Photogeneration Diffusion and Decay of Charge Carriers in Quantum-Dot Solids

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AND DECAY OF CHARGE CARRIERS

IN QUANTUM-DOT SOLIDS

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OF CHARGE CARRIERS IN QUANTUM-DOT

SOLIDS

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 16 oktober 2012 om 10 uur

door

Yunan GAO

Master of science in Optics, Peking University

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Chapter 1 Introduction

The title of this thesis is “photogeneration, diffusion and decay of charge carriers in quantum-dot solids”. But what are quantum-dot solids? Instead of the word “quan-tum dot” in the title, we start from the more easily understood word “nanocrystal (NC)”, which describes an important and big family of modern materials. Nanocrys-tals, as its literal meaning, are tiny crystals of meNanocrys-tals, semiconductors or insulating materials consisting of a few hundred to a few thousand of atoms each, and their sizes vary from several to hundreds of nanometers. In this thesis we will discuss a small group of the family: semiconductor nanocrystals.

In our colorful life, bulk semiconductors have been playing an indispensable role in modern electronics for more than hundreds of years, as their band gaps fall into the visible and near-infrared parts of the electromagnetic spectrum. They are widely used in many applications such as optical sensors, light-emitting diodes, lasers and solar cells. For bulk materials, the band gaps are mainly determined by their chemical compositions, as shown in Table 1.1 for several semiconductors. NCs open an entirely new route to tune the band gap: by simply controlling their size and shape. For example, the band gap of CdSe NCs can be tuned in the visible range, while the band gap of PbSe and PbS NCs can be tuned in the near-infrared range, as shown in Table 1.1 (ref. [1–3]). This property originates from the quantum confinement effect, which involves confinement of the wave functions of electrons by the small size of the NCs. Therefore, the NCs are also called “quantum dots (QDs)”. Because of the importance of the quantum confinement effect, we used the term “quantum dots” in the title, and will use it in the rest of this thesis.

The synthesis of QDs has been extensively studied over the past two decades, and has lead to many achievements [4, 5]: cheap and simple wet-chemical preparation; high quality, spherical dots with size variation less than 5%; more and more sophisticated structures. The properties of isolated QDs have also been heavily investigated and well documented [6–9]. Besides ongoing research into isolated QDs, a hot topic of

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current research involves assembled QDs, in the form of thin films also called QD solids, which are useful for many applications, such as photodetectors [10], field-effect transistors [11, 12], solar cells [13, 14], light-emitting diodes [15, 16], and lasers [17]. This thesis will focus on the fundamental properties of QD solids, such as photo-conductance, exciton dissociation, charge carrier motion and decay.

As individual QDs are the essential elements to assemble QD solids, in the fol-lowing section brief introductions are given to describe their synthesis and electronic properties. Critical issues related to the optoelectronic properties of QD solids are discussed in section 1.2. The outline of this thesis is presented in section 1.4.

Bulk (eV): GaN 3.4 ZnTe2.25 CdSe 1.73 Si 1.11 PbS 0.41 PbSe 0.27 QDs (eV): CdSe 1.73—2.3 PbS 0.41—1.5 PbSe 0.27—1.4

Table 1.1: Band gaps of several bulk and noanocrystalline semiconductors

1.1 Quantum dots

In this section, the synthesis and several important properties of isolated QDs, usually dispersed in solvent, will be discussed using basic models. These models capture the main characteristics of the QDs, which are closely related to the subjects studied in this thesis.

1.1.1 Colloidal quantum dot synthesis

In the family of colloidal QDs, spherical dots are the most simple objects, and have been synthesized successfully in terms of quality and monodispersity in the 1990s with the “hot-injection” method [18]. Typically, this method entails thermal decomposi-tion of metal-organic precursors in hot coordinating solvents. The synthesis generally involves the following consecutive stages: burst nucleation initialized by quick injec-tion of the precursors into hot coordinating solvents, growth of the formed nuclei, quenching of the growth as the particles reach the desired size, and isolation of the particles from the reaction mixture.

Nucleation and growth of QDs occurs in the solution phase in the presence of organic surfactant molecules, which dynamically adhere to the surface of the growing crystals. These surfactants keep the nucleation and growth rates kinetically balanced, which is required to form QDs of the scale of nanometers instead of either bulk crystals or molecular clusters. Achieving proper balance usually requires different combinations of molecular precursors, surfactants, solvents and reaction conditions

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1.1. QUANTUM DOTS

a

c

b

Figure 1.1: TEM images and schematic illustration of colloidal quantum dots [19]. a, A high resolution TEM image of a spherical PbSe quantum dot. b, Schematic of quantum dot with semiconductor core and long organic ligands. c, TEM image of a monolayer of PbSe QDs.

for QDs of different materials. For a specific type of QDs, reaction temperature and duration are usually tuned to get the desired QD size.

The colloidal QDs obtained by the above route have a typical semiconductor core of several nanometers in diameter, as shown in a transmission electron microscopy (TEM) image in Figure 1.1a, and long organic surfactants as illustrated in Figure 1.1b. Generally, the core determines the electronic properties of the QDs, while the surfactants ensure stability, and determine the solubility of the QDs. In QD solids the surfactants separate the QDs, and determine the inter-particle spacing, which has a strong impact on inter-QD electronic coupling, and hence carrier transport, and will be discussed in the following section. Figure 1.1c shows a TEM image of a drop-casted QD solid, which shows that the QDs are separated by the long organic surfactants.

1.1.2 Electronic structure of quantum dots

The two most distinguishing properties of QDs compared to bulk materials are: (1) the energy gap varies with the QD size; (2) the energy levels are discrete. These two properties stem from the size confinement effect: electrons “feel” the presence of the particle boundaries and respond to changes in the particle size by adjusting their energy spectra. The two properties can be understood with a simple “quantum box” model within the effective-mass approximation.

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m0 inside a spherical potential well of radius a: V (r) =    0 r < a ∞ r ≥ a (1.1)

The time-independent Schr¨odinger equation for this problem is:

− ~ 2m0 ∇2_{+ V (r)} Φ = EΦ (1.2)

with the following solution for the wave functions:

Φn,l,m(r, θ, φ) = C

Jl(kn,l, r)Ylm(θ, φ)

r (1.3)

where ~ is the reduced Plank constant, C is a normalization constant, Jl(kn,l, r) is

the lth_{-order spherical Bessel function, and Y}m

l (θ, φ) is a spherical harmonic.

