Chapter 9 Quantum emitters 9.1 Introduction and Basics The interaction of light with nanometer-sized structures is at the core of nano-optics. It is obvious that as the particles become smaller and smaller the laws of quantum mechanics will become apparent in their interaction with light. In this limit, contin- uous scattering and absorption of light will be supplemented or replaced by resonant interactions if the photon energy hits the energy difference of discrete internal (elec- tronic) energy levels. In atoms, molecules, and nanoparticles, like semiconductor nanocrystals and other ’quantum confined’ systems, these resonances are found at optical frequencies. Due to the resonant character, the light-matter interaction can often be approximated by treating the quantum system as an effective two-level sys- tem, i.e. by considering only those two (electronic) levels whose difference in energy is close to the interacting photon energy ¯ hω 0 . In this chapter we will consider single quantum systems that are fixed in space, either by deposition to a surface or by being embedded into a solid matrix. The material to be covered should familiarize the reader with single-photon emitters and detectors and with concepts developed in the field of quantum optics. While vari- ous theoretical aspects related to the fields emitted by a quantum system have been discussed in Chapter ??, the current chapter focuses more on the nature of the quan- tum system itself. We will adopt a rather practical perspective since more rigorous accounts can be found elsewhere (see e.g. [1-4]). 1
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Chapter 9
Quantum emitters
9.1 Introduction and Basics
The interaction of light with nanometer-sized structures is at the core of nano-optics.
It is obvious that as the particles become smaller and smaller the laws of quantum
mechanics will become apparent in their interaction with light. In this limit, contin-
uous scattering and absorption of light will be supplemented or replaced by resonant
interactions if the photon energy hits the energy difference of discrete internal (elec-
tronic) energy levels. In atoms, molecules, and nanoparticles, like semiconductor
nanocrystals and other ’quantum confined’ systems, these resonances are found at
optical frequencies. Due to the resonant character, the light-matter interaction can
often be approximated by treating the quantum system as an effective two-level sys-
tem, i.e. by considering only those two (electronic) levels whose difference in energy
is close to the interacting photon energy hω0.
In this chapter we will consider single quantum systems that are fixed in space,
either by deposition to a surface or by being embedded into a solid matrix. The
material to be covered should familiarize the reader with single-photon emitters and
detectors and with concepts developed in the field of quantum optics. While vari-
ous theoretical aspects related to the fields emitted by a quantum system have been
discussed in Chapter ??, the current chapter focuses more on the nature of the quan-
tum system itself. We will adopt a rather practical perspective since more rigorous
accounts can be found elsewhere (see e.g. [1-4]).
1
2 CHAPTER 9. QUANTUM EMITTERS
9.2 Fluorescent molecules
For an organic molecule, the lowest energy electronic transition appears between the
highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular
orbital (LUMO). Higher unoccupied molecular orbitals can be taken into account
if necessary. In addition to the electronic energy levels, larger particles, such as
molecules, have vibrational degrees of freedom. For molecules, all of the mentioned
electronic states involved in the interaction of a molecule with light have a mani-
fold of (harmonic-oscillator-like) vibrational states superimposed. Since the nuclei
have much more mass than the electrons, the latter are considered to follow the
(vibrational) motion of the nuclei instantaneously. Within this so-called adiabatic
or Born-Oppenheimer approximation, the electronic and the vibrational wave func-
tions can be separated and the total wave function may be written as a product of
a purely electronic and a purely vibrational wave function. At ambient temperature
the thermal energy is small compared to the separation between vibrational states.
C6H5 -(CH=CH)n-C6H5
1
Figure 9.1: Characteristic absorption spectra of fluorescent molecules. Left:
Linear polyenes featuring a conjugated carbon chain on which delocalized
electrons exist. Right: aromatic molecules. Electrons are delocalized over
the aromatic system. Increasing the length of the conjugated chain or the
aromatic system shifts the absorption to the red spectral region. From [5]
without permission.
9.2. FLUORESCENT MOLECULES 3
Thus excitation of a molecule usually starts from the electronic ground state with no
vibrational quanta excited (see Fig. 9.2.
9.2.1 Excitation
Excitation of the molecule can be resonant into the vibrational ground state of the
LUMO or it can be nonresonant into higher vibrational modes of the LUMO. Efficient
collisional deactivation facilitates a fast decay cascade which, for good chromophores∗,
ends in the vibrational groundstate of the LUMO†. For a fluorescent molecule the life-
time of this excited state is of the order of 1-10 ns. For resonant pumping, coherence
between the pump and the emitted light can only be conserved if the molecule is
sufficiently isolated from its environment such that environmental dephasing due to
collisions or phonon scattering becomes small. Isolated molecules in molecular beams
∗For inefficient chromophores, non-fluorescent molecules, or molecules strongly coupling to the
environment (e.g. phonons), the collisional deactivation continues to the groundstate.†This is the so-called Kasha rule.
ms
ns
S0
S1
S2
LUMO
HOMO
absorp
tion
fluore
scence
T1
T2
intersystem crossing
energ
y
phosphore
scence
Figure 9.2: Energy level diagram of an organic molecule. The electronic sin-
glet states S0, S1, S2 are complemented by a manifold of vibrational states.
