Analogies of Positronium & Quantum Dot Excitons

(1)

Analogies of Positronium &

Quantum Dot Excitons

THESIS

submitted in partial fulfillment of the requirements for the degree of

BACHELOR OF SCIENCE

in

PHYSICS

Author : Sam Woldringh

Student ID : 1810820

Supervisor : Wolfgang Löffler

2ndcorrector : Eric Eliel

(2)

Analogies of Positronium &

Quantum Dot Excitons

Sam Woldringh

Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands

June 28, 2019

Abstract

An analogy between positronium and quantum dot excitons is often invoked in the literature. The former consists of a bound

electron-positron pair in vacuum, whereas the latter consists of a bound conduction band electron and valence band hole in a quantum dot. We

investigate the extent of this analogy in terms of their atom-like properties. The properties under study are the fine and hyperfine structure, Zeeman effect, confinement, Rydberg states, Stark effect, and

entanglement of decay products. In addition, we experimentally investigate the photoluminescence of InAs/GaAs microcavity quantum

dots in cavity QED to illustrate the confined Stark effect. We find that several important analogies and differences exist.

(3)

Chapter

1

Introduction

Two bound systems exist in nature, positronium and the quantum dot ex-citon, which are qualitatively similar at first glance. Positronium consists of an electron and a positron, bound together by their opposite charges due to the Coulomb interaction. Thus quantum electrodynamics, the the-ory governing charged leptons and photons, suffices to describe it com-pletely. Quantum electrodynamics (QED) is the most well-established the-ory we have to date, but it is not particularly adept at describing bound states and is rather more suited to the description of freely interacting par-ticles. Even though it has been over 50 years since the theory was formu-lated, no general theory of bound states within QED yet exists. Positron-ium is of continuing interest to both theorists and experimentalists alike because it is an excellent testing ground for QED, and has been used to determine the values of fundamental constants to great precision. As a purely leptonic system made of particles of equal mass, its existence pro-vides a more fundamental theoretical framework than the hydrogen atom. Positronium is far more than a theoretical curiosity, however, as it has ap-plications within medical technologies and chemistry.

The exciton is a bound state that occurs in semiconductors, where an elec-tron in a filled valence band is excited into the conduction band. We speak of the creation of a positively charged hole left behind in the valence band, which can then form a bound state with the electron in the conduction band. Both systems thus consist of an electron plus its antiparticle, though the antiparticle differs in each case, and one may wonder how similar these two systems are to one another when attention is brought to

(6)

further-2

reaching aspects of their physics. The goal of this thesis is to present an ex-tensive comparison between an exciton and positronium, including their behaviour in confined environments such as semiconductor quantum dots for the case of excitons, and nanoporous materials for positronium. Such a study is currently missing in literature; we hope that it is interesting to a broad audience and useful also in education.

In this thesis, we review positronium physics as it stands today, as well as give a brief overview of the field theory underlying its behaviour. In chapter 2, we further introduce the concepts of positronium and quan-tum dot excitons. In chapter 3, we develop the description of positronium within the framework of QED and discuss how this can be used to un-derstand its decay modes. The entanglement between decay products is also discussed. We continue in chapter 4 with a discussion of the positro-nium atomic properties, such as its fine structure, the Zeeman effect, the Rydberg Stark effect, and confinement. In chapter 5 we review the semi-conductor physics of electrons, holes, and free exciton physics. In chap-ter 6, we discuss the notion of an exciton trapped within a quantum dot. The atom-like properties of quantum dot excitons are treated in chapter 7, where we again discuss the fine and hyperfine structure, the Zeeman effect, and the Stark ladder and Rydberg states, finishing with a discus-sion about the entanglement between photons produced by electron-hole recombination. To illustrate the Stark effect in excitons, chapter 8 con-tains experimental work regarding the quantum confined Stark effect on InAs/GaAs quantum dot excitons through resonant and non-resonant po-larised laser spectroscopy. In the non-resonant case, electron-hole pairs will be photo-excited in the GaAs host material after which they will be captured in the quantum dots. In a micro photoluminescence setup we will select single quantum dots and record emission from the various opti-cal transitions using a spectrometer. We conclude in chapter 10 with a care-ful comparison of the two bound systems and future research prospects.

(7)

Chapter

2

Positronium and Exciton

To start, we briefly discuss what positronium is, its history in theoretical and experimental physics, and how it is formed. Then, the existence of an exciton in semiconductor physics is explained. This chapter ends with a model of positronium and excitons such as the Bohr model in atomic physics.

2.1 What is positronium?

Positronium is the bound state of an electron and an anti-electron, also known as a positron. The two particles have equal masses, but opposite charge. Hence, a mutual attraction through the Coulomb force underlies the bound state. Positronium has been of continuous interest to both the-orists and experimentalists since its proposed existence in 1934 by Mo-horoviˇci´c [1]. This is because, unlike other bound states, positronium can be described completely using quantum electrodynamics (QED) alone. Its composite particles, the electron and positron, are fundamental point-like particles that have no inner structure. This makes positronium a more fun-damental system than the hydrogen atom, and provides a testing ground for QED itself [2].

Positronium is of such great interest because it can be called the most fun-damental bound state in nature. Unlike hydrogen and other atoms, its description is not complicated by the existence of nuclear effects. After

(8)

2.2 Positronium History and Formation 4

all, none of the two particles can be viewed as orbiting the other, as their masses are equal, and the electron has no inner structure. In addition, CP violating positronium decays may be detected with a new generation of experiments [3]. This could shed more light on the proposed existence of the axion, a possible dark matter candidate. In addition, positronium enjoys wide application in modern medicine through positron-emission-tomography, commonly known as a PET-scan.

2.2 Positronium History and Formation

Famously, the existence of antimatter was proposed by Dirac in 1927 through the equation that bears his name. It was only in 1933 that the antiparticle to the electron, called the positron, was first observed [4]. This was done in 1932 by Anderson using a cloud chamber under supervision of the famous Millikan. Soon afterwards, the existence of a bound state of a positron and electron was proposed by Mohoroviˇci´c in 1934 [1]. His article went un-noticed, and others independently developed the same idea. The term positronium was coined by Ruark in 1945 [5].

Starting in 1951 [6], experimental research into positronium was done us-ing radioactive isotopes which naturally emit high-energy positrons. Deutsch discovered that when a beam of slow positrons moves through a gas, positronium atoms are created. Ever since, positronium has been stud-ied extensively by various groups around the world. One particularly useful application is the development of the PET-scanner, which employs electron-positron annihilation in the human body to detect, for example, tumours. The development of positronium experimentation continues to this day [7, 8].

Positronium atoms are easily produced simply by bombardment of ma-terials with positron beams. Mama-terials that do this efficiently are called converters [9]. Their efficiencies range from 0.5% to 100%, and emitted positronium kinetic energies range from a few meV to thousands of meV. In this process, two distinguishable spin configurations of ground state positronium, the singlet and triplet, are produced. These have vastly dif-fering lifetimes, as we will discuss later. We highlight here nanoporous silica targets as an example of modern developments in the area of positro-nium formation.

(9)

Nanoporous silica films are bombarded by positrons, which penetrate the bulk material [10]. There, they form ground state positronium, which dif-fuses into the pores. The positronium thermalises due to collissions with the cavity walls, allowing for significant cooling. Diffusion out of the tar-get material then allows for the formation of a very dilute positronium gas. About 25% of the formed positrons are the short-lived singlet state, which is almost indistinguishable from electron-positron annihilation. There is around 30% efficiency for the formation of the triplet state, which has a longer lifetime and is thus more readily studied.