The infinite potential at the boundary r = a implies that kn,l only can have

discrete values of

kn,l=

1

aαn,l (1.4)

where αn,l is the nth zero of Jl(kn,l, r), a few of which are listed in Table 1.2. Hence

the energies of the particle are discrete as given by:

En,l= ~2k2n,l 2m0 = ~ 2_α2 n,l 2m0a2 (1.5)

The energy given by Equation 1.5 is proportional to 1/a2 and therefore is strongly dependent on the size of the sphere. The energy levels have quantum numbers n(1, 2, 3, ...) and l(0, 1, ..., n − 1). l n = 1 n = 2 n = 3 0 π 2π 3π 1 4.49 7.72 9.42 2 5.76 9.09 10.90 3 6.99 10.42 12.07

Table 1.2: Roots of the Bessel function Jl(kn,l, r)

In the above model the sphere is empty, while QDs consist of atoms forming a piece of crystalline semiconductor material. The effect of the potential due to the atoms can be taken into account by replacing the free electron mass with the effective mass, as discussed below.

We consider bulk material first and then return to quantum dots. In bulk semi-conductor materials, the highest energy states filled by the electrons at absolute zero

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1.1. QUANTUM DOTS

temperature are called valence bands. The empty states of higher energy than the valence band are called conduction bands. Between the valence band of the highest energy and the conduction band of the lowest energy, there is an energy range that has no available states, which is called energy gap. As an electron is excited from the valence band to the conduction band, a pair of one positive and one negative charge carrier (an electron-hole pair) is created. When these carriers are moving in the ma-terial, they experience a periodic potential resulting from the atoms in the material. The potential can be incorporated approximately by replacing the free electron mass with an “effective mass, m∗”. With this approximation, the energy of the conduction and valence bands are described as:

Ekc= ~2k2 2m∗ c + Eg E_kv= ~ 2_k2 2m∗ v (1.6)

where Eg is the band gap energy.

The effective-mass approximation has been extended to describe the quantum dot problem [6, 9]. Here we only discuss QDs with sizes in the strong confinement regime†, where the kinetic energy is the dominant energy contribution and the Coulomb inter-action acts as a perturbation to the kinetic energy contribution. In this case, electrons and holes can be treated independently, and the kinetic energies of the confined elec-trons and holes can be described by Equation 1.5 with effective masses of m∗_e and m∗_hrespectively. The energy Ec due to Coulomb attraction between the electron and

hole, which leads to excitons in the bulk material, is calculated by perturbation the-ory. In addition, a correction due to dielectric effects [8] is considered. Finally, the electron-hole pair state energies are written as:

Eeh= Eg+ ~ 2 2a2 α2 ne,Le 2m∗ e +α 2 nh,Lh 2m∗ h !

+ E_ehc + E_eself+ E_hself+ E_ehcross (1.7)

where E_ehc is the energy from the Coulomb interaction between electron and hole; Eself _{is the electrostatic self-energy, which is due to the interaction of a charge with}

the polarization induced by itself; Ecross is the cross-polarization energy, which arises from the interaction of a charge with the induced polarization resulting from other charges. The pair states are labeled with the quantum numbers neLenhLh. The pair

†_{There are three regimes of quantum confinement, depending on the ratio of the QD radius R and}

the Bohr radii aeof electrons and ahof holes: strong confinement regime ae, ah R; intermediate

confinement ae R ah (in case that the effective mass of the holes is much bigger than the

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state of lowest energy is written as 1Se1Sh. The correction terms have been calculated to be [8]: E_ehc = − 1.79e 2 4πεinε0a (1.8) Eself ≈ e 2 8πε0a εin− εout εinεout (1.9) Ecross ≈ − e 2 4πε0a εin− εout εinεout (1.10)

where εinand εoutare the dielectric constants inside and outside of the QD. Equation

1.8 - 1.10 show that the correction due to dielectric effect can be neglected for excitons, since Eself

h + Eeself equals −Ecrosseh . However, this correction has to be considered in

the exciton dissociation process, which will be discussed in section 1.2.2.

0.6 0.8 1 1.2 1.4 1.6 0 0.05 0.1 0.15 E nergy (eV) N or m al iz ed a bs or ba nc e E (QD)g 1S_e 1S_h e e 1P 1P_h 1D 1D_h V 1S_e1S_h 1P_e1P_h

a

b

Figure 1.2: a, Absorption spectra of PbSe QD dispersions of 6.3 nm, 4.9 nm, 4.2 nm and 2.3 nm in diameter (left to right). Three peaks in each spectrum are clearly resolved, and the first and the second ones correspond to 1Se1Sh and 1Pe1Ph

tran-sitions [20]. b, Schematic of the low discrete states and optically allowed trantran-sitions for PbSe QDs.

The discrete states in QDs are reflected in absorption spectra as individual peaks, which correspond to optically allowed transitions between the states. Figure 1.2a shows absorption spectra for several PbSe QD dispersions. The first peak is from the 1Se1Shtransition, and the second one is from the 1Pe1Phtransition, as illustrated in

Figure 1.2b. It also shows that as the diameter of QDs decreases from 6.3 nm to 2.7 nm, the 1Se1Shtransition energy shifts from 0.7 eV to 1.0 eV, indicating the effect of

quantum confinement on the QD energy gap.

The preceding model has taken into account the effect of the potential due to atoms in the lattice by using the effective-mass approximation. By doing this, the

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1.1. QUANTUM DOTS

model explains the discrete energy states and the effect of QD size on the energy gap. But the model is insufficient for accurate determination of energy levels, since in the effective-mass approximation single conduction and valence bands are considered, while quantum confinement can lead to mixing of these and other bands. A more detailed discussion about the electronic structures of QDs can be found in these books ref. [6–9]. In this thesis we mainly focus on the charge carrier properties in 1Se1Shenergy state, and the theoretical framework discussed above is sufficiently to

understand the experimental results.