Excitation of the molecule is followed by fast internal relaxation to the vi-
bronic ground state of the first excited state (Kasha rule). From here the
molecule can decay either radiatively (fluorescence, straight lines) or non-
radiatively (heat, wavy lines). Since the radiative decay often ends up in a
vibrational state, the fluorescence is red-shifted with respect to the excita-
tion (Stokes shift). Spin-orbit coupling leads to rare events of intersystem
crossing (dashed arrow) into a triplet state with a long lifetime.
4 CHAPTER 9. QUANTUM EMITTERS
and molecules embedded in crystalline matrices at cryogenic temperatures can show
coherence between resonant excitation light and the zero-phonon emission line [6]
leading to extreme peak absorption cross-sections and to Rabi oscillations (see Ap-
pendix A). Note that even if a molecule is excited resonantly, besides the resonant
zero-phonon relaxation to the LUMO, non-resonant relaxation also occurs (red-shifted
fluorescence) which leaves the molecule initially in one of the higher vibrational states
of the HOMO. This state also relaxes fast by the same process discussed before, called
internal conversion, to the vibrational ground state of the HOMO.
The strength of the HOMO-LUMO transition is determined by a transition matrix
element. In the dipole approximation this is the matrix element of the dipole operator
between the HOMO and the LUMO wave functions supplemented by corresponding
vibronic states. This matrix element is called the absorption dipole moment of the
molecule (see Appendix A). The dipole approximation assumes that the exciting elec-
tric field is constant over the dimensions of the molecule. In nano-optics this is not
necessarily always the case and corrections to the dipole approximation, especially
for larger quantum confined systems, might become necessary. Those corrections can
result in modified selection rules for optical transitions [7].
The molecular wavefunction is the result of various interacting atomic wavefunc-
tions. Since the atoms have a fixed position within the molecular structure, the di-
rection of the dipole moment vector is fixed with respect to the molecular structure.
Degeneracies are only observed for highly symmetric molecules. For the molecules
consisting of five interconnected aromatic rings and the linear polyenes of Fig. 9.1 the
absorption dipole moment approximately points along the long axis of the structure.
The emission dipole moment typically points in the same direction as the absorption
dipole. Exceptions of this rule may occur if the geometry of the molecule changes
significantly between the electronic ground state and the excited state.
9.2.2 Relaxation
Radiative relaxations from the LUMO is called fluorescence. But relaxation can also
occur nonradiatively via vibrations or collisions which ultimately lead to heat. The
ratio of the radiative decay rate kr and the total decay rate (kr + knr) is denoted as
the quantum efficiency
Q =kr
kr + knr, (9.1)
where knr is the nonradiative decay rate. If the radiative decay of the LUMO pre-
vails, the corresponding lifetime is typically of the order of some nanoseconds. The
emission spectrum consists of a sum of Lorentzians (see chapter ??), the so-called
vibrational progression, corresponding to the different decay pathways into ground-
state vibronic levels (see Fig. 9.2). At ambient temperatures dephasing is strong
and leads to additional line broadening such that the vibrational progression is of-
ten difficult to observe. However, vibrational bands become very prominent at low
9.2. FLUORESCENT MOLECULES 5
temperatures. For a molecule, the probability for decaying into a vibrational ground
state is determined by the overlap integrals of the respective LUMO vibrational state
wavefunction and the HOMO vibrational state wavefunctions. These overlap integrals
are known as Frank-Condon factors. Their relative magnitude determines the shape
of the fluorescence spectrum [6, 8].
It is important to point out that not all molecules fluoresce. Radiative decay
occurs only for a special class of molecules that exhibit a low density of (vibronic)
states (of the HOMO) at the LUMO energy. Under these circumstances a nonradiative
decay via the HOMO vibrational manifold is not likely to occur. Particularly efficient
fluorescence is observed for small and rigid aromatic or conjugated molecules, called
dye molecules or fluorophores. The same principles hold for other quantum objects
in the sense that the larger the object, the more degrees of freedom it has, and the
lower the probability for a radiative decay will be.