2.3 What are excitons?

It is a property of periodically structured materials that band structures arise [11]. This means that the dispersion relations of electrons, describ-ing the dependence between the wavevector and energy of charge carri-ers, contain gaps. This concept is illustrated in Fig. 2.1. There are en-ergies between the bands that are unavailable to the electron, no matter the wavevector. We call the ranges of energies wherein electrons can exist bands, and the voids between them band gaps.

Figure 2.1: Dispersion curve that displays band structure. Charge carriers live either in the upper band, called the conduction band, or the lower band, which is named the valence band. There is a range of forbidden energies that constitute the band gap.

The filled band of highest energy is named the valence band, whilst the lowest empty band is called the conduction band. The electrons in a ma-terial fill the bands according to the Fermi-Dirac distribution function.

(10)

Whether a material can conduct depends on how far the bands are filled: a completely filled band is called inert because it can carry no current. In the special case of semiconductors, which enjoy large applications within the computer industry, the gap between the topmost point of the valence band and the bottommost point of the conduction band is quite small. The convention is that a material with a bandgap smaller than 4 electronvolts is called a semiconductor, because an electron from the valence band can be excited into the conduction band through an energy that is not too large. This convention is chosen because light at an energy of 4eV has a wave-length of around 310 nm, which is in the ultraviolet range of the spectrum and nearly visible. For a gap of 4 eV and an experiment at room tempera-ture, the Boltzmann factor is approximately exp(−160), so there is no way that an electron is thermally excited into the conduction band. If the mate-rial is doped by adding impurities into the crystal, one may increase those odds, however.

In a semiconductor, an electron at the topmost point of a filled valence band may thus be excited into the conduction band. This can be done through the use of a laser of the appropriate wavelength. The benefit of this procedure is that the material can then conduct electricity, as there are free charge carriers present. Before this, the valence band was com-pletely filled and therefore electrically inert. Upon excitation, we are left with an electron in the conduction band and its absence in the valence band. We call this absence a hole, and treat it as a particle in its own right. The procedure is illustrated in Fig. 2.2. As may easily be verified from total charge conservation, the electron has negative charge, whereas the hole has positive charge. Thus, the hole can be considered the antiparticle to the electron in semiconductor physics. Again, their opposing charges lead to an attractive Coulomb interaction and another type of bound state emerges. The bound state of an electron in the conduction band and a hole in the valence band is known as an exciton.

(11)

Figure 2.2:An ultraviolet photon excites an electron from the filled valence band (lower, broad curve) into the empty conduction band (upper, thin curve). This leaves behind a positively charged hole in the valence band, and an exciton is formed.

Important to note is that the masses of the electron and hole are not equal to the bare electron mass m0, as is the case for positronium. By analysing

the dynamics, they are defined by the curvature of the bands in analogy with the classical expression E = p2/2m together with p= ¯hk. Taking the dispersion in the conduction band at a local minimum to be quadratic:

E(k) = Emin+αcb|k−kmin|2,

where αcb defines the curvature of the dispersion. The smaller αcb, the

broader the quadratic dispersion in the conduction band. We define the effective electron mass m∗_e to be [12]:

¯h2/m∗_e = ∂

2_E

∂k2 =2αcb. (2.1)

For holes in the valence band, the same assumption is made with αvb

de-termining the curvature of the valence band. The effective mass is by con-vention defined as being positive:

¯h2/m∗_h = −∂

2_E

∂k2 =2αvb. (2.2)

Thus, the more sharply peaked the potential, the greater α and the smaller the effective mass. A broad or "fat" curve then describes a heavier parti-cle. In general, the effective masses of the electron and hole are not equal, and herein lies the first important difference between positronium and ex-citons.

(12)

2.4 Scaling Laws 8

2.4 Scaling Laws

In atomic physics, hydrogen is first treated as if the nucleus is infinitely heavy. This approximation is justified, as the proton is around 2000 times heavier than the electron. In the case of positronium and excitons we have to take into account that the masses of the constituent particles are more comparable to one another. The Rydberg constant for an infinitely heavy nucleus orbited by an electron is given by [13]:

R∞ = mee 4

8πe2₀h3_c =13.6 eV . (2.3)

More generally, however, the mass of the electron should be replaced with the reduced mass µ. This is to correct for the fact that both nucleus and orbiting particle have a kinetic energy. The reduced mass is defined as:

µ ≡ m1m2

m1+m2

.

In addition, the permittivity of the material plays a role. The adjustment is that we must replace e0 with e0er, where er is the relative permittivity

or dielectric constant. This is an inherent property of the material, and is related to the refractive index by er = n2. Thus, we define the Rydberg

constant for a bound state of two particles as:

R= µe 4 8πe2_h3_c = µ mee2r R∞. (2.4)

The gross energy structure of atomic systems scales with the principal quantum number n in the following way:

En = −R 1

n2. (2.5)

Clasically, the radii of allowed orbits scales as r = a0n2, where the constant

(13)

2.4 Scaling Laws 9 a0 = ¯h 2 (e2_/4πe 0)me ≈0.529 Å=0.0529 nm . (2.6)

But, once again, we have to adjust this for when the approximation of a stationary nucleus is unjustified and when we consider the bound system within a material: a= ¯h 2 (e2_/4πe₎_µ = erme µ a0. (2.7)

In summary, the energies of the gross structure scale linearly with the re-duced mass and the radii scale inversely with the rere-duced mass. For the case of hydrogen, one can take the reduced mass approximately equal to the mass of the electron. For positronium it follows immediately that

µ = 1₂me, so that the Rydberg energy is 6.8 eV which is half of the

hydro-gen Rydberg constant. Similarly, it follows that the Bohr radius of positro-nium is 1.06 Å, which is twice the hydrogen Bohr radius. For excitons, however, it is the material that determines the effective masses of both electron and hole. Generally, the effective masses are much smaller than the naked electron mass. Therefore the effective mass is even smaller than that of positronium, making the energies small and the radii large. Some examples are listed in the following table [14, 15]

Material m∗e/m0 m∗_hh/m0 µ/m0 er R (meV) a (Å) GaAs 0.07 0.68 0.06 10.9 6.87 96.1 InP 0.073 0.6 0.065 9.61 9.57 78.2 InAs 0.026 0.41 0.024 11.8 2.34 260 AlSb 0.11 0.9 0.098 9.86 13.7 53.2 Positronium 1 1 1_/₂ ₁ ₆₈₀₃ _1.06

Table 2.1: Comparison of effective masses, dielectric constants, Rydberg energy, and Bohr radius for several semiconductors and positronium.

In this table, we have included positronium to allow for a swift compar-ison. Thus, we find that the binding energies of excitons are around 3 orders of magnitude below that of positronium, whereas their radii are 2-3 orders of magnitude larger.

(14)

Chapter

3

Positronium & Quantum

Electrodynamics

This chapter contains a more formal treatment of positronium in quan-tum field theory. We start with a brief discussion of the theory of bound states in quantum electrodynamics, and distinguish between para- and or-thopositronium. The entanglement between decay products is also briefly mentioned. The decay channels of the two types are discussed and com-pared to experiment.