1.1.3 Charge carrier decay mechanisms

When QDs are illuminated with photons with an energy greater than the energy gap Eg, excitons are generated in the QDs. In QD dispersions, the excitons usually

decay via three processes: radiative recombination, Auger recombination and surface trapping, as illustrated in Figure 1.3. The decay rates and required conditions are different for these paths, and they have been experimentally determined.

radiative recombination surface trapping QD energy states surface trap state Auger recombination

a

b

Figure 1.3: Schematic illustration of charge carrier decay paths: a, radiative recom-bination and surface trapping, b, Auger recomrecom-bination.

In radiative recombination, a 1Se electron recombines with a 1Sh hole, and the

energy is released by emitting a photon. This process can be observed by photo-luminescence spectroscopy (PL). Figure 1.4 shows a PL spectrum of a PbSe QD dispersion, where the emission peak is red shifted with respect to the first exciton absorption peak. The shift is related to the fine structure of the 1Se1Shexciton. For

PbSe and PbS quantum dots, the rate is typically on a ∼ µs−1 timescale [2, 21]. Auger recombination is a non-radiative process, in which the released energy is transferred to a third particle (an electron or a hole) which is re-excited to a higher-energy state (Figure 1.3b). Auger recombination has been found to be quite efficient in quantum-confined systems. In bulk materials Auger recombination is limited by momentum conservation. It has been suggested that the efficient Auger recombination in QDs is the result of relaxed momentum conservation [22, 23].

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0.8 1 1.2 1.4 1.6 1.8 0 0.1 0.2 energy (eV ) ab so rb an ce ( O .D .) PL t=1.7ms 2 4 6 8 1 2 4 a rb itr ar y un it 6 5 4 3 2 1 0 time delay (10-6 s)

a

b

Figure 1.4: a, Absorption (solid line) and photo-luminescence (dashed line) spectra of PbSe QD dispersions. b, Emission decays recorded at the peak of the luminescence, and a single-exponential decay function is applied to determine the single exciton lifetime.

In QDs, Auger recombination occurs via a sequence of “quantized” steps from N excitons to N − 1, N − 2, and so on, finally, to the single exciton. Each step has a different decay rate 1/τN. The Auger recombination rates scale up fast with

exciton numbers. For PbSe QDs, if all excitons are in the 1Se1Sh state, the Auger

recombination rate scales statistically as 1/τN ∝ N2(N − 1) [24]: the rate depends

on the product of the number of all possible 1Se to 1Sh transitions (given by N2)

and the number of carriers that can accept the energy released (given by N − 1). In addition, the Auger recombination rate strongly depends on the size of QDs. It has been observed that 1/τN scales with R−3. This scaling rule seems to be independent

of the composition and electronic structure of the material [25].

As discussed above, Auger recombination occurs when there are multi-excitons. Under optical excitation, excitons are photogenerated according to a Poisson distri-bution†:

nm/ntot =

hN im m! exp

−hN i_{, m = 1, 2, . . . , N} _(1.11)

where nm is the number of QDs with m excitons, ntot is the total number of QDs,

and hN i is the average number of photons absorbed per QD. When hN i=1, there is a 7% probability that biexcitons are formed in a QD. Hence, roughly when hN i is above 1, Auger recombination plays a role in charge carrier decay.

The biexciton Auger recombination occurs on a timescale of tens to hundreds ps that is much faster than the radiative recombination (on µs timescale). For PbSe QDs with 4.7 nm in diameter, the biexciton life time is ∼ 25 ps, while the single exciton

†_{In certain cases, one photon can produce multiple excitons. The multiple exciton generation is}

a hot topic in QD research [26–30]. In this thesis we will limit the discussion in the case where one absorbed photon generates one exciton.

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1.2. QUANTUM-DOT SOLIDS

radiative lifetime is ∼ µs. As a result, Auger recombination is the dominant decay path on short timescale and at high exciton populations, while radiative recombination is the main decay path for single exciton on long timescales.

Surface trapping is another important non-radiative process. It is strongly affected by the surface passivation of QDs. If the surfaces of QDs are not fully passivated, there are dangling bonds that may act as traps for charge carriers. Typically, after the synthesis QDs are encapsulated by long organic capping ligands. These capping ligands can very efficiently reduce the density of the trap states by binding to the surface of QDs. Note that some capping ligands can also be trap states themselves, if their molecular energy levels are in the band gap of the QD. It has been shown that surface trapping is absent on the sub-ns timescale for PbSe QDs. In this case, trapping is a slow process that can not compete with Auger recombination. However it can compete with radiative recombination resulting in a decrease of the PL quantum yield. Since the PL quantum yield is very important for applications of light emission devices and bio-labeling, surface trapping is a big issue of concern. With proper capping, the PL quantum yield can be well above 50%, and in some case can reach values of 70∼80% [31, 32].

In this subsection, we have described the main charge carrier decay paths for QDs in dispersion, which have been intensively investigated in the past decades. When QDs are assembled into QD solids, the charge carrier decay mechanisms get more complex, as more factors need to be considered, such as that carriers may diffuse, that surface passivation may be poor, and that the electronic environment is more complicated. The charge carrier decay mechanisms in QD solids get increasing attention now, since the solid form is relevant to many applications. Chapters 4 and 5 present our studies on this topic.

1.2 Quantum-dot solids

Quantum-dot solids are arrays of quantum dots. They can be assembled by solution deposition of colloidal quantum dots. The assembly of QDs is briefly described in section 1.2.1. QD solids are able to retain the quantum confinement properties of the individual elements. They can have new properties due to the interactions between QDs. In this section, we discuss several key issues related to optoelectronics devices: exciton dissociation, inter-particle electronic coupling and carrier transport.

1.2.1 Quantum-dot solid assembly

QD solids can be assembled as ordered superlattices or as amorphous (glassy) struc-tures. In the case of ordered superlattices, the position of particles, the inter-particle

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distances and packing density are well defined throughout the whole structure, while in glassy solids, there are variations in the inter-particle distance and local organiza-tion. Both types of assemblies are under intense investigations.