Due to non-negligible spin-orbit coupling in molecules (mainly originating from
heavy elements) there is a finite torque acting on the spin of the electron in the excited
state. This results in a small but significant probability that the spin of the excited
electron is reversed. This process is known as intersystem crossing and, for a good
chromophore, it typically occurs at a rate kISC much smaller than the excited state
decay rate. If a spin flip happens, the total electronic spin of the molecule changes
from 0 to 1. Spin 1 has three possible orientations in an external magnetic field leading
to a triplet of eigenstates. This is the reason why the state with spin=1 is called a
triplet state as opposed to a singlet state for spin 0. The energy of the electron in the
triplet state is usually reduced with respect to the singlet excited state because the
exchange interaction between the parallel spins increases the average distance between
the electrons in accordance with Hund’s rule. The increased average distance leads to
a lowering of their Coulombic repulsion. Once a molecule underwent an intersystem
crossing into the triplet state it may decay into a singlet ground state. However, this
is a spin forbidden transition. Triplet states therefore have an extremely long lifetime
on the order of milliseconds.
Because of triplet-state excursions, the time-trace of fluorescence emission of a
molecule shows characteristic patterns: the relatively high count rates associated
with singlet-singlet transitions are interrupted by dark periods of a few milliseconds
resulting from triplet-singlet transitions. This fluorescence blinking behavior can eas-
ily be observed when studying single molecules and we will analyze it quantitatively
later in this chapter.
Frequently, blinking is also observed on longer timescales. Long dark periods of
seconds are mostly attributed to fluctuating local environments or transient interac-
tions with other chemical species such as oxygen. Typically, a molecule eventually
ceases to fluoresce completely. This so-called photobleaching is often due to chemical
reactions with singlet oxygen: a molecule residing in its triplet state can create singlet
6 CHAPTER 9. QUANTUM EMITTERS
oxygen in its immediate environment by triplet-triplet annihilation.‡ This reactive
singlet oxygen then attacks and interrupts the conjugated or aromatic system of the
molecule [9].
9.3 Semiconductor quantum dots
The use of colloidally dispersed pigment particles for producing colorful effects is
known since ancient times. In the early 1980s experiments were done with colloidal
solutions of semiconductor nanocrystals in the laboratory of L. Brus at Bell labora-
tories with applications towards solar energy conversion. It was found that colloidal
solutions of the same semiconductor showed strikingly different colors when the size of
the nanocrystals was varied. This observation is attributed to the so called quantum
confinement effect. The excitons in semiconductors, i.e. bound electron-hole pairs,
are described by a hydrogen-like Hamiltonian
H = −h2
2mh∇2
h −h2
2me∇2
e −e2
ε |re − rh|(9.2)
where me and mh are the effective masses of the electron and the hole, respectively,
and ε is the dielectric constant of the semiconductor [10]. The subscripts ’e’ and ’h’
denote the electron and the hole, respectively. Once the size of a nanocrystal ap-
proaches the limit of the Bohr radius of an exciton (see problem 9.1), the states of the
‡The ground state of molecular oxygen is a triplet state.
1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4
no
rma
l. I
nte
nsity
Energy [eV]
800 700 600 500 400
1.47 470
1.85 950
2.5 2350
3.05 4220
Reff
(nm) #atoms
Wavelength [nm]
2 nm 8 nm
Figure 9.3: Spectral response of semiconductor nanocrystals with different
size. Left panel: Emission and excitation spectra. Right: Emission from a
series of nanocrystal solutions with increasing particle size excited simulta-
neously by a UV lamp. Courtesy of H.-J. Eisler.
9.3. SEMICONDUCTOR QUANTUM DOTS 7
exciton shift to higher energy as the confinement energy increases. In semiconduc-
tors, due to the small effective masses of the electrons and holes, the Bohr radius can
be on the order of 10 nm, which means that quantum confinement in semiconductor
nanocrystals becomes prominent at length scales much larger than the characteristic
size of atoms or fluorescent molecules. The confinement energy arises from the fact
that according to the Heisenberg uncertainty principle the momentum of a particle
increases if its position becomes well defined. In the limit of small particles, the
strongly screened Coulomb interaction between the electron and the hole, c.f. the last
term in Eq. (9.2), can be neglected. Both, the electron and the hole, can consequently
be described by a particle-in-the-box model, which leads to discrete energy levels that
shift to higher energies as the box is made smaller. Therefore, a semiconductor like
CdSe with a bandgap in the infrared yields luminescence in the visible if sufficiently
small particles (≈3 nm) are prepared. The quantum efficiencies for radiative decay
of the confined excitions is rather high because both the electron and the hole are
confined to a nanometer sized volume inside the dot. This property renders quantum
dots extremely interesting for optoelectronic applications.
In order to fully understand the structure of the electronic states in a semicon-
ductor nanocrystal several refinements to the particle-in-the-box model have to be
considered. Most importantly, for larger particles the Coulomb interaction has to be
taken into account and reduces the energy of the exciton. Other effects, like crystal
field splitting and the asymmetry of the particles as well as an exchange interaction
between electrons and holes also have to be taken into account [11].
