Thermoelectric Effects in Nanowire Quantum Dots

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Universit`

a degli Studi di Pisa

DIPARTIMENTO DI FISICA

Corso di Laurea Magistrale in Fisica della Materia

Tesi di Laurea Magistrale

Thermoelectric Effects in Nanowire Quantum Dots

Candidato:

Domenic Prete

Relatori:

Prof. Stefano Roddaro Dott. Francesco Rossella

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Introduction

In the last century an increasingly strong interest has been given to low dimensionality physical systems and to the exploitation of their reduced di-mensions in order to experimentally access the ’quantum’ features of nature. As a matter of fact, in the last decades a plethora of growing complexity tech-niques has been developed in order to realize peculiar systems such as het-erostructured semiconductor nanomaterials, 2D layered materials, nanowires and nanotubes. In these systems, phenomena like integer [1] and fractional [2] quantum Hall effects, quantization of conductance [3] and quantum Coulomb blockade [4] were experimentally observed, allowing a deeper comprehension of nature and the development of more detailed and profound theoretical models.

Our work mainly concerns zero dimensional systems (quantum dots -QDs) obtained by growing InAs/InP heterostructured nanowires in which two thin (∼5 nm) InP barriers isolate a ∼20 nm InAs island. The relative position of conduction bands of these materials induce a confinement (Figure 1) which induces energy spectrum quantization in the nanostructure. Fur-thermore, the confinement induced in this way is much stronger compared with electrostatically-defined QDs (in which the confinement is induced by means of electric fields), allowing us to explore higher and uncharted working temperatures with our systems. The goal of this thesis was the fabrication and characterization of nanowire quantum dots based devices, with an opti-mized design for investigation of thermoelectrics of this kind of systems.

The physics and phenomenology of thermoelectric effects in bulk mate-rials have been known and exploited for decades and have found important applications such as energy harvesting (exploiting the thermovoltage induced

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Figure 1: Transmission electron micrograph of an heterostructured InAs/InP quantum dot. The red line represents the profile of the bottom of the con-duction bands along the heterostructure which defines an InAs island at the center of the structure.

by the Seebeck effect), refrigeration (trough the Peltier effect) and thermom-etry (thermocouples and Seebeck effect can be exploited to measure temper-ature). Nonetheless, achieving a thermodynamically efficient thermoelectric conversion is still very challenging today, since it requires a maximization of the ’figure of merit’ ZT = σS2_T

k and thus to maximize σ (electrical

conduc-tivity) and S (Seebeck coefficient) while minimizing k, which is the thermal conductivity which includes the contribute of both electrons and phonons. This turns out to be hard in general, as these quantities are interconnected: for example, improving σ usually also improves the electron thermal conduc-tivity due to the Wiedeman-Franz law.

The issue of optimizing the figure of merit in thermoelectric materials may be faced shifting from bulk materials to nano-sized materials. As a matter of fact, going to the nanoscale has two main advantages: first, quantum confine-ment on a nanometer scale can be exploited to tailor the electronic density of states and to enhance both the Seebeck coefficient and the conductivity σ; the second advantage is linked to the mean free path: nanoscale systems can, in principle, have a dimension which is smaller than the mean free path of

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phonons while being still larger than the mean free path of electrons; nanos-tructuring can thus reduce the thermal conductivity, which represents a pure efficiency loss in the thermoelectric conversion, without affecting electrical transport [5]. Several studies in the past years have demonstrated that con-finement in low-dimensional nanostructures do have an enhancement effect on thermoelectric figure of merit [6]. In particular, QDs have proved to be a promising platform for the development of new and efficient thermoelectric materials, thanks to their reduced size and sharp density of states. How-ever, experiments on QD thermoelectrics were performed predominantly in the sub-Kelvin temperature regime, due to the weak confinement usually present in typical QDs. Our interest is to fill the gap in such a promising research field by exploring QD thermoelectrics in higher temperature regimes exploiting heterostructured semiconductor nanowires.

The experimental work included both the nanofabrication of the device and the characterization of its properties through low temperature transport measurements. These activities have been performed at NEST (National En-terprise for nanoScience and nanoTechnology) laboratory in Pisa. Moreover, simulations of the system thermoelectric properties were carried out by using models developed in collaboration with the group of Prof. Fabio Taddei in Scuola Normale Superiore of Pisa.

The thesis is structured as follows:

• In chapter 1 main results obtained in experiments on semiconductor nanowires and heterostructured semiconductor nanowires are briefly described.

• In chapter 2 an introduction to the studied physical systems is given: Coulomb blockade and thermoelectric effects in quantum dots are briefly explained. In this chapter we also present numerical calculation for quantities we measured, in order to give an insight on theoretical ex-pectations.

• In chapter 3 details on fabrication techniques used in this work are given; these include nanowire growth by Chemical Beam Epitaxy and the variety of technical steps which are used to produce the samples

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used in the experiment. In the last part of this chapter the experimental setup used is described.

• In chapter 4 measurements and data analysis is presented, along with the interpretation of data.

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Chapter 1 State of the Art

1.1 Semiconductor nanowires

Historically low dimensionality systems were designed and realized fol-lowing a so-called “top-down” paradigm, meaning that a higher-dimensional system gets nanostructured in order to achieve a lower-dimensional one. In particular, one dimensional (1D) or zero dimensional (0D) systems can be easily obtained by a two-dimensional electron gas1 _{(2DEG) mainly through}

two different top-down approaches, namely chemical etching and electrostatic gating.

Through lithographic techniques one can design and create a protective mask on a surface, which then allows to selectively etch material from the higher-dimensional system to create a lower-dimensional architecture. Using this kind of techniques, 0D systems such as vertical quantum dots can be fab-ricated; despite its simplicity, low-dimensional systems fabrication through chemical etching presents some issues linked to etching rates and homogene-ity, giving limits to characteristic length scales: in fact, structures as big as 100 nm can be obtained, giving relatively weak confinement. Also electro-static gating leads to a weak confinement, given by local gates which are obtained by lithographic techniques (typically electron beam lithography) that induce a local confinement potential on a below standing 2DEG.

Opposed to this top-down fabrication logic there are bottom-up

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niques, which basically consists in using self-assembled nanostructures in-stead of creating them by “de-assembling” bigger systems. Clear examples of bottom-up systems are semiconductor nanowires, a very versatile physical system that allows to observe interesting phenomena in almost every research branch.

The first step towards the use of self-assembled nanostructures has been realized by Wagner et al., who demonstrated that crystal growth was possible through the so-called ”Vapor-Liquid-Solid deposition technique” [7], in which metal nanoparticles were used as ”seeds” for crystal growth, as schematically shown in figure 1.1.

Figure 1.1: Vapor Liquid Solid deposition conceptual scheme. Blue arrows indicate the flow of particles promoting the self-assembling of the nanostruc-ture.