QD superlattices are very attractive for both practical applications and fundamen-tal studies due to their well-defined structures. Long-range ordered superlattices have been prepared via self-assembly of colloidal QDs, upon slow evaporation of solvent or gentle destabilization of the colloidal solution. It has been shown that different types of long-range ordered single- and multi-component QD superlattices can be prepared by this approach [33]. The assembly of QD superlattices is still facing many chal-lenges. First of all, precise control of assembling large QD superlattices with good reproducibility is still challenging. A further challenge is to control the properties of these self-assembled superlattices. The self-assembly and the properties of QD su-perlattices both strongly depend on the capping ligands of QDs. As a result, it is very hard to control both at the same time. QD superlattices with significant inter-particle electronic coupling have not been reported. Although the studies of ordered structures are predominantly focusing on the physics of colloidal self-assembly, it is expected that new materials with novel properties will come out from this approach. On the other hand, for many applications ordered structures are not really needed. Amorphous structures can be assembled easily with all kinds of solution deposi-tion techniques, such as drop-casting, spin-coating, or layer-by-layer dip-coating, etc. Thick, smooth and large-scale films can be fabricated in these ways. The properties of QD solids, such as charge carrier mobilities, types of conductance etc, can be tuned by thermal annealing or chemical treatments, during which the capping ligands are removed or replaced (see section 1.2.3).

In this thesis, we study the optoelectronic properties of amorphous QD solids. The Layer-by-layer (LBL) dip-coating method and chemical treatments with ligand exchange are used to fabricate glassy, smooth PbSe QD solids. Figure 1.5 shows a photograph, SEM images and a cross-sectional TEM image of typical films, which indicate the smooth and crack-free surface in all length scales.

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Quartz

PbSe

QDs

CdSe

QDs

a

b

d

c

Figure 1.5: Photograph and images of layer-by-layer dip-coated quantum-dot solids. a, Photograph of a solid on a transparent quartz substrate. The shadow of tree leaves is visible through the smooth and homogeneous film. b, c, SEM images at intermediate magnification demonstrating the excellent homogeneity of the film. d, Cross-sectional TEM image shows a smooth and sharp interface between the substrate and layers of PbSe QDs and CdSe QDs.

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1.2.2 Exciton dissociation

QDs dispersed in a solvent are separated from each other and isolated by the sol-vent molecules, hence charge carrier transport between QDs does not occur. In QD solids, the spacing between the QDs can be engineered to be between zero and several nanometers. This decreased distance results in enhanced electronic coupling between QDs, and may result in charge charge transport. Before examining charge carrier transport (Section 1.2.3), first we have to consider whether the charge carriers (elec-trons and holes) can dissociate from each other.

The dissociation rate kdiss of an exciton is determined by the energy ∆Ediss

re-quired to dissociate the electron-hole pair, and is approximated by [34]:

kdiss= k0exp

−∆Ediss

kBT

(1.12)

where k0is a pre-factor related to the carrier hopping rate, hence the carrier mobility,

which will be discussed later (Section 1.2.3). The dissociation energy can be estimated using Equation 1.7.

If an electron and a hole are spatially separated and do not interact, the direct Coulomb interaction term and the cross-polarization energy in Equation 1.7 are zero. By comparing to the energy of the exciton, the dissociation energy is obtained as

∆Ediss = −Eehc − Eehcross

= 1.79e 2 4πεinε0a + e 2 4πε0a εin− εout εinεout (1.13)

Equation 1.13 tells us that the dissociation energy strongly depends on the dielectric constants of the QDs and their surroundings. For instance, PbSe has a static dielectric constant of ∼ 250, while CdSe has one of ∼ 6.2, therefore a much lower dissociation energy for an exciton in a PbSe QD solid than in a CdSe QD solid is expected. In a QD solid εout is determined by the capping ligands and neighboring QDs, and is

higher than εout in dispersion, therefore a lower dissociation energy is expected.

Following the same procedure as above, we estimate the dissociation energy for a carrier to dissociate from any combination of multiple carriers. The site energy, including the first and second terms of Equation 1.7, and electrostatic self-energy, will not contribute to the dissociation energy, so for simplicity we temporarily only consider the direct Coulomb interaction term Ec

eh and the cross-polarization energy

Ecross

eh . The sum of these two terms is referred to as E0.

Figure 1.6a shows E0 _{for an exciton. Figure 1.6b shows E}0 _{for a trion (eeh),}

where we have to assume that the interactions between two charge carriers are not influenced by the presence of other charge carriers. As a result, in Figure 1.6b the interaction between particle 1 and 2 leads to an energy of Ec

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between 1 and 3 leads to another Ec eh+E

cross

eh , and the interaction between 2 and 3

gives E_hhc +E_hhcross. Between two electrons or two holes, the Coulomb repulsion and cross polarization have the same absolute values as for an electron and a hole, but the signs are opposite. To be clear, it is written down here as E_hhc =E_eec=-E_ehc and Ecross

hh =E cross

ee =-Eehcross. By doing this, E

0 _{can be calculated for any carrier}

combina-tion. By comparing E0 before and after charge carrier dissociation, the dissociation energy ∆Ediss can be calculated, which is presented in Table 1.3 for various cases of

charge carrier dissociation.

E’=E +E

(1,1)

a

eh c eh cross 1 2 3 (1,2)

b

E’=2(E +E )+E +E

_ehc _ehcross _hhc _hhcross

=E +E

_ehc _ehcross

Figure 1.6: A Schematic illustration of how E0, i.e. the sum of direct Coulomb in-teraction energy Ec _{and the cross-polarization energy E}cross_{, is calculated. a, E}0 _for

an exciton. b, E0 for a trion (ehh) including contributions from interactions between particles 1&2, 1&3, and 2&3.

initial occupation final occupation E_initial0 E_final0 ∆Ediss

A(0,1)B(0,0) A(0,0)B(0,1) 0 0 0

A(1,1)B(0,0) A(1,0)B(0,1) E_ehc +E_ehcross 0 -(E_ehc +E_ehcross) A(2,0)B(0,0) A(1,0)B(1,0) -(Ec

eh+E cross eh ) 0 E c eh+E cross eh

A(2,1)B(0,0) A(1,1)B(1,0) E_ehc +E_ehcross E_ehc +E_ehcross 0 A(2,1)B(0,0) A(2,0)B(0,1) Ec eh+E cross eh -(E c eh+E cross eh ) -2(E c eh+E cross eh )

A(1,1)B(1,1) A(0,1)B(2,1) 2(Ec_eh+E_ehcross) E_ehc +E_ehcross -(E_ehc +E_ehcross) Table 1.3: The dissociation energy ∆Ediss is calculated for various cases of

charge-carriers dissociation. A(n, m)B(n, m) presents two QDs with or without charge car-riers present, with electron and hole numbers of (n, m) (also can denote n holes and m electrons). For clarity only Ec

eh and E cross

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1.2.3 Charge carrier transport

Charge carrier transport in QD solids requires sufficient inter-particle electronic cou-pling. However the coupling should not be so strong as to lose the distinct properties due to quantum confinement. Here, we will restrict our discussion to a weak cou-pling regime, where quantum confinement is retained and carrier transport occurs by hopping, and moreover, we only discuss carrier transport due to hopping between adjacent QDs†.

