realized.

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Chapter 3

Experimental results

The samples fabricated during my thesis work have been characterised in terms of electrical and optical properties in a broad range of temperatures. In this chapter I shall present the experimental setup and the results demon-strating the correct operation of the LEDs. In the last section I shall de-scribe the details of the HBT setup that I have realised to characterise the second-order correlation function.

3.1 Electrical properties of the LEDs

In order to test their electrical properties, the devices were mounted on a chip carrier designed to allow high-bandwidth characterisation, shown in Fig.3.1, and connected to the chip-carrier pads by gold-wire bonding.

Figure 3.1: High-bandwidth chip carrier.

The current-voltage characteristics of the junctions were determined by means of a source-monitor unit (Keithley K2600) by applying a voltage to the n-type contact while keeping the p-type contact grounded and measuring

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the current injected by the source-monitor unit.

The voltages applied to the LEDs (−5 ÷ 5 V) and the measured currents (−5 ÷ 5 µA) were well within the specifications of the Keithley K2600.

All the devices showed the expected rectifying behaviour of p-n junctions in the entire range of temperature explored, from room temperature down to 5 K. Figure3.2shows two representative curves at room temperature and at 19 K.

The derivative of the I-V curve below -2.1 V saturates to a constant value corresponding to ∼ 2 kΩ at 19 K, due to the 2DHG resistance as it is shown in the inset of Fig.3.2.

Figure 3.2: Plot in logarithmic scale of the absolute value of the current as a function of the voltage applied to the n-type contact of one of the devices under test at room temperature and at 19 K. The device showed the expected rectifying behaviour. Inset: device resistance calculated by differentiating the I-V curve. Below -2.1 V the resistance saturates to a costant value of ∼ 2 kΩ at 19 K, due to the 2DHG resistance.

At cryogenic temperatures, the current as a function of voltage shows a double-threshold behaviour, increasing abruptly at V1 ∼ −0.68 V and at

V2 ∼ −1.46 V, as indicated by the arrows in Fig.3.2. The threshold at V2 is

consistent with the bandgap (∼ 1.52 eV) of the GaAs QW and reflects the expected behaviour for the forward conduction of a p-n diode. The sudden

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is due to Schottky conduction from portions of the n-type contact that – due to less-than-perfect lift-off or to lateral diffusion of the metal during the annealing process – lay on the mesa, where the 2DHG is present. The value of V1 is indeed consistent with half of the bandgap of the QW. The

presence of the Schottky conduction in my devices has no impact on their optical properties because electrons are injected from the metal directly into the valence band. This process does not lead to light emission and I shall not take it into account in the following.

3.2 Optical properties of the LEDs

To reduce the thermal energy and maximise the impact of Coulomb inter-actions between electrons in the emission spots I have cooled the device at liquid-helium temperatures in a Oxford Instruments Optistat CF-V cryo-stat, shown schematically in Fig. 3.3. The devices were placed on a sample holder and inserted in the cryostat, thermally isolated from the environment by a vacuum chamber with a base pressure in the 10−5÷ 10−7 _{mbar range.}

A continuous flow of liquid helium, controlled by means of a needle valve, is injected into the sample holder from a storage dewar through a low-loss transfer tube. The temperature of the sample, cooled by thermal conduction through a heat exchanger, can be controlled by means of an ITC temper-ature controller with a guaranteed stability of ±50 mK. The optical access to the sample is provided by a large spectrosil-B window.

The junctions were investigated by means of the setup shown in Fig.3.4. The light emitted by the LEDs is focused by the aspheric lens and the 50x Mitutoyo objective into an optical fibre with a diameter of 100 µm and guided into a ISA jobin Yvon Spex HR320 monochromator, that disperses it on a CCD camera. The CCD camera is thermoelectrically cooled down to -70◦C to reduce the dark counts down to a value, given by the CCD specifications, of 0.002 electrons/pixel/hour.

The diffraction grating of the monochromator can be rotated to select the desired range of wavelengths. I have set the centre wavelength to ∼ 810 nm, which corresponds to emission from a GaAs layer and calibrated the CCD by means of a Ne lamp in order to be able to associate a wavelength to each pixel of the CCD. The calibration procedure was carried out by collecting the spectrum of the lamp by means of the CCD and comparing the acquired spectrum with the tabulated emission lines of Ne to identify the line wavelengths. A linear fit, y = A · x + B, of the position of the lines on the CCD and their wavelenght yielded A = −(500 ± 4) · 10−4 nm/pixel and B = 838.1 ± 0.2 nm. This pixel-to-wavelength conversion is applied to all the spectra reported in the following.

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(a) (b)

Figure 3.3: (a) Picture of the Optistat CF-V cryostat mounted on a support with three micropositioners. (b) Schematic view of the cryostat: the chip-carrier is held in front of the optical window by the heat exchanger which also guarantees a good thermal contact between it and the liquid helium inserted by means of a low-loss transfer tube through the entry port. A copper-plated plastic cover shields the low-temperature parts of the cryostat from room-temperature radiation. The sample is held in vacuum during normal operation of the cryostat.