Starting from this concept, various other deposition and epitaxy tech-niques were developed, allowing easy an access to structures such as nanowires and superlattices ([8], [9]). In particular, the technique used to grow the het-erostructured nanowires used in this work, developed by Tsang et al., is the so-called chemical beam epitaxy (CBE [10]); details on this particular tech-nique, however, will be given later in the chapter dedicated to experimental methods.

Using these novel technologies, a new class of 1D system was made ac-cessible to investigation: with respect to previously developed 1D architec-tures (obtained by local gating of 2DEGs), semiconductor nanowires have the feature to physically constrain transport in one dimension, allowing to have stronger and more effective confinement. A huge effort was made in

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order to investigate transport in semiconductor nanowires, having allowed to experimentally observe exciting phenomena like ballistic transport [12] and thermoelectric effects in 1D semiconductors [13].

1.2 Quantum dots

As anticipated in the previous section, nanoscaled devices (including quantum dot based nanostructures) were first devoloped following the top-down logic, thus ”carving” them from bigger structures and obtaining to lower dimensionality systems. An example of quantum dot (QD) obtained with this logic is a micropillar-embedded QD. Thanks to this kind of tech-nology, it was possible for the first time to obtain an artificial atom (as effectively reviewed in [14]). Interestingly, using lithographic techniques to give micropillars different geometrical shapes one can reproduce electronic orbital wavefunctions symmetries.

Moreover, it was also possible to study Coulomb blockade in these nanos-tructures (Figure 1.2) by totally depleting QDs and investigating trans-port[15].

Figure 1.2: Coulomb Blockade peaks, image from [14]. Coulomb blockade regime is clearly visible between current peaks, and the inset show the depen-dence of the energy required in order to add a new electron in the dot with respect to the number of electron already present in the QD.

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years not only for Coulomb blockade measurement, but mostly because this kind of systems has a variety of possible applications:

• First, quantum dots can serve as spin filters and spin sensors. Inter-esting research in this field was done, for example studying spin filling in quantum dots [16] and spin manipulation in semicondutor nanowire quantum dots [17]. Also, it turned out that quantum dot physical dimensions (and thus the strength of the confinement) had a strong influence on electrons’ effective g-factor [18]. This naturally brings quantum dot systems to be highly of interest in quantum computation, as single electron spin manipulation becomes possible [19][20].

• The discreteness of energy levels in quantum dots also allows to exploit optical properties of such systems for photonics and photon quantum information. In particular, quantum dots were used as entangled and coherent single-photon sources[21],[22].

• Finally, sharp density of states features make quantum dot systems appealing for thermoelectric applications. Being this the frame in which this work is inserted, we find more appropriate to dedicate the next section to nanoscale thermoelectrics.

1.3 Thermoelectrics in nano-sized systems

A huge boost to the interest in thermoelectric effects in nanostructures has been given by the farseeing intuition of some authors in the early 90s looking for materials with high thermoelectric figure of merit (ZT = σS2_T

k ).

As already explained in the introduction, in bulk materials this quantity is quite hard to optimize, being S,k and σ physically linked one to another. Things can change in lower dimensionality systems: it was, as a matter of fact, demonstrated that highly improved thermoelectric figure of merit can be obtained in layered 2D materials [6], 1D conductors and quantum wires [23]. A more general calculation was made by Mahan and Sofo [24], who derived electrical conductivity, thermopower and thermal conductivity as functions of the system density of states distribution. They found that a

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delta-shaped density of states distribution gives the highest theoretical figure of merit. Thus the use of nanostructures, in which electronic density of states can be engineered easily and effectively, in thermoelectrics has become the most appealing path to follow; moreover, the narrowest electronic density of states that can be obtained is achieved using quantum dots, where the presence of energy levels (due to heavy confinement) leads (theoretically) to a delta-shaped density of states.

In this conceptual frame a huge variety of researches live: thermopower in a quantum dot subject to a temperature bias ∆T between source and drain contacts has been theoretically [25] and experimentally [26] studied in the linear regime (i.e. with kb∆T eVbias,∆); nonlinear regime has also been

studied [27], and an interesting inversion in the sign of the thermocurrent has been observed. Exploring the physical nature of such inversion brought to interesting theories which involve interaction between electrons and the surrounding environment, pointing out how exploring nanoscale thermoelec-tric effects has important consequences in exploring fundamental condensed matter of physics phenomena. As a matter of fact, studying this kind of phe-nomena brought to the investigation of disordered systems, in particular on what effect disorder has on the thermopower [28][29]; furthermore, phonon drag [30] and correlation between heat and charge current fluctuation has been explored [31].

Nanoscale thermoelectrics does not simply have relevant impact on fun-damental research, but also has heavy technological implications: recently the effect of heat currents on superconducting qubits has been investigated [32]. Furthermore, the role of quantum dots in the field of nano-sized ther-moelectrics is fundamental [33]: this kind of systems, as a matter of fact, can be naturally used, thanks to their physical nature, as ultra-sensitive ther-mometers [34], low-temperature nanorefrigerators [35] and high efficiency heat engines [36][37].

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Chapter 2 Theoretical Models and

Concepts

The scope of this section is to give the reader a brief introduction on theoretical models used to interpret the experimental data. We will face the general problem describing Coulomb blockade and thermoelectrics in quantum dots. The objective is to give the reader an introduction on the theory of transport and thermoelectrics in quantum dot systems work.

2.1 Transport in quantum dots: the constant

interaction model

In general, the quantum-mechanical problem to solve for a quantum dot is complex and takes into account interaction between electrons, which in-troduces many-body components in the hamiltonian of the system making it difficult to solve explicitly; here we will describe an approximation (the capac-itance model) which severely simplifies the problem, approximating electron-electron interaction modeling the dot as a metallic island with discrete energy spectrum (due to confinement effects), and assuming capacitative coupling with gates1_.

1_{Here with the term gate we are generally referring to every element of the system} apart form the dot.

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The interaction part of the hamiltonian of the system (i.e. interaction between electrons on the dot and gate potentials) are taken into account by a capacitance matrix C. In this way, charge on the metallic objects is related to gate voltages with the following formula2_:

Qi = n

X

j=0

CijVj (2.1)

Where the index i(j)=1,2,...,n goes over the number n of gates. The index i=0 is taken as reference for the dot itself; furthermore, we are assuming that all capacitances do not depend on the number of charges present (constant interaction approximation).

Being interested on dot’s electrostatic energy, we need to compute the potential V0 of the metallic island, which is in principle not known. What

is known, nonetheless, is that charge on the dot is quantized, i.e. it is an integer multiple of the elementary charge. We can therefore write

V0(Q0) = Q0 CΣ − n X j=1 C0j CΣ Vj, (2.2) where CΣ = C00.