In this subsection, first we introduce two popular approaches to increase the inter-particle electronic coupling; in what follows, we discuss the relations between the coupling energy, the hopping rate and the carrier mobility; we end up with a discussion about the impact of disorder and the Miller-Ambrahams model.

Inter-particle electronic coupling

If we say that exciton dissociation is like the desire of jailed carriers to escape from the Coulomb interaction, inter-particle electronic coupling is the pathway. As shown before (Figure 1.1), in a drop-casted film QDs are separated by the organic capping ligands with several nm long hydrocarbon chains, which behave as long dielectric tunneling barriers and lead to very weak electronic coupling. The tunneling rate decreases exponentially with the barrier width according to the Wentzel-Kramers-Brillouin (WKB) approximation, for resonant tunneling, as

Γ = Γ0exp − 2m∗_E bar ~2 1/2 R ! (1.14)

where m∗_{is the effective mass of the charge carrier, R is the width of the barrier, ~ is} the reduced Planck constant, and Ebar is the barrier height. Γ0 is the rate in case of

vanishing barrier. As a result, the films with long capping ligands are insulating, and carrier transport through the film does not occur. A straightforward way to increase the coupling, hence the carrier transport, is to decrease the ligand length.

Thermal annealing is a way that has been used to remove the insulating ligands, resulting in reduction of inter-particle distances and improvement of film conductance by several orders of magnitude [37–39]. However, there are several weaknesses to this approach. The removal of capping ligands leaves dots bare with many dangling bonds on the surface, which behave as charge carrier traps. Annealing removes the organic

†_{When we say “weak coupling regime”, we mean that the electron wave function is localized on}

a quantum dot. The energy overlap integral between neighboring QDs would be much smaller than the interaction energy of two electrons of opposite spins (Mott’s model) and the energy disorder in QD dot solids (Anderson’s model) [35]. In certain cases, such as low temperatures, hopping can occur between sites that are not closest. This process is pointed out by Mott, and is termed Variable range hopping [36].

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R

DE

_bar

decrease R

Figure 1.7: A Schematic to illustrate that by decreaseing the width of the barrier the inter-particle electronic coupling increases, and carrier transport can occur.

ligands, but in some situation also leads to QD sintering, forming large crystals. Additional problems are poor control of properties of the annealed solids and low reproducibility [39]. In addition, high-temperature operation is not favorable for low cost device fabrication.

Another approach is to use a chemical treatment, where short ligands are used to replace the native long insulating ligands from the synthesis. Significant progress has been achieved in recent years by this approach. Various short organic ligands with dif-ferent anchor groups, such as amine, thiol or carboxylic acid and with difdif-ferent lengths have been used as replacing ligands [11, 40–54]. These works have shown enhanced inter-particle electronic coupling and high carrier mobilities of several cm2_V−1_s−1_.

Recently, replacing ligands have been extended to inorganic ligands like Br−_{, Cl}−_,

I−, S2−_{, HS}−_{, etc [55, 56]. With some of these inorganic ligands even higher carrier}

mobilities of 16 cm2_V−1_s−1 _{are obtained[55].}

Electronic-coupling energy, dissociation rate, hopping rate and carrier mo-bilities

So far, we can understand that the inter-particle electronic coupling together with the exciton dissociation rate determines carrier transport. However it is hard to calculate the values directly, since information about the micro organization of the QDs in the solids is unavailable, especially for amorphous solids. In practice, it works the other way around: the coupling energy, the dissociation rate and the hopping rate are estimated from the carrier mobilities, which are determined experimentally.

The coupling energy β is estimated by [57]:

β = hΓ/4 (1.15)

and the hopping rate is estimated using the Einstein-Smoluchowski equation:

Γ = µkBT /(ea2) (1.16)

where e is the elementary charge, a is the hopping distance taken as the center-to-center distance, kB is the Boltzmann constant, T is the temperature, and µ is the

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carrier mobility.

The dissociation rate can be estimated by the Onsager-Braun model [34]:

kdiss= 3eP µ 4π<ε>ε0a3 exp −∆Ediss kBT (1.17)

where <ε> is the effective dielectric constant of the film.

Only the average values over the whole solid are estimated, and local details and variations are ignored.

Disorder and Miller-Ambrahams model

Disorder decreases the carrier mobilities, and affects the energy distribution of carri-ers, as has been shown in length for a variety of disordered materials [58–60].

Here we discuss two types of disorder in QD solids, energy disorder and electronic coupling disorder. Energy disorder in QD solids is unavoidable, since that QDs have a finite dispersion in their size and shape. Advanced synthetic techniques allow for size dispersions within a single atomic layer [61] which still leads to a standard deviation of several percent. In QD solids, there is an additional source for energy disorder: a variation in local QD density will result in a variation of surrounding dielectric constant <ε>out, which influences the site energies through the self energy of the

charges (Equation 1.7). The electronic-coupling disorder is due to variation of the spacing between neighboring QDs. The disorder is clearly reflected by a relatively broad exciton absorption peak in dispersion as well as in a QD solid. In dispersion, the broad peak is mainly due to energy disorder, while in solids it is a combined effect of the both types of disorder.

In 1960 Miller and Abrahams have derived a model [62] to describe charge trans-port in doped inorganic semiconductors. Later, this model has been extended and widely used to describe the hopping rates in organic materials, and is currently ap-plied to QD solids. In the Miller-Abrahams formalism, the hopping rate Γi→j from i

site to j is expressed as:

Γi→j= Γ0exp (−2βRij)      1, Ei≥ Ej exp Ei− Ej kBT , Ei< Ej (1.18)

where Γ0 is an attempt hopping rate, β is the overlap factor, Rij, is the separation

between sites i and j, and Ei and Ej is the relative energy of the sites. The first

exponential term describes the dependence of the electronic coupling on the inter-particle spacing, and the second term describes the impact of energy disorder.