3.1 Bound State Quantum Electrodynamics

One of the most celebrated accomplishments of the 20th century is the velopment of Quantum Field Theory (QFT), which offers a satisfying de-scription of the fundamentals of Nature. As one may recall, the Maxwell equations in vacuum naturally lead to the existence of photons. Photons can be seen as quantised excitations of the field, or particles. In an analo-gous philosophy, QFT takes all fundamental particles to be excitations of associated fields [16].

Various particle processes, such as scattering and decays, are pictorally represented by Feynman diagrams. These are schematic representations of the underlying physics of particle interactions, which contain the ingo-ing and outgoingo-ing particles, as well as unobserved intermediate particles.

(15)

3.1 Bound State Quantum Electrodynamics 11

From a given Feynman diagram, one can easily find an expression for its associated amplitude, which is a measure of the likeliness of the process to occur. From an amplitude, the cross-section and lifetime can be calcu-lated. A scattering or decay is thus a black box: we cannot observe the in-termediate steps, but only its input and output. In principle, one can draw infinitely many Feynman diagrams for a given process. This is known as a perturbation series in interaction strength, in which higher order terms correspond to increasingly more complicated inner workings of the black box. Luckily, for most purposes we only require the first few terms in this series, as the amplitudes for more elaborate diagrams quickly fall off. Quantum Electrodynamics (QED) is the field theory that describes elec-trons, posielec-trons, light, and the interactions between them. It is thus re-sponsible for most day-to-day occurences, from human vision to microwav-ing food to radio communications.* _{QED should naturally be suitable to}

describe positronium, as it consists of an electron and positron. There is an important caveat, however. We would like to describe positronium as a bound state, but QED is appropriate for free particles that can scatter or decay. The problem is that no satisfactory Bound State QED exists as of yet, though the problem is of continuous interest [17, 18]. The reason is that in order to write the aforementioned perturbation series for a process, an expansion parameter must be chosen. For positronium three choices can be made, though each is plagued by its own issues [2]:

The most obvious expansion would be in the coupling strength α. This expansion is then identical to the one for free particle QED, where each power of α signifies an increasingly more complicated Feynman diagram. This procedure is not ideal because bound state calculations only require contributions from the first few terms in the expansions series. It should be noted that this choice of expension paramater is usually made, and the calculation of higher order amplitudes is of continuous interest to some theorists [19–22].

Alternatively, one could expand in the Coulomb strength Zα. The expan-sion is unfortunately not well-behaved, as the limit Zα → 0 describes ex-actly an unbound system which is useless to us. This leads to nonanalyti-cal behaviour, which in turn yields terms logarithmic in α.

Lastly, the mass ratio between nucleus and orbiting particle, m/M may

*_{For a wonderful discussion on quantum electrodynamics, see the book ’QED: The}

(16)

be chosen. Again, this suffers from the limit m/M → 0, which signifies a bound massless particle. As before, logarithmic terms show up.

There is much work still to be done in the field of Bound State QED, but we will set these problems aside and continue the rest of this chapter with a discussion on the decays of positronium. These can be calculated using QED through α-expansions. This is justified because the binding energy is small compared to the constituent masses.

3.2 Singlet and Triplet Positronium

Positronium, being a composite system of two spin-½ fermions, can exist in two distinguishable spin configurations. There exists a singlet, which has a total spin S=0, and a triplet with total spin S=1. The spin 0 singlet is called parapositronium, often abbreviated as pPs, and the spin 1 triplet is named orthopositronium, abbreviated as oPs. One of the most impor-tant features of positronium is that it can decay from its ground state into a number of photons through matter-antimatter annihilation. This has prompted some to call the vacuum the true ground state of positronium, as a ground state is typically the most stable state. This is a matter of per-sonal taste and not physics, however, and so we’ll simply regard the n =1, l =0 state of positronium as the ground state.

To see which decays can occur for ground state positronium, an impor-tant selection rule exists. Intuitively, parapositronium must decay into an even number of photons and orthopositronium into an odd number of photons due to the photon having spin 1 and the conservation of angular momentum. Though factually true, the dependence on angular momen-tum conservation is a common misconception [23]. The true selection rule is based on the existence of positronium as a charge-parity eigenstate. The charge conjugation operator transforms a particle into its antiparticle. It is then obvious to see that positronium is indeed an eigenstate of this opera-tor. For two fermions with orbital quantum number L and total spin S, the charge parity eigenvalue is equal to:

C = (−1)L+S.

This result arises from three factors. Firstly, a factor (−1)S+1 arises from the nature of the singlet and triplet formation: charge conjugation can be

(17)

interpreted as an exchange of the electron and positron. The singlet(S =

0)configuration is antisymmetric under this exchange, whereas the triplet

(S =1)state is symmetric. Second, charge conjugation is equivalent to the parity operator from the perspective of the positronium center of mass. The eigenvalue of the parity operator is(−1)L. Lastly, an extra factor(−1)

arises from the opposite intrinsic parity of fermions and anti-fermions. For n photons, the charge conjugation is:

C = (−1)n,

which comes from field theory [24]. In short, it is due to the invariance of the electromagnetic interaction under charge conjugation C. The inter-action scales with the potential Aµ = (φ,A~), which obviously is

trans-formed into−AµunderC. To preserve the charge conjugation invariance,

the charge conjugation of the photons must have the eigenvalue (−1)n. Thus, the following selection rule naturally arises:

(−1)L+S = (−1)n. (3.1)

We see that for ground state parapositronium, an even number of photons are produced. On the other hand, an odd number of photons is produced for ground state orthohydrogen. Notably, the photons are entangled, and we dedicate some words to this in the following.

Both para- and orthopositronium decay into entangled photons. When parapositronium decays into two photons, the photons are of opposite cir-cular polarisation [25] and propagate in directly opposite direction. There-fore, when one of the two is detected one can measure its (previously un-determined) polarisation and thereby also the polarisation of the second photon. Thus, we say that the photon polarisations are maximally entan-gled. In addition, it was found in [26] that the decay of orthopositronium also produces entangled photons, though the correlations depend on the two angles between the three particles. For a decay into three particles, the energies and angles are no longer singularly determined. It was this fact that lead to the discovery of the neutrino [27]. In [28], generalisations of two-particle Bell states to three-particle states are presented, and the types of entanglement that can emerge from the orthopositronium decay are investigated.

(18)

3.3 Parapositronium Decays 14

3.3 Parapositronium Decays

We have seen that the singlet state parapositronium decays into an even number of photons. Thus, the two primary decay modes are pPs→ γγ

and pPs→ γγγγ, where pPs is an abbreviation for parapositronium and

each γ represents a photon. The lowest order Feynman diagrams for these two processes are shown in the figure below.

Figure 3.1: Feynman diagrams for the first-order diagrams of a decay into two and four photons.

The conventions for Feynman diagrams used here are that fermions are represented by straight lines, photons by wavy lines, the arrow of time moves from left to right, and a fermion in the opposite direction is an antifermion. From these diagrams, the amplitudes can be read off and one finds the following decay rate into two photons [29, 30]:

Γtheory(pPs→γγ) =7989.50 µs−1. (3.2)

This result compares well to experiment, where the observed decay rate is:

Γexp(pPs →γγ) =7990.9 µs−1. (3.3)

Taking the decay into four photons into account, one finds the branching ratio:

BR(pPs) ≡ Γ(pPs→γγγγ)

Γ(pPs →γγ) =1.49·10 −6

. (3.4)

(19)

Γ(pPs) = 7989.6178(2)µs−1. (3.5)

This agrees well with experiments, though the uncertainty on the theoret-ical value is much smaller than the uncertainty of the experimental result. It is hoped that with the new generation of detectors, notably J-PET [7], greater accuracy will be achieved.