Now we can easily obtain the electrostatic energy of the dot, Ees, i.e. the

energy needed to add N electrons to the island Ees(N ) = Z −|e|N 0 dQ0V0(Q0) = e2_N2 2CΣ + |e|N n X j=1 C0j CΣ Vj. (2.3)

We can at this point write the total energy for the dot, assuming to know levels of the discrete energy spectrum of the the island3

E(N ) = N X n=1 n+ e2_N2 2CΣ + |e|N n X i=1 C0i CΣ Vi. (2.4)

2_{We are assuming that in the case V}

j= 0 for every j there is no residual charge on the object

3_{Which are obtained by solving the problem for the single particle part of the starting} hamiltonian

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It is now possible to write down the expression for a fundamental quantity, representing the energy needed to add an electron on the island, i.e. the chemical potential µN = E(N ) − E(N − 1) = n+ e2 CΣ (N − 1/2) + |e| n X i=1 C0i CΣ Vi. (2.5)

It is important to notice that this value depends on the number of electrons present in the dot; for this reason chemical potential in quantum dots has quantized values, and transport is mediated by means of single resonant levels and not by a complex multi-level spectrum.

In the formula for µN we can identify every term with a contribution

accounting for different effects:

• n is the energy of the quantum level the electron has to occupy;

• e2

CΣ(N − 1/2) is the energy needed to overcome electrical repulsion

be-tween the Nth _{charge and the other (N − 1) charges already present on}

the dot. The quantity Ec = e

2

CΣ is conventionally called the charging

energy; • |e|Pn

i=1 C0i

CΣVi is the energy cost given by the electrostatic potential

imposed by device gates. The quantity −C0i

CΣ = αi is conventionally

called the lever arm of the ith _{gate, and quantifies the gate coupling;}

This picture we have given is strikingly simple compared to the complex initial problem we had to solve; nonetheless, it allows to achieve a basic but effective comprehension on transport in quantum dot systems. There are, anyway, some experimental features we will not be able to explain with this model and for which we will have to give further information:

• Having assumed simple capacitative coupling between dot and gates, we are not taking into account tunneling barriers through which elec-trons flow; this has the drawback to now allow us to quantify the width of conductance peaks, which can be explained considering resonant tun-neling electronic transport. Why conductance peaks occur is explained in section 2.1.2;

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• The constant interaction approximation we worked with surely holds when N is not small; anyhow we often find ourselves to work in com-pletely depleted systems, where this approximation does not hold. This has effects which will be described in section 2.1.3, where charge sta-bility diagrams for quantum dots are explained;

• We have considered leads (source and drain) as electron reservoirs with zero temperature, with a fixed chemical potential µS(D). In general,

and most importantly in order to study thermoelectric effects, we have to consider Fermi-Dirac distributions for electrons living in the leads and the effect of temperature on these distributions. We will give more information about this topic in section 2.2.4;

Having described the basis of the capacitance model and having computed the fundamental quantity µN, it is now possible to describe transport through

quantum dots.

We will consider a simplified system, pictorially represented in Figure 2.1:

Figure 2.1: Scheme for the model we are using; the dot is assumed to be a metallic island with discrete energy spectrum.

In this 3 terminal configuration two terminals (source and drain) serve as electron reservoirs used for providing charges to the dot. They are assumed to

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be coupled to the dot via tunneling junctions, which provide both a tunneling resistance and a capacitance (taken into account in the model we are using). The last (gate) is capacitively coupled to the dot and is used to manipulate its discrete spectrum.

2.1.1 Current-voltage relation in the Coulomb

block-ade regime at a fixed V

gate

configuration

In the considered configuration, suppose we fix Vgate while varying Vbias

from negative to positive values. Energy levels are fixed and depend only

Figure 2.2: Transport through a Quantum Dot with fixed Vgate, with (a)

neg-ative, (b) zero and (c) positive Vbias.

on the number of electrons in the island and on the (fixed) gate potential, while chemical potential in source and drain is shifted by Vbias. In 2.2(a) the

dot is populated by N electrons, but µN +1 ∼ µD so that electrons can flow

from drain to source passing through the (N + 1)th _{level, allowing transport.}

In 2.2(c) a similar reasoning works, except for the inversion of sign in Vbias

and thus of the direction of electrons. What happens in 2.2(b) is different. The Nth _{level is occupied, so that no electrons can flow through that level,}

but electrons in the lead do not have sufficient energy to allow transport through the (N + 1)th _{level: the system is the so-called Coulomb blockade}

regime, conduction is suppressed as no electrons can enter the dot because of the combined effect of Coulomb repulsion and energy spectrum quantiza-tion (both inducing a spectrum of quantized values for accessible chemical potential levels).

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How this regime reflects in the I − Vbias curve is quite intuitive: there is a

plateau (i.e. no current variation) in the blocked zone, i.e. in the Vbias values

interval (bias window) for which there is no unoccupied level accessible to mediate transport through the QD. The dimension of the bias window allows us to obtain information on QD energetics, as it is connected to the energy distance between two consecutive resonant levels. Figure 2.3 shows a pictorial example of a I − Vbias curve in which both Coulomb blockade and normal

conduction regimes are accessed.

Figure 2.3: Scheme for current flowing through the QD in the examined case. Letters are related to energy schemes represented in figure 2.2

2.1.2 Coulomb blockade: fixed V

bias

Another important characteristic curve which is frequently considered when studying quantum dot systems is the transconductance, i.e. I − Vgate

curve with fixed Vbias4. In this case the chemical potential of the source and

drain leads is fixed, while potential applied by the capacitively-coupled gate shifts the position of the resonant level of the dot. Depending on the value of this potential the resonant level can be moved in and out the bias window, thus allowing or blocking transport through the dot. Figure 2.4 provides a visual explanation of the phenomenon.

4_{We will assume for simplicity that V}

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Figure 2.4: Transport through a quantum dot with fixed Vbias. The Coulomb

Blockade regime is represented in panel (b)

In Figure 2.4 ϕgateis the Gate’s electrostatic potential, and thus −|e|ϕgate

represents the energy shift imposed to dot levels.

Transport in the configuration we are considering is described as follows: • Starting from µS ∼ µN ∼ µD (as in 2.4(a)) transport is allowed as

shown. There is a net current flowing from source to drain leads; • Increasing ϕgate, one arrives at the configuration where no levels are

included in the bias window: there cannot be any electrons traveling from source to drain contacts, being the Nth _{level occupied and the}

(N +1)th_{not accessible. The system is in the so called Coulomb blockade}

regime (fig. 2.4(b));

• Further increasing ϕgatekeeps the system in a blocked state, until µS ∼

µN +1∼ µD and transport gets allowed again (fig. 2.4(c));

Using equation 2.5, assuming negligible Vsource and Vdrain and that a

conduc-tance peak is expected when µN +1 = µS = µD, we can write gate voltage

value for which a conductance peak occurs: Vg(N + 1) = 1 eαg N +1+ e2 CΣ N − |e|X i αiVi− µS ! . (2.6)

If distance between peaks has to be computed: ∆Vgate = Vg(N + 1) − Vg(N ) = 1 eαg ∆ + e 2 CΣ , (2.7)

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where ∆ = N +1− N is the difference between quantum levels and e

2

CΣ is

the charging energy.