For metal nanoclusters, e.g. made from gold, confinement of the free electrons
to dimensions of a few nanometers does not lead to notable quantum confinement
effects. This is because the Fermi energy for conductors lies in the center of the
shell
organic layer
core
Figure 9.4: Structure of a typical colloidal semiconductor nanocrystal. Cour-
tesy of Hans Eisler.
8 CHAPTER 9. QUANTUM EMITTERS
conduction band and, upon shrinking of the size, quantization effects start to become
prominent at the band edges first. However, if the confinement reaches the level of
the Fermi wavelength of the free electrons (≈0.7 nm), discrete, quantum-confined
electronic transitions appear as demonstrated in Ref. [12]. Here, chemically prepared
gold nanoclusters are shown to luminesce with comparatively high quantum yield at
visible wavelengths determined by the gold cluster size.
9.3.1 Surface passivation
Since in nanoparticles the number of surface atoms is comparable to the number of
bulk atoms, the properties of the surface strongly influence the electronic structure of
a nanoparticle. For semiconductor nanocrystals it is found that for ’naked’ particles
surface defects created by chemical reactions or surface reconstruction are detrimen-
tal for achieving a high luminescence quantum yield since any surface defect will lead
to allowed electronic states in the gap region. Consequently, nonradiative relaxation
pathways involving trap states become predominant. This leads to a dramatic reduc-
tion of the quantum yield for visible light emission. In order to avoid surface defects,
nano crystals are usually capped with a protective layer of a second, larger-bandgap
semiconductor that ideally grows epitaxially over the core such that the chemical com-
position changes abruptly within one atomic layer. These structures are designated
as Type I. If the capping layer has a lower band gap, then the excitons are prefer-
entially located in the outer shell with less confinement. These structures are then
designated as Type II. For CdSe nanocrystals usually a high-bandgap ZnS capping
layer is applied. With this protective shell, it is possible to functionalize the particles
by applying suitable surface chemistry without interfering with the optical properties.
The overall structure of a typical semiconductor nanocrystal is shown in Fig. 9.4. The
implementation of such a complex architecture at the nanometerscale paved the way
for a widespread application of semiconductor nanocrystals as fluorescent markers in
the life sciences.
Another widespread way to produce semiconductor quantum dots different from
the wet chemistry approach is to exploit self assembly during epitaxial growth of semi-
conductor heterostructures. Here, the most common way to produce quantum dots
relies on the so-called Stranski-Krastanow (SK) method. Stranski and Krastanow
proposed in 1937 that island formation could take place on an epitaxially grown sur-
face [13]. For example, when depositing on a GaAs surface a material with a slightly
larger lattice constant, e.g. InAs, then, the lattice mismatch (≈7% in this case) in-
troduces strain. The first few layers of GaAs form a pdeudomorphic two-dimensional
layer, the so-called wetting layer. If more material is deposited, the two-dimensional
growth is no longer energetically favorable and the material deposited in excess after
the wetting layer is formed organizes itself into three-dimensional islands as shown in
Fig. 9.5. These islands are usually referred to as self-assembled quantum dots. The
size and the density of the quantum dots produced this way can be controlled by
9.3. SEMICONDUCTOR QUANTUM DOTS 9
the growth parameters which influence both, the kinetics and thermodynamics of the
growth process. To complete the structure, the quantum dots have to be embedded
in a suitable capping layer, similar to the case of colloidal nanocrystals. The capping
layer also has to be carefully chosen to guarantee defect-free quantum dots with high
luminescence quantum yield.
9.3.2 Excitation
The absorption spectrum of semiconductor nanocrystals is characterized by increasing
absorption strength towards smaller wavelengths. This behavior originates from the
density of states which increases towards the center of the semiconductor conduction
band. The very broad absorption spectrum allows different-sized nanocrystals to be
excited by a single blue light source as illustrated in the right panel of Fig 9.3. Similar
to fluorescent molecules, semiconductor nanocrystals excited with excess energy first
relax by fast internal conversion to the lowest-energy excitonic state from which they
decay to the ground-state by photon emission. Different to molecules, multiple exci-
tons can be excited in the same quantum dot at high excitation powers. The energy
necessary to excite the second exciton is lowered by the presence of the first exciton.
Fig. 9.3 shows the excitation and emission spectra of a range of CdSe nanocrystals
of varying size. Apart from some fine structure near the band edge, the increasing
absorption for blue excitation can be clearly observed independent of the particle size.
For the emitted light, a shift of the emission (red curve) towards the blue spectral
region can be observed as the particle size is reduced.
Because of the symmetry of a nanocrystal, its dipole moment is degenerate. CdSe
nanocrystals are slightly elongated in direction of the crystal axis. Fig. 9.6 (a) sketches