3.4 Orthopositronium Decays

We have seen that ground state orthopositronium can only decay in an odd number of photons. There is an important caveat: the decay into a single photon is obviously forbidden by conservation of energy and momentum. Thus, the foremost decay mode of orthopositronium (abbreviated as oPs) is into three photons: oPs→ γγγ. The lowest-order Feynman diagram for

this process is shown in the figure below.

Figure 3.2:Decay of oPs into three photons

The second decay is the decay into five photons, and its lowest-order dia-gram is now easily visualised. The rate of the above decay is [29, 30]:

Γ(oPs →γγγ) = [7.0382+0.39·10−4·BO]µs−1, (3.6)

and the branching ratio is given by:

BR(oPs) = Γ(oPs→γγγγγ)

Γ(oPs →γγγ) =1.0·10

−6_, _(3.7)

which is comparable to that of parapositronium. Immediately noticable is that the decay rate is about three orders of magnitude smaller than that

(20)

of parapositronium. Thus, we see that orthopositronium is indeed much longer lived than parapositronium. One also immediately notices that a strange factor BO appears in Eq. 3.6. This factor is not one determined

on physical grounds, but is a quantitative measure of the long-standing orthopositronium lifetime puzzle. Several groups had attempted to mea-sure the oPs lifetime, but their meamea-surements differed tremendously. Mea-surements of BOrange from−185 to 338. Strangely enough, it appears as if

the decay rate depended on the medium wherein the experiment was con-ducted. Proposed solutions are a weak decay into neutrinos (oPs→ν ¯ν), or

decays into exotic particles. These were quickly ruled out as the branching ratio of the weak decay would amount to an insignificant factor 6.2·10−18, and the ratio of exotic decays many orders of magnitude smaller. The so-lution came in 2003, when it was finally found that experimenters had not taken into account a number of positronium that scattered back in the di-rection from which the positron beam came [31, 32]. The total decay rate, which agrees with experiment, is now known to be [3]:

(21)

Chapter

4

Atom-Like Properties of

Positronium

Positronium, though short-lived, displays atom-like properties. However, we cannot call these atomic properties, because positronium is not an atom in the traditional sense. In the first section the fine and hyperfine structure of the positronium system are discussed, followed by the Zeeman effect in the second section. In the third, positronium Rydberg states are cov-ered with particular emphasis on Rydberg Stark states. The chapter ends with a section on the effects of confinement on the atom-like structure of positronium.

4.1 Fine and Hyperfine Structure

In a course on quantum mechanics, one learns that the fine structure of hydrogen consists of a relativistic correction and the spin-orbit coupling [33, 34]. The former using the next-to-leading-order term in the expansion of the relativistic kinetic energy E = p(pc)2_{+ (}_m

ec2)2−mec2, whereas

the latter is a correction due to the interaction between the magnetic field of the proton with the magnetic dipole moment of the electron (in hydro-gen). The proton magnetic field is proportional to the orbital angular mo-mentum L, whilst the dipole moment of the electron is proportional to its spin angular momentum Selectron. Consequently, the spin-orbit correction

(22)

is proportional to their product L·S. The expectation value of this oper-ator can be found by defining a momentum J ≡ L+S and making use of the eigenvalues of squared momentum operators. Having worked out these corrections, one moves on to hyperfine structure, which is the spin-spin coupling. The spin-spin-spin-spin coupling arises from the interaction between the dipole moment of the proton with the dipole moment of the electron, and is thus proportional to the product of the two spins Sproton·Selectron.

For positronium the stage is completely different. First to note is that the Born-Oppenheimer approximation, which can be made for hydrogen, is maximally violated due to the equal masses of the electron and its an-tiparticle. Neither of the two could possibly be seen as the nucleus and the other as its orbiting particle, but we can say that both orbit their center of mass which is between them. One says that positronium suffers from max-imum recoil effects, and the kinetic energy is doubled. This doubling of the kinetic energy is taken into account by making the replacement me → µ.

The kinetic Hamiltonian is, to first order:

Ekinetic =

p2 2µ =

p2

me , (4.1)

where µ is the reduced mass of positronium which is seen to equal half the electron mass. The relativistic Hamiltonian yields the following lowest-order correction [33] to hydrogen:

E0_kinetic = −R 2 ∞ 2mec2n4 4n l+1_/₂ −3 . (4.2)

The factor that matters for the generalisation to positronium is R2_∞/me,

which must be replaced by R2_Ps/µ. After application of eq 2.4, one finds that the correction for positronium is only half that of hydrogen. For the ground state, one finds that the correction amounts to a mere−0.4525 meV, which is negligible compared to the binding energy of 6.8 eV.

The second important result for positronium is that the fine and hyper-fine structure are of the same order, because neither of the two particles can be regarded as the nucleus and spin-orbit and spin-spin coupling are equivalent. Thus, one says that positronium has no hyperfine structure. However, one still speaks of a hyperfine splitting for historical reasons. By

(23)

this is then meant the energy difference between the ground state singlet and triplet, which is equal to [35]

∆EHFS =E(1S0) −E(3S1) ≈203 GHz . (4.3)

Comparing this to the mere difference of 1.4 GHz in hydrogen, we can see that a variety of atom-like effects are enhanced for positronium. This is one of the reasons why its study is fruitful: the enhanced effects allow for better measurements. These measurements then allow one to determine the values of fundamental constants such as α ≈ 1/137, which has been done to great precision [2].

Third, we have to use QED in order to analyse the properties of positron-ium. For example, the spin-spin coupling in hydrogen is due to the mag-netic dipole interaction of nucleus and electron,*_{which in positronium can}

only be made sense of as the exchange of virtual photons. Two contribut-ing diagrams for this virtual photon exchange are drawn in Fig. 4.1a. In addition, the possibility of virtual annihilation greatly shifts both fine and hyperfine structure [6]. In particular, this virtual annihilation amounts to around half of the hyperfine splitting. Two diagrams to illustrate this prin-ciple are drawn in Fig. 4.1b. This last contribution is obviously not present in ordinary atomic physics.

Figure 4.1: Four diagrams that contribute to the fine structure of positrium. Fig-ure a) shows two processes involving the exchange of virtual photons. FigFig-ure b) shows two virtual positronium annihilation events.

The final result for the fine structure corrections up to order α4was worked out by Ferrell [38]. The energy differences with respect to the gross energy structure are

(24)

4.2 Zeeman Effect 20 ∆En,l,s,j = α 4_m ec2 n3 11 64n − (1+e/2) 2(2l+1) . (4.4)

Most symbols speak for themselves, with the exception of e, which is a function that depends on the spin state. For the singlet S = 0, e ≡ 0. For the triplet state, one finds:

el,j ≡ − 7 3δl,0+ [1−δl,0]        −(3l−4) (l+1)(2l+3) j=l+1 1 l(l+1) j=l 3l−1 l(2l−1) j=l−1 (4.5)

The δ’s here are Kronecker deltas. Ferrell himself had made an error and did not include the factor7_/₃_{in the first term, which was corrected by [10].}

In the different cases, one recognises the similarity to the different spin states that contribute to the singlet and triplet state.