It is important to notice that Coulomb blockade is first of all a classical phenomenon which can be realized even if the is no discreteness in the island’s energy spectrum, as the electrochemical potential gets discretized because of electron-electron repulsion. Calculations performed in chapter 2.1 are valid also for the classical Coulomb blockade phenomenon, with the only differ-ence of the absdiffer-ence of the term n in the expression for the electrochemical

potential. Figure 2.5 provides examples of Coulomb blockade measurements both in the classical and quantum regime.

Figure 2.5: Example of current peaks in a dot entering and exiting Coulomb Blockade regime. 2.5.a is an example of classical Coulomb blockade, while 2.5. is quantum Coulomb blockade conductance peaks (taken from [15]). The difference between the two systems is just the strength of confinement in the dot (the second being more confined).

What discerns quantum Coulomb blockade from classical Coulomb block-ade is the presence of a discretized energy spectrum in the dot. This explains why in classical coulomb blockade all peaks are equally spaced, while this does not happen in the quantum regime. That is, the charging energy depends on dimensional and material system properties (contributing to its total capac-itance) and it is constant, while energy levels are determined by confinement and level spacing is not, in general, constant. Furthermore, in the quantum case one usually observes level spin degeneracy (as in figure 2.5(b)), so that one has to “pay” ∆ + e2

CΣ and

e2

CΣ alternatively in order to introduce a new

electron in the dot.

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blockade is and on why this phenomenon occurs; however they don’t explain the lineshape of conductance is what is shown in Figure 2.5. Instead, it would seem that conductance should set itself as a series of delta functions, centered at the energies of resonant levels.

Using a more formal approach and studying resonant tunneling through the barriers separating the dot from the source and drain leads, it is possi-ble to derive that the transmission probability of electrons through a single quantum dot is a lorentzian centered at the resonant level’s energy and hav-ing a width Γ = Γl+ Γr, where Γland Γr represent the left and right barriers

tunneling rates. We can write the current using the Landauer-B¨uttiker model I = 2e

h Z

[fL(E, µl, T) − fR(E, µr, T)] τ (E − Er,Γ)dE, (2.8)

where fL and fR are the left and right lead Fermi distributions depending

on energy E, lead chemical potential µl(r) and temperature T , τ represents

the transmission probability and Er is the resonant level energy. In the

approximation of small applied bias voltage and hΓ kBT, we obtain

G= I Vbias = e 2 h ΓlΓr Γl+ Γr 1 4kBT cosh2(µ_2kD_B−µ_Tl) , (2.9)

where µD is the electrochemical potential of the dot. A peak in the

conduc-tance if found whenever the electrochemical potential of the dot is aligned with the electrochemical potential of the leads; the peak is thermally broad-ened and decays exponentially when getting further from the maximum.

2.1.3 Coulomb diamonds

Apart from considering current with fixed Vbias and Vgate, one can

ex-plore the whole Vbias− Vgate space identifying configurations in which there

is conduction through the dot or Coulomb blockade: for every Vgate value,

as a matter of fact, levels in the dot are shifted and one has to change Vbias

(i.e. chemical potential in the leads) accordingly in order to keep the system conducting (or blocking). These Vbias−Vgateplots are often referred as charge

stability diagrams, as one can see in a glance what is the state of the system knowing applied voltages.

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From now on we assume that a voltage −f Vbias is applied to the drain

and (1 − f )Vbias to the source, where 0 ≤ f ≤ 1 is introduced for generality:

if f = 0, drain contact is considered as the reference potential for the circuit and all bias voltage is considered applied to the source lead; for f = 1/2 a symmetric bias is applied (−Vbias/2 on drain and Vbias/2 on source). As

both these configurations happen to be used in general in experiments, it is convenient to be general on whatever kind of bias is applied (symmetric or asymmetric). Furthermore, we need a reference for Vgate, namely Vgate(0), for

which we have5_:

µN(Vgate(0)) = N +

e2

CΣ

(N − 1/2) − eαgVgate(0) = µS/D = 0.

It should be clear from previous sections that conditions for transport in a quantum dot to be allowed correspond to chemical potentials alignment; if one focuses on the transition from the N-electrons stable state to a (N+1)-electrons there are basically 4 equations to study, namely:

µN = µS,

µN = µD,

µN +1 = µS,

µN +1= µD.

Leading to the following equations relating Vgate and Vbias:

Vgate = − 1 αg [αs(1 − f ) − αdf + (1 − f )] Vbias, (2.10) Vgate = − 1 αg [αs(1 − f ) − αdf + f )] Vbias, (2.11) Vgate= 1 eαg ∆ + e 2 CΣ − 1 αg [αs(1 − f ) − αdf + (1 − f )] Vbias, (2.12) Vgate = 1 eαg ∆ + e 2 CΣ − 1 αg [αs(1 − f ) − αdf − f)] Vbias. (2.13)

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Plotting these lines in a Vgate − Vbias graph, the typical “Coulomb

dia-mond” structure is obtained, as represented in Figure 2.6. Here, the gray diamonds represent stable charge states. In this picture single electrons tun-neling through the dot generate current: putting the system where the red dot is in figure 2.6 the state with N charges will be stable and thus coulomb blockade regime holds; shifting Vgate in order to get to the green dot will

get the system conducting, and the number of electrons in the dot will keep oscillating from N to N+1, thus giving transport.

Figure 2.6: Ideal charge stability diagram for a dot.

Referring to equations 2.10-2.13, we can notice that 2.10 and 2.12 ex-pressions, as well as 2.11 and 2.13 exex-pressions, represent lines with the same slope. Furthermore, we can notice that:

|∆m| = |m1− m2| =

1 αg

, (2.14)

so that we can retrieve a fundamental parameter for our problem, i.e. gate’s lever arm, by knowing coulomb blockade diamonds boundary lines slopes. This will allow us to convert experimental data (i.e. gate voltages) to energy values.

We have previously outlined that the failure of the constant interaction approximation is visible thanks charge stability diagrams; indeed, if we allow

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capacitances (and thus lever arms αi) to depend on the number of charges

living in the dot and considering equations 2.10 to 2.13 we immediately see that dimensions of coulomb blockade diamonds can severely depend on how many electrons are on the island. This, however, does not have a negative effect on our physical insight of the system, but makes data understanding more complex.

2.2 Thermoelectrics

In this section our goal is to briefly explain thermoelectric phenomena, first introducing a general approach used in bulk materials. We then pro-ceed in treating thermoelectrics in nano-scaled semiconductors, focusing on quantum dots.