To apply this model to QD solids, ideally, we would need to know the distributions of both the inter-particle spacing and the energy levels, rather than a combined profile

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1.3. EXPERIMENTAL METHODS

reflected by absorption spectra. However, this detailed knowledge is not available. In addition, the dissociation energy discussed in section 1.2.2 and the degeneracy of the energy states have to be considered. Nevertheless, by including these terms with proper assumptions, the Miller-Ambrahams model can give us a platform for carrying out numerical simulations, which is shown in Chapter 4 to be a powerful tool in understanding the physics behind the experimental results obtained in this thesis.

1.2.4 Charge carrier decay kinetics

In QD solids, the charge carrier decay has been found to be mainly due to radiative recombination, surface trapping and Auger Recombination. The impacts of these processes change dramatically as compared to the situation in QD dispersions. In solids, the surface of QDs is less well passivated compared to the surface of QDs in dispersions, resulting in surface trapping that dominates over radiative recombination. In addition, in solids with significant electronic coupling, excitons can dissociate before radiative recombination. Auger recombination could occur at low carrier density, if carriers can diffuse through the solids. Few studies have been carried out on charge carrier decay in QD solids, and parts of this thesis present our detailed studies on this topic.

1.3 Experimental methods

1.3.1 Quantum dot synthesis and quantum-dot solid assembly

We used an oleylamine-based PbSe QD synthesis process [63] that resulted in weakly bound ligands that could be easily replaced. Oleylamine (14 ml) and 0.38 g PbCl2

were degassed under vacuum at 100◦C, followed by heating to 110∼160◦C (depends on aimed sizes) under nitrogen. Next, 50∼750 ml Sn[N(SiMe3)2]2(depends on aimed

sizes) dissolved in 2 ml trioctylphosphine (TOP) and 6 ml of 1 M Se in TOP were mixed in a syringe and immediately injected into the PbCl2/oleylamine solution.

The reaction mixture was kept at the temperature after the injection for 2 min, and cooled with a water bath. The resulting solution was filtered with a 200 nm pore size filter to remove residual PbCl2. The QDs were precipitated with butanol and finally

dried under vacuum. The QDs were dispersed in tetrachloroethylene for absorption measurements and in hexane for dip-coating.

Quartz substrates were silanized with (3-aminopropyl) triethoxysilane [64]. PbSe QD solids were produced by a mechanical dip coater (DC Multi-8, Nima Technology) mounted inside a nitrogen glove box. A silanized quartz substrate was dipped

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alter-nately into a solution of PbSe QDs in hexane for 60 s, a 0.1 M solution of replacing ligands in methanol for 60 s, and a rinsing solution of methanol for 30 s. The dipping procedure was repeated 15 times, resulting in typical film thicknesses of 30-40 nm and a surface roughness of 5 nm, as determined with a Veeco Dektak 8 step-profilometer.

1.3.2 Absorption spectra

Optical absorption spectra were recorded with a Perkin-Elmer Lambda 900 spectrom-eter equipped with an integrating sphere. The spectra were corrected for scattering and reflection.

1.3.3 Transient absorption spectroscopy

The samples were excited and monitored by pump and probe pulses from a chirped-pulse amplified laser system (Mira-Legend USP, Coherent Inc.), running at 1 kHz and delivering pulses of 60 fs, 2.2 mJ and at a wavelength of 795 nm. Tunable infrared and visible pulses (<100 fs) were generated by optical parametric amplification (Topas-800-fs and Opera, Coherent Inc.). Pump and probe beams were imaged onto InGaAs pin photodiodes (Hamamatsu G5853-23, G8605-23). The two beams were spatially separated downstream of the sample. Orthogonal polarization of the beams allowed further separation by means of a polarizer.

1.3.4 Terahertz spectroscopy

The ultrafast photoconductance of the QD solids was investigated by time-domain terahertz spectroscopy. The samples were photoexcited with a 60 fs laser pulse from the same laser described above and probed by a picosecond terahertz pulse. Tera-hertz pulses were generated by optical rectification in a ZnTe crystal. The excitation beam (diameter,∼ 5 mm) was aligned on the sample with the focused terahertz beam (diameter, <2 mm). The terahertz field transmitted through the sample was detected by means of the Pockels electro-optic effect in a 0.5 mm ZnTe crystal. The relative change in terahertz transmission, ∆E/E, is proportional to the photoconductance, or, equivalently, to Φ(t)Σµ, where Φ(t) is the yield of charge carrier photogeneration and Σµ is the sum of the electron and hole mobilities.

1.3.5 The Time-resolved Microwave Conductivity Technique

(TRMC)

The photoconductance of QD solids was investigated using the TRMC technique [40, 65, 66]. The samples can be mounted in an X-band microwave cavity (8.4 GHz) at the

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1.4. OUTLINE OF THE THESIS

position of maximum electric field (100 V/cm) or in an open cell. The temperature of the cavity or the open cell can be varied between 90 K and 350 K using liquid nitrogen and power resistors. Photoexcitation laser pulses with a duration of 3 ns were obtained by pumping an optical parametric oscillator with the third harmonic of a Q-switched Nd:YAG laser (Opotek Vibrant 355 II). The sample was excited at 1.8 eV. The photon fluence I0 was varied between 1×1011 and 1×1017 photons/cm2/pulse, using neutral density filters and focusing lenses. The average number of photons absorbed per QD was obtained as <Nabs>=I0σ, where σ is the absorption cross-section taken from

ref. [3].

Upon photoexcitation, the change in microwave power reflected was measured. For small photo-induced changes in the real conductance of the sample, ∆G(t), and negligible change in imaginary conductance, the relative change in microwave power is

∆P

P = −K∆G(t) (1.19)

K is a sensitivity factor that has been determined previously [66]. The photoconduc-tance ∆G(t) can be expressed as

∆G(t) = eβI0FaΦ(t)Σµ (1.20)

where e is the elementary charge, β is the ratio between the broad and narrow inner dimensions of the waveguide, I0is the photon fluence in the laser pulse, and Fa is the

fraction of light absorbed by the sample. The time resolution of the TRMC setup is limited to ∼17 ns by the response time of the cavity when resonant cavity is used, and is limited by the 3 ns laser pulse when the open cell is used.