In addition, there is an effect called the Lamb shift, and the announcement of its discovery by Lamb and Retherford at the first Shelter Island con-ference ushered in the subject of quantum field theory [39]. It is caused by the system jumping to an excited state, emitting a virtual photon, and the immediate reverse of this procedure [40]. Ferrell worked out that the positronium Lamb shift is about one half the Lamb shift in hydrogen, and thus has a negligible effect on the positronium fine structure. We will not discuss the Lamb shift further here.

4.2 Zeeman Effect

An important phenomenon in the study of atomic physics is the Zeeman effect, which is displayed once a magnetic field is applied. The Hamilto-nian is given by:

ˆ

H = Hˆ0+HˆZ. (4.6)

Here, ˆH0 is the unperturbed Hamiltonian that includes the fine structure,

and ˆHZ is the Zeeman Hamiltonian arising from a magnetic field ~B =

(25)

Figure 4.2: Classical depiction of positronium and an applied magnetic field in the z-direction.

The unperturbed basis states are represented by nSl J M_J. The Zeeman Hamiltonian is given by [13]:

ˆ

HZ = (ge−_µ_Bˆs_e−)B_z− (g_e+_µ_Bˆs_e+)B_z =g_e_µ_B(ˆs_e− −ˆs_e+)B_z. (4.7)

The factor ge represents the gyromagnetic ratio, which is equal for

elec-tron and posielec-tron and approximately equal to 2. The Bohr magneton is written as µB, and ˆse represents the electron and positron spin operators.

Diagonalisation of this Hamiltonian with the appropriate basis yields:      E1,0 =1/2[E(3S1) +E(1S0)] +1/2Eh f s √ 1+x2 E0,0 =1/2[E(3S1) +E(1S0)] −1/2Eh f s √ 1+x2 E1,1 =E1,−1 =E(3S1) (4.8)

The energy subscripts denote the quantum numbers ml for the electron

and positron. Here, the function x is introduced for simplicity and is de-fined as:

x(Bz) ≡ 2e¯h

Eh f s

Bz ≈0.276·Bz. (4.9)

In these expressions Eh f s represents the hyperfine splitting energy, given

(26)

Figure 4.3: The Zeeman effect in positronium. The three lines correspond to the three in Eq. 4.8. Adapted from [41].

One can see that the Zeeman effect provides a non-linear splitting for small magnetic fields in positronium. For large magnetic fields, it is approxi-mately linear. This is different to hydrogen, where there are four distin-guishable curves in the Zeeman splitting. Two of these are nonlinear for weak fields but linear for strong, such as two of the curves for positronium. However, hydrogen has two globally linear lines, whereas positronium has only one with an energy completely independent of the field strength. Important to note is that the effect causes mixing between the positronium ground states, as evident by the first term in Eq. 4.8. This influences the annihilation rates, as the triplet and singlet have significantly differing de-cay rates. In addition, dede-cays that would normally be forbidden are now permissible.

In addition to the effect of a magnetic field on positronium, we also discuss the effects of an electric field. However, this effect is negligible for ground state positronium. It is more easily studied when considering Rydberg states, which are highly excited states and are the subject of the following section.

(27)

4.3 Rydberg Stark States 23

4.3 Rydberg Stark States

In addition to the Zeeman effect, which arises when a magnetic field is present, we must also consider what happens to positronium when an electric field is applied. Because the positron and electron are considerably closer to one another for the ground state than for higher excited states, the effect of an electric field in the ground state is difficult to observe due to screening. The degree of separation and the electric dipole moment scales with n2, so the Stark effect is more clearly visible when we consider highly excited states [42]. Such highly excited states are called Rydberg states, and as such we consider here so-called Rydberg Stark states of positron-ium. Another added benefit is that the lifetime of Rydberg positronium is greatly increased, allowing for better experimentation. This is because the annihilation rate of positronium is proportional to the amplitude at the center of mass, which decreases when the degree of separation increases. In this case, it is useful to solve the Schrödinger equation in parabolic co-ordinates [10]. This is because in a semiclassical picture, one can imag-ine the circular orbits of the particles to be stretched into an ellipse when a force is applied. This was not necessary for the magnetic field, as in that case the force is simply radially isotropic. The states are then charac-terised by two parabolic quantum numbers, imaginatively labeled n1, n2.

One needs two quantum numbers because an ellipse can be characterised by two numbers, whereas a circle needs only one. The difference between these (k ≡ n1−n2) serves as an index that characterises the Stark states.

One finds that in an electric field~F= (0, 0, Fz)the Stark energies are given

as EStark = 3 2nkeaPsFz− n4 16(17n 2₋ 3k2−9m2+19) e 2_a2 Ps 2hcRPs F_z2+ O(F_z3). (4.10) And the higher order terms are even less friendly. In this expression, e is the electron charge, aPs is the positronium Bohr radius, and RPs is its

Ry-dberg constant which we derived in chapter 2.4. This equation is plotted for n=14 positronium in Fig. 4.4.

(28)

4.4 Confined Positronium 24

Figure 4.4: Stark effect in n = 14 positronium where each state is labeled by k≡n1−n2. Figure adapted from [43].

One immediate feature is that the Stark effect is very nearly linear. We can thus pose that there is an energy shift of the dipole:

EStark = −~µelec· ~F . (4.11)

Comparing this expression to the equation above, we conclude that the electric dipole moment of positronium is equal to:

µelec= −

3

2nkeaPs ≈ −nk·2.54·10

−29_Cm_{≈ −}_nk_·_{7.61 D .} _(4.12)

The units used are the Coulomb-meter and the Debye. We thus confirm that the Stark effect is rather difficult to observe for ground state positron-ium, but becomes increasingly significant the higher the excitation.

4.4 Confined Positronium

Nanoporous silica was mentioned earlier as a converter for the produc-tion of positronium. Foreshadowing quantum dot excitons, which must

(29)

be considered as confined systems, the effects of confinement on the en-ergy of positronium are discussed.

Because the mass of positronium is small compared to atomic masses, the de Broglie wavelength of the center of mass can be considerably large as it scales inversely with mass. Hence, larger cavities such as the ones in nanoporous or mesoporous silica allow for strong interaction between positronium and the cavity walls. It is for this reason that the effects of confinement are of a considerable magnitude for positronium. Due to this interaction with the confining potential, the positronium dynamics are slowed down, and this technique can be used to study Ps-Ps interaction by creating a gas of high density.

The confinement also leads to a shift in the positronium 13S1−23PJ

transi-tion resonance frequency and a narrowing of its linewidth. The linewidth is narrowed because of the reduced freedom of positronium and the thereby reduced Doppler broadening [13]. This has been measured experimentally and the results can be seen in Fig. 4.5 below.

Figure 4.5:Measurement of the 13S1−23PJtransition for positronium in vacuum

(a) and in a mesoporous structure (b). Figure taken from [9]. The experiment itself is detailed in [44].

One may argue that the line narrowing depicted in the figure above is not evident. An explanation provided by the original researchers is that this

(30)

is because the newly measured peak probably consists of many narrowed peaks of slightly differing shifts. Due to statistical reasons, a Gaussian emerges.

One problem here is that the pores in the silica are interconnected and of irregular shapes, so an analytical solution for such cavities is impossible. Theoretically, we can consider the idealised case of an infinite spherical potential. This potential leads to a confinement energy of [9]:

Econfinement = π¯h2

8meb2

≈750 meV/b2, (4.13)

where b is the dimensionless radius of the pore expressed in nanometers. Thinking back to the early days of a first education on quantum mechan-ics, we recognise that the scaling of the confinement energy on the radius of the confining potential is the same as that of the one-dimensional infi-nite potential well.