2.2.1 Thermoelectric effects in bulk materials

2.2.2 Boltzmann transport equation

Considering a single isolated conduction band E(k) in thermodynamic equilibrium, state occupation probability is given by the Fermi-Dirac distri-bution: f0(k) = 1 e E(k)−µ kB T _{+ 1} . (2.15)

With the application of external electric fields or temperature gradients we have f0(k) = f0(k, r, t). An electron living in this conduction band evolves

in the phase space from (r, k) at time t to (r + vkdt,k +F_¯_hdt) at time t + dt,

where vk = 1_¯_h∂E(k)_∂k is the electron band velocity and F = d¯_dthk is an external

force applied on the electron. Remembering that f (k, r, t)drdk

4π3 is the number of electrons which occupy

the volume drdk around the point (r, k) in the phase space at time t, we want to use Liouville’s theorem, for which the volume in the phase space should not change for t → t + dt, we have:

f(r + vkdt,k +

Fdt

¯h , t+ dt) = f (r, k, t) + ∂f

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where we have assumed that in the motion of the electron in the band collision processes (due interaction with lattice vibrations or impurities) can cause a change ∂f

∂t|coll of the number of electrons in the volume drdk.

Expanding 2.16 in Taylor series: ∂f ∂rv + ∂f ∂k F ¯h + ∂f ∂t = ∂f ∂t|colldt. (2.17)

If the deviation of f from f0 because of collisions is small, we can write ∂f

∂t|colldt = − f −f0

τ , where τ is the so-called relaxation time of the electron

because of collisions6_.

In the case of a system in which at t = 0 F = 0,∂f_∂r = 0, the solution of equation 2.17 is:

f(k, t) = f0+ [f (k, t = 0) − f0]e−

t

τ. (2.18)

Knowing this, we can obtain non-equilibrium occupation only by knowing the equilibrium distribution function f0 and the relaxation time τ

2.2.3 Using Boltzmann equation for thermoelectrics

Using results we have just obtained, we can now define important quan-tities needed to study transport, namely electron current density:

J = 2

(2π)3

Z

−evkf dk (2.19)

and energy flux density:

U = 2

(2π)3

Z

Ekvkf dk. (2.20)

Furthermore, we need to remember that, in general: f0(k, r) = 1 e E(k)−µ(r) kB T (r) _{+ 1} , (2.21) where µ(r) = µ(T (r), n(r)), so that: ∂f0 ∂r = ∂f0 ∂E −E T ∂T ∂r − T ∂(_Tµ) ∂r , (2.22)

6_τ _{represents the time that passes between two collisions for one electron. This} ap-proximation is called relaxation time apap-proximation

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∂f0

∂k = ∂f0

∂E¯hv. (2.23)

Introducing an electric field as source for the external force F, we have for Boltzmann transport equation 2.18:

∂f ∂rv + 1 ¯h ∂f ∂k(−eE) = − f − f0 τ = − f1 τ . (2.24)

Now, saying that E and ∂T

∂r are small we can put f = f0 in the first member

of 2.24. Using this and substituting 2.22 and 2.23 in 2.24, we obtain: f1 = − ∂f0 ∂Eτ h −eE − T ∇µ T i · v −∂f0 ∂Eτ(−E) ∇T T · v. (2.25)

Finally, using this function in equations 2.19 and 2.20 for current densities: J = eK0 h eE + T ∇µ T i + eK1 ∇T T , (2.26) U = −K1 h eE + T ∇µ T i − K2 ∇T T , (2.27) where Kn= 1 4π3 Z τ(e · v)2_En −∂f0 ∂E dk = = 1 12π3 Z −∂f0 ∂EE n_dE Z E=const τ v2 |∇kE(k)| dS, (2.28) where n = 0, 1, 2, e = E

|E| is the versor pointing in the direction of the electric

field and |∇kE(k)| = ¯hv.

It is now convenient to introduce the generalized conductivity, noting that J = σE and using equation 2.26:

σ(E) = e 2 12π3_¯h Z E=const. τ vdS, (2.29)

obviously σ(EF) = σ0 is the standard conductivity. With this new quantity

equation 2.28 takes the form: e2Kn= Z −∂f0 ∂E Enσ(E)dE. (2.30)

We can use Sommerfield expansion for the second member of the equation: Z ∞ 0 −∂f0 ∂E g(E)dE = g(µ) + π 2 6 k 2 BT 2 d2g dE2 E=µ + O(T4_), _(2.31)

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so that we have, for transport coefficients Kn: e2K0 = σ(µ) + π2 6 (kBT) 2_σ00 (µ), (2.32) e2K1 = µσ(µ) + π2 6 (kBT) 2_[2σ0 (µ) + µσ00(µ)] , (2.33) e2K2 = µ2σ(µ) + π2 6 (kBT) 2_{2σ(µ) + 4µσ}0_{(µ) + µ}2_σ00_{(µ) .} _(2.34)

Where the derivatives are calculated at the Fermi energy µ.

It is finally possible to write down expressions for current densities J and U so that it will be easier to interpret thermoelectric phenomena. We thus manipulate equations 2.26 and 2.27:

J = e2K0 E + 1 e∇µ − S(T )∇T , (2.35) U = K1 (−e)K0 J − ke∇T, (2.36)

where we have introduced the Seebeck Coefficient S(T ) = π 2 3 k2_BT (−e) σ0(µ) σ(µ), (2.37)

and the electron thermal conductivity: ke= π2 3 k2 B e2T σ0. (2.38)

Here we can see that the ratio between thermal conductivity and electrical conductivity is linear in T7 _{(Wiedemann-Franz Law):}

ke σ0 = π 2_k2 B 3e2 T = LT (2.39)

It is immediate to notice that equation 2.35 basically consists in three terms: • e2_K

0E ∼ σ0E is a drift term, thus the ”standard” electrical transport

term given by conductivity;

7_{This relation has been already qualitatively introduced when talking about the} diffi-culties in optimization of the thermoelectric figure of merit ZT

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• eK0∇µ is a term given by inhomogeneity of chemical potential (r

de-pendence) in equation 2.21; • e2_K

0S(T )∇T is the term given by the presence of a temperature

gra-dient.

The dissipated per unit time and unit volume E·J contains an always positive term J2_/σ

0, which is irreversible heat dissipated, and two terms linear in J,

which represent reversible heat as they can be both positive or negative terms. We can now use expressions 2.35 and 2.36 to understand several phenom-ena involving thermal transport:

Seebeck effect

Consider a conductor in open circuit configuration (i.e. J = 0) subject to a uniform temperatur gradient ∇T (Figure 2.7).

Figure 2.7: Configuration for the observation of the Seebeck effect: a bulk material sample is subject to a uniform temperature gradient in open circuit condition.

In order to mantain J = 0 an electric field E is naturally generated, and thus a thermovoltage is measured between sample’s extremities A and B only as a consequence of a difference of temperature. E can obtained using

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equation 2.35: J = e2K0 E +1 e∇µ − S(T )∇T = 0 =⇒ E = −1 e∇µ + S(T )∇T. (2.40) Thomson effect

We now want to consider how heat is reversibly released or absorbed in an homogeneous material in which an electric current flows, which is commonly called Thomson effect. Changing the sign of the electrical current has the effect to reverse the sign of this effect: this is the main difference with the well known joule heating, which is exclusively a dissipative phenomenon.