1.4 Outline of the thesis

This thesis describes “the life and fate” of photogenerated charge carriers in strongly coupled PbSe QD solids: from the first moment of charge carrier photogeneration, to carrier thermalization and carrier diffusion through mid-life, and finally to carrier decay. Figure 1.8 presents a schematic summary of all processes that charge carriers can undergo.

Chapter 2 describes studies of carrier photogeneration and carrier transport, as shown in panels a and b of Figure 1.8. Chapter 3 reports the fast carrier thermal-ization processes in the first few ps after the initial photogeneration, including both intra-QD and inter-QD thermalization (Figure 1.8c). In Chapter 4 and Chapter 5, the charge carrier decay processes are studied, including diffusion-mediated Auger recombination and carrier trapping (Figure 1.8d, e). Chapter 5 presents a systematic study of the influence of the chemical nature of surface passivation ligands on the photoconductivity in QD solids.

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isolated QD

t~1ps to t~3ps carrier thermalization

diffusion mediated Auger Recombination t~1ns to t~ 1 ms_{first order trapping}

a, energy state structure c, carrier inter QD thermalization

d, carrier Auger Recombination

tdiffusion tAR e, carrier trapping E electron trap hole trap assembled QD solid b, carrier generated at 100%

yield and with high mobility

Figure 1.8: Schematic illustration of the “life and fate” of photogenerated charge carriers in strongly coupled PbSe QD solids. a, With the reduction of the capping ligand length, the electronic coupling is increased. b, In QD solids, charge carrier transport can occur with high carrier mobilities, and photogenerated excitons can efficiently dissociate to form free carriers. c, After photogeneration, a fast carrier thermalization follows in a few ps. d, Carriers decay via diffusion mediated Auger recombination on a ∼ 100 ps timescale. e, Finally, on a longer timescale of a ns to ms, charge carriers get trapped by surface trapping and recombine to valence band.

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Chapter 2 Unity quantum yield of

photogener-ated charges and band-like transport

in quantum-dot solids

Appendix Solid films of colloidal quantum dots show promise in the manufacture of photodetectors and solar cells. These devices require high yields of potogenerated charges and high carrier mobilities, which are difficult to achieve in quantum-dot films owing to a strong electron-hole interaction and quantum confinement. Here, we show that the quantum yield of photogenerated charges in strongly coupled PbSe quantum-dot films is unity over a large temperature range. At high photoexcitation density, a transition takes place from hopping between localized states to band-like transport. These strongly coupled quantum-dot films have electrical properties that approach those of crystalline bulk semiconductors, while retaining the size tunability and cheap processing properties of colloidal quantum dots.

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2.1 Introduction

Colloidal semiconductor nanocrystals, also called quantum dots (QDs), have been subject to a great deal of research because of their tunable optical and electronic properties, as well as their facile and cheap solution-based synthesis and processing. These properties make them interesting for use in field-effect transistors [1], biosen-sors [2], photodetectors [3], light-emitting devices [4], lasers [5] and solar cells [6–11]. In most devices, colloidal QDs are assembled into thin films, known as QD solids. These QD solids were initially found to exhibit low carrier mobilities of, <1 × 10−4

cm2_V−1_s−1_{(refs. [12, 13]) as a result of the long insulating organic ligands present at}

the QD surface after synthesis. The introduction of shorter organic ligands dramati-cally enhanced the carrier mobility [1, 13] up to the currently highest reported values of 0.9 cm2V−1s−1 for electrons and 0.2 cm2V−1s−1 for holes in PbSe QD solids [1]. Inorganic metal chalcogenide complexes may be used instead of the organic ligands, and these complexes can subsequently be sintered in an annealing step. This ap-proach has very recently resulted in electron mobilities as high as 16 cm2_V−1_s−1 _and

the observation of band-like transport [14].

Films of Pb chalcogenide QDs have been used as the active material in solar cells [6, 7, 9–11, 15], producing a record power conversion efficiency of 5.1% (ref. [11]). Disagreement exists on the mechanism of operation of these solar cells. Several au-thors claim that excitons diffuse to electron or hole extracting interfaces where they dissociate [6, 7], while others argue that electrons and holes are separated by means of a Schottky [9, 10]. In the case of weak coupling between QDs, photogenerated electron-hole pairs will be confined within a single QD and are unlikely to separate into free charge carriers, unless a large electric field is applied to induce charge sepa-ration [16]. However, if the QDs are strongly coupled, photogenerated excitons may dissociate efficiently, similar to the situation in many bulk semiconductors.

Here, we present measurements of the time-resolved photoconductance (using tera-hertz spectroscopy and the time-resolved microwave conductivity technique (TRMC)) as well as the transient interband absorption in high-mobility PbSe QD solids. Both terahertz spectroscopy and the TRMC technique determine the AC photoconductivity of the films without the need for contacting electrodes, thus eliminating complicating issues related to band bending [9], trapping and recombination at the QD-electrode interface [17]. Replacement of the original long ligands with 1,2-ethanediamine results in a sum of the electron and hole mobilities of 3 cm2_V−1_s−1_{. The interband}

absorp-tion and the real and imaginary terahertz conductivity show identical decay kinetics, demonstrating directly that the quantum yield of charge carrier photogeneration is unity, even in the absence of electric fields or concentration gradients. We further show that in these strongly coupled QD solids, the photoconductivity becomes

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in-2.1. INTRODUCTION

dependent of temperature at high photoexcitation densities. This implies that a transition takes place from hopping between localized states to band-like transport. In such an, “Anderson transition”, the wave-functions of electrons and holes become extensively delocalized over the QD solid.

To the best of our knowledge, this is the first time that a unity exciton dissocia-tion efficiency has been reported for QD solids. This is of great importance for the design and further optimization of QD-based solar cells. In these PbSe QD solids, every exciton dissociates into free charge carriers. Internal quantum efficiencies sig-nificantly lower than unity must therefore be caused by inefficient carrier extraction. The use of donor-acceptor geometries to increase the exciton dissociation efficiency is unnecessary. Instead, the generation of more charge carriers requires the inclusion of concepts such as carrier multiplication [18], which is known to occur efficiently in PbSe QDs [19, 20].

a b

c d

Figure 2.1: Characterization of PbSe QD solids. a, Photograph of a PbSe QD solid (brown) on a transparent quartz substrate. The shadow of tree leaves is visible through the smooth and homogeneous film. b, Absorption spectra of a QD dis-persion and a QD solid. c, SEM image at intermediate magnification demonstrating the excellent homogeneity of the film. d, High-resolution TEM image.