(31)

Chapter

5

Basics of Excitons

Though the concepts of electrons, holes, and excitons have already been established in chapter 2, the intent of this chapter is to expand this knowl-edge and to develop more in-depth concepts that are necessary for the fol-lowing discussions. Before speaking of quantum dot excitons in the next chapter, we first discuss free excitons.

5.1 Electrons and Holes

To reiterate, a filled valence band means that all available states in that band are occupied by electrons. Only a finite number of states is available due to the exclusion principle. If such a band is nearly filled it makes sense to consider the empty states as quasiparticles. These quasiparticles are called holes, which will be denoted as h+ and can be created by exciting an electron from the valence band into the conduction band. Important to note is that the wavevector and spin of the thusly created hole are opposite to the wavevector and spin of the electron that was removed [45]. These two properties follow from a consideration of conserved quantities. Before the creation of the hole, a filled valence band has a sum total momentum and spin equal to zero. Therefore, in order to have zero total momentum and spin after the photonic excitation, the properties of the hole must be the opposite of those of the removed electron. We say that both conduc-tion band electron and valence band hole are quasiparticles because they

(32)

5.1 Electrons and Holes 28

can occur only within crystals and have no meaning in vacuum. The an-tiparticle to the electron in vacuum is the positron, not the hole, and the band gap for the creation of a "vacuum exciton" amounts to twice the elec-tron mass: 1.022 MeV. This is enormous compared to the exciton creation energy, for the band gaps in semiconductor are only a few electron volts. The concept of the effective masses of electrons and holes was previously defined as:

¯h2 m∗ ≡

∂2E

∂k2 , (5.1)

but this is an oversimplification. It fails to take into account the nonspher-ical symmetry of the crystal, and thus the direction of~k must be taken into consideration. The correct definition is:

¯h2 m∗ ≡

∂2E ∂ki∂kj

i, j∈ {x, y, z}, (5.2)

enhancing the notion of effective mass to a rank two tensor. This means that the effective mass dependens on direction, and its origin lies in the anisotropy and strain of the crystal [45].

Another detail that was omitted in the basic discussion was the distinction between two separate types of holes. There is often a degeneracy in the valence band near points of symmetry in the Brillouin zone [46], meaning that the two dispersion curves share a maximum but have different cur-vatures. This yields the aforementioned distinction: a hole living in the broader curve is called a heavy hole, whereas a hole living in the narrow curve is dubbed a light hole. This is illustrated in Fig. 5.1 below.

(33)

Figure 5.1: The band structure of a semiconductor with a valence band degener-acy resulting in two different types of holes

5.2 Free Excitons

Due to the opposite charges of the electron and hole, an attractive Coulomb force emerges and the bound state is called an exciton. Often, excitons are denoted by the letter X. For now we will only consider free excitons and discuss excitons confined within quantum dots in the following chapter. In order to keep the discussion simple, we also limit the subject to exci-tons that arise from direct band gaps that can appropriately be described by simple parabolic dispersion curves, such as the ones depicted in Fig. 5.1.

Using these assumptions, one can separate the equations of motion into one for the centre of mass and one for the relative motion between electron and hole. This then leads to the following expression for the energy [45]:

EX =Egap− RX

n2 +

¯h2~K2

2M . (5.3)

Here, RX represents the effective Rydberg constant of an exciton, which

is defined in Eq. 2.4. The symbol n is an integer which represents the principal quantum number, and ~K = ~ke +~kh and M = me +mh are the

(34)

exciton energy is due to the energy of the band gap, the atom-like gross structure, and overall motion of the exciton. This results in the binding energy spectrum sketched below.

Figure 5.2: Exciton energies in a direct band gap as a function of total wavevector

~

K = ~ke+~kh. Curves are shown for various values for the principal quantum

number n, which in the limit n → ∞ converges to the unbound energy E =

Egap+Ekinetic.

There is also an important distinction to be made here between two differ-ent types of excitons. One has, on the one hand, so-called Frenkel excitons, whose wavefunctions are contained within one unit cell of the crystal. On the other hand, Wannier excitons exceed this size, and are the type under consideration here. We describe such excitons with a wavefunction of the form [14]:

ΨX =N−1/2φe(re)φh(rh)Φ(Re, Rh). (5.4)

The factor N−1/2 is a normalisation constant, which is not of importance here. The functions φeand φhrepresent the Wannier functions for the

elec-tron and hole. The Wannier functions are wave packets of Bloch waves, which describe electrons propagating through crystals. The envelope func-tionΦ is of the form:

(35)

Φ(R, r) = N−1/2ei~K·~RRnl(r)Ylm(θ, φ), (5.5)

which immediately reminds one of the wavefunction of the electron in a hydrogen atom in that Rnl is the associated Laguerre polynomial and Ylm

is the spherical harmonic. Through this method, one can check that the dispersion relation found above is recovered.

If one strives for a more field-theoretical approach to excitons, one finds that an exciton creation operator can be defined as proportional to cre-ation operators for an electron and a hole. If the commutator of this opera-tor with itself is calculated, one finds that the exciton obeys Bose-Einstein statistics, though there is a dependence on density in a correction term. Around room temperature and at low exciton densities, excitons then ap-proximately obey Boltzmann statistics and can be viewed in the same way as a positronium gas. The creation of an exciton Bose-Einstein condensate, however, is difficult due to the previously mentioned density-dependent term [45, 47].

(36)

Chapter

6

Quantum Dot Excitons

Though free excitons are interesting in their own right, of particular impor-tance are excitons trapped within quantum dots. Quantum dots are spe-cially engineered nanostructures that employ the properties of semicon-ductors to create potential wells. These quantum dots can trap excitons, and their peculiar properties have prompted their nickname of ’artificial atoms’. First, we discuss self-assembled quantum dots, which are partic-ularly useful and available to us. Experimental work on these is done in chapter 8. We then discuss how quantum dots can act as confining struc-tures for single electrons or holes. Lastly, we conclude this chapter by a discussion on the use of quantum dots to confine excitons and a few words on entangled photons produced from exciton recombination.

6.1 Self-Assembling Quantum Dots

Imagine a semiconductor material A of bandgap EAimmersed in another

semiconductor B of a different bandgap EB. Making an appropriate choice

of A and B such that EA < EB and that the bands of the materials are

suitably aligned, one sees that a three-dimensional potential well is con-structed. The situation is sketched in figure 6.1.

(37)

Figure 6.1: Two semiconductors A and B of different bandgaps form a confining potential well.

If the bands are not directly spatially aligned, the electron and hole are confined to different layers. The latter situation is not under consideration, and our treament covers only heterostructures where the conduction band minima and valence band maxima are spatially aligned. This is then a practical manner of constructing finite potential wells which can be used to confine both electrons and holes. If the size is of the order of the electron wavelength, such a structure can be called a quantum dot.

In addition, the density of states for such a quantum dot consists of a series of sharp peaks [14]. This is because the movement of particles within an infinitely large crystal is described by Bloch waves. For a quantum dot, these must be quantised due to the confinement in a manner analogous to particles in a quantum-mechanical box. Hence, the density of states is modified due to the reduced dimensionality of the system. For this reason a quantum dot is sometimes called an artificial atom.