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Figure 3.4: Optical setup used for the characterisation of the spectral emis-sion. The LEDs are cooled down to liquid-helium temperature in the Opti-stat CF-V cryogenic system. The emission, collected by an aspherical lens and by a Mitutoyo 50x objective, is guided to a Jobin IVO ISA HR 320 monochromator through an optical fibre. The dispersed light is collected by a SPEC 10:256 CCD camera.

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The emission spectra of one of my devices, collected at 9.9 K are shown in Fig.3.5at different values of the n-p bias, ranging from -1.45 V down to -1.95 V in steps of -0.05 V. The intensity and lineshape of the emission are observed to vary as a function of the voltage applied. At low forward biases the emission is centred at ∼ 818 nm, and is characterised by a double-peak structure. At increasing biases, the smaller-wavelength peak grows faster than the one at longer wavelength, and, eventually, a third peak appears at lower wavelengths (∼ 796 nm). At higher wavelengths, a peak at ∼ 830 nm is observed due to carbon defects in the GaAs, typical of MBE-grown heterostructures.

Figure 3.5: Spectra of a representative device at different bias voltages, from -1.45 V to -1.95 V in steps of -0.05 V at a temperature of 9.9 K. The exposure time for each spectrum was 3 s. The arrow marked with X indicates the emission due to recombination of neutral excitons. The arrow labelled with X+ _{indicates the emission due charged excitons. Emission from carbon}

defects is marked by an arrow at ∼ 830 nm. A peak originating from the recombination of electron in the first excited subband is observed at ∼ 796 nm.

To identify the origin of the observed spectral features, I have performed a best-fitting procedure of the double-peak structure at ∼ 818 with a sum of two Lorentzian curves. The area of each Lorentzian as a function of the

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of the injected current in semi-log scale. A linear fit of the data yields slopes Alow = 2.28 ± 0.28 and Ahigh = 0.83 ± 0.1 of the evolution of the

high-wavelength and low-high-wavelength peaks respectively. This behaviour allows identifying the two peaks as originating from the recombination of neutral excitons (X) and charged excitons (X+). Indeed, the intensities of the X and X+ emission lines have a characteristic evolution: the former grows quadratically as a function of the injected current, while the emission from X+ recombination increases linearly [1], consistently with the observed be-haviour. Qualitatively, the formation of a X+ exciton, composed of one electron and two holes, is favoured with respect to the formation of X ex-citons due to the high concentration of holes available in the 2DHG when the number of injected electrons is small. Increasing the injected current leads to a decrease (increase) of the hole (electron) density, favouring the formation of X excitons.

The distance in energy between the peak at 796 nm, shown in Fig. 3.5, and the X peak was found to be 34 meV, which is consistent (here I am neglecting the differences in the excitons binding energies, which are ex-pected to be of the order of 1 meV) with the calculated separation of 32 meV between the first two electronic subbands, 1E and 2E (see Tab. 2.1) suggesting that this feature originates from the 2E-1H transition. This is generally forbidden in symmetrical QWs because of the selection rules for optical transitions in two-dimensional systems. Nevertheless, in the case of my QW, both transitions from 1E and 2E to 1H are allowed. This is due to the built-in electric field, perpendicular to the QE, occurring because of charge transfer from the acceptor layer to the QW, that renders the electron and hole envelope wavefunctions asymmetric with respect to the centre of the QW. The transition probabilities P1E−1H and P2E−1H in the

envelope-function approximation, obtained from the calculated waveenvelope-functions (shown in Fig. 3.7), were found to be of the same order of magnitude, confirming that both the transitions are allowed:

P1E−1H ∝ |hψ1E| d dz | ψ1Hi| 2 _{= 5.1 · 10}23_m−4 _(3.1) P2E−1H ∝ |hψ2E | d dz | ψ1Hi| 2_{= 1.1 · 10}23_m−4 . (3.2)

3.3 Electrical setup

In order to be used in a quantum-cryptography setup, a source has to be able to emit single photons on demand, for example in response to an electrical or optical pulse. To test the time-resolved behaviour of my LEDs, I have realised and optimised a setup allowing to excite and study the emission of the devices in the nanosecond range.

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Figure 3.6: Blue and red squares: areas of the emission peaks at ∼ 818 nm derived from a best-fitting procedure with a sum of two Lorentzian on the emission spectra shown in Fig. 3.5 as a function of the diode current. The linear fits, shown as dashed lines, of the X (X+) areas as a function of the current yielded slope and intercept values of AX = 2.28 ± 0.28 and BX =

2.83 ± 0.23 (A_X+ = 0.83 ± 0.1 and B_X+ = 3.95 ± 0.08), consistently with

the expected quadratic (linear) increase of the area of the neutral (charged) exciton.

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Figure 3.7: Envelope functions of the fundamental hole subband 1H (blue line) and of the first two electronic subbands 1E (black line) and 2E (red line). The dark green lines represent the conduction-band bottom and the valence-band top. The asymmetry of the wavefunctions with respect to the centre of the QW allows both the 1E-1H and 2E-1H transitions.