In order to study this effect, we consider a simple case of a cylinder with length dl and section dΣ in which J solely depends on the x direction, and which has extremities A and B at temperatures TA and TB. We will

furthermore consider that temperature is kept constantly equal to TA if x ≤

xA and equal to TB for x ≥ xB:

Figure 2.8: Configuration for the observation of the Thomson effect: a bulk material subject to a current flow releases or absorbs reversibly heat.

Considering that ∇T = 0 at points A and B (as T = TA(B) for x ≤ (≥

)xA(B)), we can use equation 2.36 and write:

UA = − K1(TA) eK0(TA) J UB = − K1(TB) eK0(TB) J. (2.41)

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Writing the heat generated in time dt in the considered sample:

δQ= dU + δL (2.42)

Where dU is the energy given by unbalance between energy entering and exiting the cylinder, while δL is the work performed by the electric field.

Using equation 2.41 we have: dU dtdΣ = UA− UB = − K1(TA) eK0(TA) − K1(TB) eK0(TB) J, (2.43)

while using equation 2.35: δL dtdΣ = Z B A J · Edl = Z B A J 1 σ0 J − 1 e∇µ + S(T )∇T · dl = 1 σ0 J2dl −J e(µB− µA) + J Z TB TA S(T )dT. (2.44)

Now, putting together equations 2.42,2.43 and 2.448_:

δQ dtdΣ = [TASA(T ) − TBSB(T )] J + J Z TB TA S(T )dT = −J Z TB TA T dS(T ) = −J Z TB TA TdS dTdT. (2.45)

We thus have demonstrated that reversible heat is produced in a sample subject to an electric field and a temperature gradient.

Peltier effect

Heat can be reversibly generated not only in presence of a temperature gradient with the presence of an electric field (and thus a current density J), but also in a condition with ∇T = 0, given that we are considering not a single sample, but a junction. In fact,if we consider a junction with two different conductors (and thus two different Seebeck coefficients SA and SB),

we can use equation 2.45 to write: δQ

dtdΣ = −JT [SB(T ) − SA(T )] = ΠABJ (2.46)

8_{We also don’t consider the Joule heating term} J2dl

σ0 , which is totally ineffective for

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Where

Π(T ) = T S(T ) (2.47)

Is the so-called Peltier Coefficient. Obtaining reversible heat by current flow-ing, which is somehow the opposite phenomenon with respect to the already treated Seebeck effect, is called Peltier effect, and has several important ap-plications in refrigeration systems.

2.2.4 Thermoelectric effects in quantum dots

We now want to show how thermoelectric signals, e.g. thermovoltage or thermocurrent, change when the system investigated is nano-sized. In par-ticular, we will focus on thermoelectrics in quantum dot systems.

Figure 2.9: Fermi-Dirac distribution smearing caused by temperature (curves at 1◦K and 10◦K are compared).

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transport measurements, given that we consider that being in a finite tem-perature regime smears Fermi-Dirac distributions in the leads (Figure 2.9).

Thus, if a ∆T between source and drain contacts is established, a ther-mocurrent naturally flows through the quantum dot even if no bias voltage is applied, as shown in figure 2.10.

Figure 2.10: Effect of Fermi distribution smearing in inducing a thermocur-rent flow.

Figure 2.11: Thermovoltage generated by Fermi distribution smearing in open circuit configuration.

Moreover, the shift of distributions’ Fermi level given by temperature smearing causes a potential difference Vth to rise between source and drain

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contacts naturally if an open circuit configuration is realized. This thermo-voltage is generated as in open circuit configuration the condition J = 0 has to hold and, in order to deny current flowing from source to drain, the cold lead’s Fermi level is energetically shifted by the quantity eVth, as shown in

figure 2.11.

Even if this pictorial model gives an understanding of how transport is described with the application of a thermal bias, a more formal way to face thermal transport in quantum dots is given by using Landauer-B¨uttiker for-malism[38][39]. In this frame we write the current flowing through the dot:

I = 2e h

Z ∞

−∞

τ() [fL(, µL, TL) − fR(, µR, TR)] d (2.48)

Where µL(R) represents chemical potential in leads, TR = TL+ ∆T is the

right lead’s electronic temperature, which is higher than left lead’s electronic temperature. τ () is the transmission function through the quantum dot, which is ideally a delta-function τ () = δ( − µ), but more realistically can be approximated with a lorentian-shaped function

τ() = ΓlΓr Γl+ Γr

(Γ/2)2

( − µ)2_{+ (Γ/2)}2, (2.49)

where µ is the energy of the resonant level involved in transport and

Γ = Γl+ Γr (2.50)

is the width of the lorentian, related to the tunneling rates Γl and Γr of the

left and right InP barriers.

Assuming that ∆T << T and |eVbias| << µ, µ being the equilibrium

chemical potential, we can write:

f(, µL,R, TL,R) ∼ f0(, µ, T ) ± (µ − µL,R) ∂f0 ∂ ∓ TL,R− T T ∂f0 ∂ (2.51) Where f0 is the equilibrium Fermi distribution for ∆T = 0 and Vbias = 0.

If we compute the formula for the thermocurrent 2.48 within this ap-proximation we obtain a typical oscillating behavior with respect to resonant levels energies (Figure 2.12).

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Figure 2.12: Computed thermocurrent on two resonant levels. For this sim-ulation environment temperature has been set to 4◦K, ∆T = 1.5◦_{K, µ} _{= 0}

and Ec= 3 eV

Apart from the analytical formula, there also is a more qualitative way to understand this oscillating behavior. Referring to figure 2.10 (right-side), the application of a thermal bias produces a positive thermocurrent as transport is enabled by Fermi distribution smearing in the left lead. If one increases gate voltage (i.e. ”shifts” shown level’s energy down) thermocurrent will diminish while getting close to right Fermi level and then will reverse its sign, because temperature smearing of the Fermi distribution did shift its Fermi level higher in energy, but also had the side-effect to diminish occupation probability at lower energies. In this way, electrons in the right side actually see empty states in the left distribution and will flow from right lo left, thus giving a negative thermocurrent.

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equation 2.48 to 0, and remember that S := Vth ∆T: S = Vth ∆T = − 1 e(T + ∆T /2) R ( − µ)∂f0 ∂τ()d R ∂f0 ∂τ()d (2.52) Plotting the expression for Seebeck coefficient we have just derived, consid-ering the existence of two levels separated by the charging energy Ec which

we already discussed, we obtain the typical sawtooth shaped plot, shown in Figure 2.13.