The remarkable thing is that for some choices of semiconductors, these incredibly useful structures can assemble themselves. Even better, these structures form without any defects, and are therefore ideally suited for experimental study. Of note is that the absence of any defects means that the only possible recombination of electron and hole far from the sur-face boundaries is a radiative one. If defects are present, recombination channels involving phonons would be available. We say that the optical quantum efficiency of quantum dots is very high [48]. Take as an exam-ple the InAs/GaAs quantum dot. These dots are grown by first creating

(38)

a substrate of GaAs and then depositing InAs on top of it layer by layer. There is a slight difference in the lattice constants of these materials, and so the InAs is put under strain. The first few layers are called the wetting layer (sometimes abbreviated as WL), and subsequently deposited layers of InAs suffer more and more from the stress. Eventually, InAs forms tiny islands to minimise their energy under strain. These islands are capped by more layers of GaAs. The islands, or quantum dots formed are pancake-shaped, with diameters of ~5-50 nm and a height of about 2.5 nm [49, 50]. The growth procedure is illustrated in Fig. 6.2. Typically, the vertical di-rection in the figure is denoted the growth didi-rection or the z-didi-rection.

Figure 6.2: Self-assembling InAs/GaAs quantum dot growth procedure. In the first step (left) a layer of InAs is deposited onto GaAs. This layer is called the wetting layer (WL). Secondly (middle), more layers of InAs are deposited which form into droplet-shaped islands under strain. These islands are called quantum dots. Lastly (right), more GaAs is grown atop the wetting layer and quantum dots. Adapted from [51].

We can now imagine how these self-assembling quantum dots act as po-tential wells for charge carriers. The electrons excited into the conduction band of GaAs move freely, but fall down into the potential well of the InAs, which has a smaller energy gap. The holes do so similarly, and fall upwards into the same quantum well. Several different structures can be imagined and manufactured, though we will only here be interested in the simplest case where the band gaps are directly aligned as previously mentioned. This is illustrated in Fig. 6.3.

(39)

Figure 6.3: An InAs/GaAs quantum dot acting as a potential well to confine charge carriers. In the top picture, a difference in band gap energies between InAs and GaAs results in a finite potential well. In the middle picture, it is shown how non-resonant excitation of the surrounding GaAs can lead to the creation of an electron in the conduction band and a hole in the valence band. These then relax into the quantum dot, as represented by the curved arrows. In the bottom figure, the wavefunctions of an electron (red) and hole (blue) are sketched. They can be seen to spill out of the potential well.

We continue with a discussion of what exactly the effects of this confine-ment mean for both electrons and holes individually.

6.2 Single Particle Confinement

We now discuss the confinement of single charge carriers to a quantum dot potential. To go any further, we approximate the quantum dot to be of a particularly simple form, following the line of reasoning in [14]. We assume the confining potential caused by the quantum dot to be a finite potential well of size L and depth V0:

(40)

6.2 Single Particle Confinement 36 V(z) = ( 0 z ∈ [−1_/₂_L,₊1_/₂_L_] V0 elsewhere . (6.1)

An electron in such a potential can be written as a sum over Wannier func-tions multiplied by envelope funcfunc-tions. The Wannier funcfunc-tions are de-noted φ with a subscript referring to material A or B. To reiterate what was said at the end of section 5.2, Wannier functions are wave packets of Bloch waves and depend on the difference between the coordinate r and lattice vector Ri. The envelope functions, as previously defined in Eq. 5.5,

are here denoted by Φ with a similar subscript, and depend on the lattice vector.

Ψ(r) =

(

N−1/2_{∑ Φ}A·φA inside the well

N−1/2_{∑ Φ}B·φB outside the well

. (6.2)

Note that there is an abrupt change in the effective mass of the charge car-riers when transitioning from the bulk material to the quantum dot. Inside the well, the envelope function obeys the following equation of motion:

" − ¯h 2 2m∗_A ∂2 ∂~R2 # ΦA(~R) ≈ EΦA(~R). (6.3)

Here, the effective mass is assumed to be independent of the direction (isotropic) and is represented by m∗_A. An analogous equation exists outside of the well with the subscripts replaced through the permutation A → B. By the reasoning of Yu, Cardona [14], this differential equation is separable because the potential depends only on one coordinate, namely the growth direction. The separation to be made is thenΦ(~R) = ϕ(x, y) ·ψ(z), where

the potential is taken along the z-direction as in Eq. 6.1. The function

ϕ(x, y) is that of free particles due to the absence of a strongly confining

potential, and is therefore approximately proportional to exp(ikxx+ikyy).

This approximation is justified because the diameter of our quantum dots is about an order of magnitude larger than the height. One finds the con-finement energy of the electron:

En(kx,ky) = ¯h 2

2

∑

_i

π2n2_i

(41)

Here, the summation over i ranges over the three different directions. One recognises this as the confinement energy of a ’particle in a box’ where the box has three different dimensions. This expression is only approximately true, as the true shape of the quantum dots is elliptical and these energies are based on a box-like dot.

For holes, the principle is similar though the distinction between the two separate types of holes must be made. Light holes have angular momen-tum Jz = 1/2, whereas heavy holes have angular momentum Jz = 3/2.

The exchange interaction Hamiltonian between holes and electrons can be written as [14]: H = ¯h 2 2m∗ " (γ1+5/2γ2)∇2−2γ2 Jx ∂ ∂x +Jy ∂ ∂y +Jz ∂ ∂z 2# +V(z) +Hexternal. (6.5) The γ’s in this equation are dimensionless constants, the so-called Lut-tinger parameters that parametrise the band structure. Their values for InAs and GaAs are given in table 6.1.

Material γ1 γ2 InAs 19.67 8.37

GaAs 7.65 2.41

Table 6.1:Luttinger parameters used in the hole Hamiltonian for InAs and GaAs. The Luttinger parameters are dimensionless constants and parametrise the band structure. Data from [14].

The quadratic term containing Ji∂i signifies that the equation of motion is

not immediately separable due to the mixing terms, but can be made sep-arable by the approximation that off-diagonal terms are negligible. This assumption means that there is not mixing between angular momenta in different directions. With this approximation, the equation can be sepa-rated in the same fashion as for the electrons.

The γ2terms in the Hamiltonian then add together, and the Jz = ±1/2hole

behaves as if it has a lighter effective mass than the Jz = ±3/2 hole. The

confinement energy is thus higher for the light hole than for the heavy, as the energy scales inversely with effective mass. Once the off-diagonal

(42)

6.3 Exciton Confinement in a Quantum Dot 38

terms are taken into account through the use of perturbation theory, a mix-ing between the two hole states occurs. The external part of the Hamilto-nian will be considered in the next chapter of this thesis. The HamiltoHamilto-nian presented above is in reality a specialised form of the one presented in [52].

We continue this chapter with a short discussion on what happens when an exciton forms in a quantum dot, as opposed to the single confinement of either a hole or electron.