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QD solids were prepared using a layer-by-layer (LbL) procedure using 3.9 nm diameter PbSe QDs and 1,2-ethanediamine as the capping molecules (see Chapter 1). This procedure results in glassy [21], smooth films with a typical thickness of 30-40 nm and a roughness of 5 nm. Figure 2.1 shows a photograph of a film produced in this manner, demonstrating its excellent homogeneity. Absorption spectra of a dispersion of PbSe QDs and a QD solid are shown in Figure 2.1b. The first exciton peak of the film is redshifted by 30 meV with respect to that of the QD dispersion. This redshift is typical for QD solids with short interparticle distances, and has been attributed to polarization effects that result from the change in dielectric environment [22]. The full-width at half-maximum of the peak increases from 100 meV to 260 meV. The width of the broadened first exciton transition reflects the electronic coupling [23, 24] and the disorder in site energies. Scanning and transmission electron microscopy images (SEM and TEM, Figure 2.1c and d, respectively) show that the quantum dots do not sinter during the LbL process. They retain their spherical shape and their crystal lattices remain distinct, while the interparticle separation is reduced to several angstroms, in line with the 0.38 nm length of the 1,2-ethanediamine capping molecules.

2.2 Unity quantum yield of photogenerated charges

Figure 2.2 shows transients of the interband optical absorption normalized by the photon fluence in the laser pulse, I0. The sample is excited at 800 nm and probed at

the first exciton resonance. The excitation density is expressed as the average number of absorbed photons per laser pulse per QD, <Nabs>, which is obtained as I0σ, where

σ is the absorption cross-section taken from ref. [25]. At the lowest excitation density, the bleach of the interband absorption is almost constant. As the excitation density is increased, a faster decay component develops. This higher-order decay component is attributed to Auger decay taking place between two excitons or between three charges (two electrons and a hole, or two holes and an electron). We have previously observed similar behaviour for films of CdSe QDs [26, 27]. The long-lived signals at low fluence indicate that first-order recombination processes such as charge trapping or geminate recombination are relatively slow.

The bleach of the interband absorption may result either from excitons or from separated mobile electrons and holes. It can be shown that the total absorption bleach of an exciton is indistinguishable from that of separated charges [28]. For this reason, we also carried out an investigation using terahertz spectroscopy. Excitons do not lead to absorption of terahertz radiation; they only produce a phase delay. This implies that excitons do not contribute to a real terahertz conductivity, as demonstrated by the absence of such a signal for QDs in dispersion, and only result in a change in the

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2.2. UNITY QUANTUM YIELD OF PHOTOGENERATED CHARGES TA ∆T /T ( 10 -3) 520 510 500 -2.0 -1.5 -1.0 -0.5 0.0 TH z ∆ E/ E (1 0 -3 ) 30 20 10 0 8 6 4 2 0 TA THz Real THz Imaginary, scaled b a ∆T /( TI0 ) (10 -18 cm 2) incr easing <N abs >

Figure 2.2: Ultrafast response of QD solids to optical excitation. a, Interband absorp-tion transients of a QD solid at excitaabsorp-tion densities kNabsl of 0.002 (black curve) to 0.3 (red curve) absorbed photons per QD and probed at the first exciton resonance. b, Comparison of interband absorption transients (TA, red curve, left axis) and the real and imaginary terahertz conductivity transients (THz, black and green curves, respectively, right axis) at identical excitation densities of 0.15 absorbed photons per QD.

imaginary terahertz conductivity [29]. Mobile charge carriers on the other hand do absorb terahertz radiation, resulting in a change in the real terahertz conductivity in photoconductive QD solids [30]. In the presence of barriers for charge transport (such as grain boundaries in QD solids), mobile charge carriers also show a change in the imaginary terahertz conductivity. In this case the decay kinetics of the real and imaginary terahertz conductivity are identical for pump-probe delays exceeding the terahertz waveform (∼1 ps) [31], assuming the carrier mobility does not change with time.

Figure 2.2b shows a comparison of the interband absorption transient (red curve) and transients of the real and imaginary terahertz conductivity (black and green curves, respectively), measured on the same sample and at identical excitation densi-ties. From this figure it is clear that the transient interband bleach and the real and imaginary terahertz conductivity have identical decay kinetics except for the first 2 ps. This is true at different pump fluences (see Appendix A.2). If a significant frac-tion of excitons were present upon photoexcitafrac-tion, these would decay with different kinetics. This would arise from a difference in the diffusion coefficients of the excitons and charges, as well as the fact that two excitons can undergo Auger recombination, making this a second-order process, but three charges are required for the Auger pro-cess, making charge Auger decay a third-order process. Second-order and third-order decay processes exhibit different decay kinetics.

Photogeneration Diffusion and Decay of Charge Carriers in Quantum-Dot Solids

AND DECAY OF CHARGE CARRIERS

IN QUANTUM-DOT SOLIDS

OF CHARGE CARRIERS IN QUANTUM-DOT

SOLIDS

Proefschrift

Contents

Chapter 1

Introduction

1.1

Quantum dots

1.1.1

Colloidal quantum dot synthesis

a

c

b

1.1.2

Electronic structure of quantum dots

a

b

1.1.3

Charge carrier decay mechanisms

a

b

a

b

1.2

Quantum-dot solids

1.2.1

Quantum-dot solid assembly

Quartz

PbSe

QDs

CdSe

QDs

a

b

d

c

1.2.2

Exciton dissociation

E’=E +E

a

b

E’=2(E +E )+E +E

=E +E

1.2.3

Charge carrier transport

R

DE

1.2.4

Charge carrier decay kinetics

1.3

Experimental methods

1.3.1

Quantum dot synthesis and quantum-dot solid assembly

1.3.2

Absorption spectra

1.3.3

Transient absorption spectroscopy

1.3.4

Terahertz spectroscopy

1.3.5

The Time-resolved Microwave Conductivity Technique

(TRMC)

1.4

Outline of the thesis

References

Chapter 2

Unity quantum yield of

photogener-ated charges and band-like transport

in quantum-dot solids

2.1

Introduction

2.2

Unity quantum yield of photogenerated charges