Figure 2.13: Computed Seebeck coefficient sawtooth behavior taking into ac-count two resonant levels. For this simulation environment temperature has been set to 4◦K and ∆T = 1.5◦_{K, while µ}_{= 0 and E}

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Chapter 3 Experimental Methods

In this chapter we describe techniques used in order to fabricate devices used in our experiments; we will briefly explain how InAs/InP heterostruc-tured nanowires are grown by Chemical Beam Epitaxy techniques1_{. We}

furthermore will introduce uv lithographic processes used for chip fabrica-tion and electron beam lithography techniques used for device fabricafabrica-tion. Finally, we will give a brief description of measurement instruments. All manufacturing steps of our devices have been carried out in clean room fa-cilities at the NEST in Pisa.

3.1 Nanowire growth

The principle of Au nanoparticle-catalyzed deposition technique is rep-resented in Figure 3.1. The starting point for nanowires (NW) growth is an InAs substrate on which a thin Au film is evaporated; after this, by thermal annealing, atoms from substrate can diffuse in the gold film, forming liquid InAs/Au alloys which distribute on the substrate as separated nanoparticles (as sketched in Figure 3.1 panels (a) and(b)). When semiconductor precur-sors are injected in the chamber, they get absorbed by these nanoparticles until they reach a supersaturated regime; in these conditions, in order to

1_{Nanowire were grown by Lucia Sorba’s group at NEST; Chemical Beam Epitaxy} growing technique will be briefly described in this chapter, even if details about NW growth go beyond the scope of this work

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restore equilibrium, the precursors tend to crystallize in the solid-liquid in-terface between the nanoparticle and the substrate, thus growing a single NW layer just under the Au nanoparticle (Figure 3.1 panels (c) and (d)). In this way, we can build our NWs one layer at a time. Growth stops when precursors are not injected in the chamber anymore. Moreover, it is possible to grow different materials in the same NW by switching between different precursors during growth.

Figure 3.1: Sketch of the nanoparticle-assisted nanowire growth process. As we have already pointed out in chapter 1.1, nanowires used in this the-sis are grown using Chemical Beam Epitaxy (CBE) technique, which has been identified as the most efficient epitaxial growing technique for III-V semicon-ductors compared to previously developed techniques such as MOCVD and MBE2_{. A schematic picture of a CBE chamber is proposed in figure 3.2.}

Semiconductor precursor compounds used in the process are collected in bottles with individual heaters before being injected in the chamber. A preliminary step performed is to stabilize gas lines from bottles to the cham-ber by regulating their pressure through dedicated needle valves. Every gas line can be opened or interrupted using shutters and regulated through flow meters. The group III precursors, trimethylindium (TMIn), triethylgallium

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Figure 3.2: Scheme of a typical Chemical Beam Epitaxy system; image taken by [10].

(TEGa) and trimethylaluminium (TMAl), are inserted into the chamber by using low temperature (50◦C) line injectors and they are pyrolyzed at the heated substrate. Group V precursors, tertiarybutylarsine (TBAs),

tertiary-butylphospine (TBP), trimethylantimony (TMSb) and tris(dimethylamino)antimony (TDMASb), are injected by high temperatures (1000◦_{C) injectors.}

Figure 3.3 shows a tilted Scanning Electron micrograph which allows to appreciate the shape and dimensions of CBE-grown InAs nanowires. Het-erostructured nanowires used for this work are grown using the following pro-cedure: a thin Au film is evaporated on the (111)B InAs substrate surface and thermal annealing at 520◦_{C is used in order to produce Au}

nanopar-ticles used as growing catalysts. Subsequently the chamber temperature is set to 425◦_{C and TMIn with 0.15 mbar pressure and TBAs at 1.5 mbar is}

introduced into the chamber. To grow InP barriers TBAs flux is interrupted using its gas line’s shutter, while TBP flux is started. When the newly-grown InP barrier have reached the desired width, TBP flux is blocked and TBAs was introduced in the chamber again. Figure 3.4 shows an example of an heterostructured nanowire grown in our facilities.

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Figure 3.3: 45◦ tilted scanning electron micrograph of CBE-grown InAs NWs

Figure 3.4: Scanning transmission electron micrograph of an InAs/InP het-erostructured NW

not trivial and requires an accurate calibration study, whose description goes beyond the purposes of this thesis.

After growth, a small piece of the substrate on which NWs were grown is cleaved and put in isopropyl alcohol (IPA). Nanowires are detached from the substrate by sonication and then the solution containing NWs suspended in IPA is stored until they are ready to be used to fabricate devices.

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3.2 Device fabrication

In this section we will describe device fabrication processes we were in-volved in; devices were fabricated on p-doped Si/SiO2 substrates (with 285

nm oxide thickness) using InAs/InP heterostructured nanowires in IPA so-lution.

3.2.1 Ultraviolet lithography

Two inches Si-SiO2 wafers were patterned in of 2 × 2 mm2 devices, each

containing a frame of 24 metallic contact pads surrounding an active area for device fabrication of 160×160 µm2_{. To this end, wafer undergoes lithographic}

processes and is subsequently cut into smaller chips.

Figure 3.5: UV lithography procedure conceptual scheme.

Being the dimensions of connection pads are of the order of few microm-eters, we use ultraviolet (UV) lithographic techniques. This procedure is schematically depicted in figure 3.5. This typically consist in applying a photosensitive polymer solution (resist) on the samples by spin-coating and subsequently exposing them to UV light (using for example an mercury-base lamp), covering parts not to be exposed with a previously prepared mask

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(Figure 3.5, panel (a)). In this way the pattern on the mask is exactly trans-ferred to the resist. Assuming the use of a positive resist (which is our case), immersing the treated sample in a development solution makes the exposed resist to dissolve (Figure 3.5, panel (b)), allowing to evaporate metal (or any substance one wants to use) in order to produce desired pattern on the sample surface (Figure 3.5, panel (c)). The evaporation of the metal clearly covers the whole area, but while metal only sticks the surface where there is no resist left, metal evaporated on resist will be removed in the so-called lift-off process (Figure 3.5, panel (d)).

Figure 3.6: Single frame in a chip: every sample we fabricate has 4 frames. Alignment references are highlighted with red circles.

For chip fabrication, we spin coated a bilayer of LOR-3A (Microchem) and S1813 (Microposit), spinned at 3000rpm and 5000rpm, respectively, and softbaked according to datasheets. We then exposed the sample to UV light using the appropriate mask. The samples were then developed in MF319 for 1 minute and rinsed in water. The sample was placed in the thermal evapo-rator, where vacuum up to 10−6 _{mbar allows ballistic trajectories for metal}

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near boat-shaped containers and high accuracy control of the height of de-posited material was possible thanks to a quartz-based scale. We evaporated Ti/Au (5/50 nm). We later proceeded to lift-off the sample, immersing it in Acetone (ACE) heated to 50◦_{C for 5 minutes. The patterned substrate was}

then cut in 4mm × 4mm chips, containing 2x2 frames for device fabrication. The final result is shown in figure 3.6, in which markers used for alignment during Electron Beam Lithography are highlighted.