6.3 Exciton Confinement in a Quantum Dot

We can trap valence band holes and conduction band electrons in quan-tum dots because both electron and hole relax into the potential. Prac-tically, the substrate material can be illuminated at the right frequency, exciting an electron above the band gap energy. The excited electron and hole both relax into the quantum dot and form a bound exciton through their mutual Coulomb attraction. This process is illustrated in Fig. 6.2. One important effect is that the ground state energy of the quantum dot exciton is blue-shifted [53] due to the added confinement energy as a re-sult of the reduced freedom. Two important limits on the quantum dot must be considered. If the Bohr radius of the exciton in the bulk material is much smaller than the radius of the quantum dot, then it can be approx-imated as nearly free and the confinement energy is rather small. In the opposite limit the confinement energy plays a significant role: one must treat the electron and hole as seperate particles both confined to the po-tential. However, this still assumes that the potential well formed by the quantum dot is infinitely deep. A finite barrier height must be accounted for, resulting in the wavefunction spilling out of the dot.*

Generally, the theory of finite potential wells allows for only a finite num-ber of bound states. No matter how shallow the well, there is always at least one bound state [33]. If the energy is high enough to liberate a par-ticle from such a well the bound state is destroyed and the parpar-ticle moves freely. In the reverse case where a particle is not trapped but initially free,

*_{For a lucid discussion on square potential wells, see ’Quantum Theory’ by Bohm,}

(43)

6.4 Entanglement 39

a wave packet coming in from the left does not always scatter off the po-tential well and the transmittivity is then maximised. This occurs when the energy of the wavepacket plus the depth of the potential well equals the energies of an infinite potential well [34].

Lastly, we must mention that charged multi-exciton states can also exist in quantum dots. Firstly, one can create a charged exciton by adding an extra hole or electron. These can then also relax into the quantum dot and assume a different spin configuration due to the Pauli exclusion principle. If both an extra hole and electron are present we speak of a biexciton. Of note is that the biexciton can decay into two photons, which is a property it shares with singlet positronium. In addition, these photons are entangled [54]. A few words are dedicated to this entanglement in the following section.

6.4 Entanglement

In the research of Benson et al., the recombination of electrons and holes in a micropillar structure containing an InAs/GaAs quantum dot is studied. It is found that the electron-hole recombination results in circularly po-larised light along the direction of the micropillar. If there is not an exciton, but a biexciton trapped in the quantum dot, the resulting two photons are polarisation entangled with one another, and the previously mentioned device can act as a reliable and fast source of entangled photon pairs. This is because the first electron recombines with a hole and produces light of one circular polarisation, and the second recombines into light of the op-posite circular polarisation. The reason is that the two electron spins are necessarily opposite when they are confined to the quantum dot due to the Pauli exclusion principle. The two photons are in a maximally entangled Bell state, though it is of note that their energies are not equal and differ by about 4 meV. For an overview of research into entangled photons from single quantum dots, see the recent review by Shields [55].

(44)

Chapter

7

Atom-Like Properties of QD

Excitons

In this chapter we discuss the various atom-like properties that excitons display. We start with both the fine and hyperfine structure in the first section and the Zeeman effect in the second. Though we are primarily interested in excitons confined to single quantum dots, the Stark effect will be discussed in the context of multiple quantum dot excitons spaced close together. The Stark effect on a single quantum dot exciton will be experimentally covered in the next chapter. In addition, we discuss some properties of highly excited Rydberg state excitons in the last section of this chapter, though we will see that this is of little relevance to quantum dot excitons. It must be mentioned that these topics, though not ideally suited to a discussion with primary focus on quantum dot excitons, are necessary in a later detailed comparison between positronium and quan-tum dot excitons.

7.1 Fine and Hyperfine Structure

As mentioned before in the context of positronium, the fine structure con-sists of a relativistic correction and a spin-orbit correction. In Eq. 4.2, we see that the first-order relativistic correction is proportional toR2

X/µ. Upon

(45)

7.1 Fine and Hyperfine Structure 41 R2_X µ = µ me 1 e4r R2_∞ me . (7.1)

From the values in table 2.1, we find that for InAs the relativistic correction is only 1.2·10−6 times that of hydrogen, and for GaAs this factor is 4.3·

10−6. This leads to a completely negligible energy correction for ground state excitons in the nanoelectronvolt range.

The coupling of hole spin to the electron spin leads to the exciton fine structure. Our discussion of this effect follows that of [56]. The coupling of the heavy hole spin to other spins is twofold. Firstly, there is the short-ranged interaction between the heavy hole and electron spins, which is the type primarily under consideration here. This short-ranged interaction splits the exciton states into dark and bright excitons. The long-ranged interaction will be seen to cause a splitting of the bright excitons in asym-metric quantum dots. If one writes the electron spin as S and the angular momentum of the hole as J, the short-ranged exchange interaction is writ-ten as a coupling between these in the following effective Hamiltonian [57, 58]: HFS = −

∑

i h aiJiSi+biJi3Si i . (7.2)

Here, the parameters ai, biare coupling constants and the summation runs

over the x, y, z directions. The angular momentum operators Ji for the

j =3_/₂_{holes are given by:}

Jx = ¯h₂     0 √3 0 0 √ 3 0 2 0 0 2 0 √3 0 0 √3 0     , Jy= i¯h₂     0 −√3 0 0 √ 3 0 −2 0 0 2 0 −√3 0 0 √3 0     Jz = ¯h₂     3 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −3     . , (7.3)

(46)

We can label the states of the exchange interaction by |Jz, Szi, which can

assume the following eight forms: ±3 2,± 1 2 , ±1 2,± 1 2 . (7.4)

Using these states, we rewrite the Hamiltonian in matrix form and intro-duce some definitions on its entries for ease of writing:

HFS= Ahh Ahh−lh A∗_hh₋_lh Alh . (7.5)

The entries A are 4×4 matrices, which are indexed by a shorthand for the type of hole (light or heavy) that they couple to. We start with the definition of Ahh, which couples heavy holes to the electron:

Ahh =     ∆0 ∆1 0 0 ∆1 ∆0 0 0 0 0 −∆₀ ∆₂ 0 0 ∆2 −∆0     , (7.6)

where we have tacitly defined:

∆0 ≡ 3₄az+27₁₆bz,

∆1 ≡ −3₈(bx−by),

∆2 ≡ −3₈(bx+by).

(7.7)

Now we see that the matrix Ahh separates into two 2×2 matrices. The

upper-left one of these couples to excitons with total angular momentum S = ±1 and the lower-right one of which couples to excitons with a total S= ±2. Due to the optical selection rule for dipole transitions, the former of these are dubbed bright excitons and the latter of these dark excitons. Thus, the heavy hole submatrix splits directly into a part for bright and one for dark excitons. We now see that the dark and bright excitons are split by twice the matrix entry ∆0. The entry∆1 couples the bright heavy

hole excitons, and∆2couples the dark heavy hole excitons.

Analogies of Positronium & Quantum Dot Excitons

Analogies of Positronium &

Quantum Dot Excitons

Analogies of Positronium &

Quantum Dot Excitons

Sam Woldringh

Abstract

Contents

Chapter

1

Introduction

Chapter

2

Positronium and Exciton

2.1

What is positronium?

2.2

Positronium History and Formation

2.3

What are excitons?

2.4

Scaling Laws

Chapter

3

Positronium & Quantum

Electrodynamics

3.1

Bound State Quantum Electrodynamics

3.2

Singlet and Triplet Positronium

3.3

Parapositronium Decays

3.4

Orthopositronium Decays

Chapter

4

Atom-Like Properties of

Positronium

4.1

Fine and Hyperfine Structure

4.2

Zeeman Effect

4.3

Rydberg Stark States

4.4

Confined Positronium

Chapter

5

Basics of Excitons

5.1

Electrons and Holes

5.2

Free Excitons

Chapter

6

Quantum Dot Excitons

6.1

Self-Assembling Quantum Dots

6.2

Single Particle Confinement

∑

6.3

Exciton Confinement in a Quantum Dot

6.4

Entanglement

Chapter

7

Atom-Like Properties of QD

Excitons

7.1

Fine and Hyperfine Structure

∑