3.2.2 Nanowire deposition, imaging and CAD

design-ing

Figure 3.7: Scanning electron micrograph of deposited InAs/InP NWs used in our work (a). Zoom on a single NW (right). The arrow indicates the gold nanoparticle catalyst (b)

After chips have been fabricated, we proceeded to fabricate devices which we actually measured; to do so, the first step we needed to perform is the deposition of the nanowires on the chip by drop casting.

In order to deposit the nanowires stored in IPA, we gently shaked the bottle containing the solution and then deposited a ∼ 10µL drop on the chip

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using a pipette. We then rinsed the chip in IPA and dried with nitrogen afterwards. This procedure removes any residues and nanowires which did not stick on the substrate due to van der Waals interaction. This operation had to be repeated several times until the density of nanowires deposited on the chip is enough to fabricate the desired quantity of devices.

We then acquired scanning electron microscope (SEM) images of the sam-ple, both of the entire frame and isolated nanowires we considered viable for following fabrication steps (figure 3.7); frame image was used for CAD de-signing (example given in figure 3.8) and beam alignment during electron beam lithography.

Figure 3.8: An example of device CAD project.

3.2.3 Electron beam lithography

We could now proceed to device fabrication using Electron Beam Lithog-raphy (EBL), as we now needed much more precision in the lithographic

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process (order of tens of nm) compared to chip fabrication (for which we used UV lithography). Figure 3.9 depicts the sequence of EBL steps. The working principle is very similar to what we have explained for UV lithog-raphy: the main difference here is that we do not use a mask to expose the resist, but rather a precisely collimated electron beam. In this way we can literally ’draw’ a pattern on the resist layer using electrons, thus granting nanometric precision. Another difference between the two procedure relies on the precision they are capable of. In the case of UV lithography, photon beam diffraction represents the limiting factor on procedure precision. On the other side, in EBL the limiting factors are polymeric chain length in the resist and secondary electron scattering. Beam diffraction does not limit this process, as electrons have a much shorter wavelength compared to photons.

Figure 3.9: Electron beam lithography procedure conceptual scheme.

For our devices, we preliminarily placed our sample (i.e. chip with de-posited nanowires) on the hot plate set at 120◦_{C for 5 minutes, in order}

to let any solvents residuals and humidity evaporate. We then spin-coated AR679.04 PMMA-based resist at 4000 rpm for 60 seconds. We afterwards baked the deposited resist layer at 120◦_{C for 15 minutes, in order to let}

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re-sist’s solvent evaporate. After this, we proceeded to the actual lithographic process. Here a crucial step is accurately retrieving alignment markers’ co-ordinates using scanning electron miscroscope’s image and relating them to coordinates in the corresponding CAD project, so that the pattern will be precisely drew on the sample.

Figure 3.10: Scanning Electron Micrograph of the final device. a) Whole frame. b) Zoom on single device.

For each frame, lithographic process takes approximatively 4 minutes. After we completed EBL, we proceed in developing the sample, immersing it in AR600-56 developing solution at 15◦_{C for 2 minutes. Before evaporating}

metal in order to deposit device contacts and gates, two additional steps are performed: descum and passivation. Descum is a process based on Oxygen plasma etching at low power (about 10-20 W) and it is used to remove resist residues that remain in the exposed regions. Afterwards, we proceed with the passivation procedure, that is necessary to remove the native oxide on the InAs surface and to obtain a good ohmic contacts. In this phase we immersed our sample a standard ammonium polysulfide (N H4)2Sx solution at 44◦C

for 1 minute: when immersed in this compound, the native oxide (which naturally forms on nanowires’ surface when exposed to air) is replaced by sulfur chains. The sample is tehn lifted from the solution, washed throughly in de-ionized water and dried with high pressure nitrogen flow.

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Having passivated the sample, we can now introduce it in the evaporator and deposit 10 nm Cr and 110 nm Au. After a lift-off our device’s fabrication is completed and we can proceed to bonding phase.

Figure 3.10 shows a device fabricated following the explained procedure.

3.2.4 Device packaging

After fabrication, the chip was attached on the die pad of a Dual In-Line (DIL) chip carrier, using silver conductive paste. The Oxide on the back of the Si chip is previously damaged by scratching it with a diamond scriber, in order to ensure a good electrical contact between the DIL die pad and the backside of the sample. We need this electrical contact so that we can apply voltage to device’s backside and operate backgating. We also connect frame’s pads to DIL pads, so that we can electrically contact specific leads. In order to create these connections, we use the so-called bonder machine, shown in figure 3.11, which melts thin gold wires’ extremities using ultrasound pulses. In figure 3.11 an example of device ready for measurement is also shown.

Figure 3.11: Ultrasonic wedge bonder machine and bonded device.

3.3 Transport measurements setup

In this section we are going to discuss measurement setup we have used in order to perform measurements we will present in the next chapter. We will first describe how we can obtain low temperature regimes in which we

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need to perform our measurements using cryogenic systems. Afterwards, we will list and briefly describe electrical measurement devices we used in our facilities.

3.3.1 Cryogenics

Low temperature measurements were performed in a HelioxVL cryostat by Oxford instruments. The system consists in a vacuum loaded 3_He

cryo-stat, which allows us to perform measurements at temperature down to a lower limit of 250mK and magnetic fields up to 12T. The sample under test is placed in a sample socket located in the bottom of the insert. During op-eration the cryostat is inserted in a liquid helium storage dewar, consisting in several chambers, as represented in figure 3.12:

Figure 3.12: Schematic representation of the dewar hosting the insert. a) Front view. b) Top view.

The outer vacuum chamber (OVC) serves as a thermal isolation for in-ternal chambers, in order to avoid having liquid nitrogen in thermal contact with the external environment, thus causing huge gas evaporation and con-sequent loss. Furthermore, the chamber containing liquid nitrogen isolates (and pre-cools) the area containing liquid helium, which is as a matter of fact our main thermal bath (4.2◦_{K temperature), by minimizing radiative}

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heat-ing. This means that, without following any additional step, we can achieve sample temperatures down to 4.2 ◦_{K by simply dipping the insert (which is}

represented in Figure 3.13) in the dewar.

Figure 3.13: Picture showing helioxVL insert we use in our facilities.

Moreover, it is possible to achieve temperatures down to 250mK in our system by proceeding with the so-called condensation procedure. This mainly relies on the fact that by pumping over vapor in thermodynamic equilibrium with a liquid we are basically removing ”hot” particles and thus cooling down our system. Here, this is implemented using3_He_{vapor and a sorption pump,}

as shown in figure 3.14. Pumping on 4_He _{vapor here is needed in order to}

condensate3_{He, whose condensation temperature is 3.3}◦_{K, lower than liquid} 4_He _temperature.

The sorption pump (from now on we will call it sorb) will absorb gas when cooled below 40 K; moreover,the amount of gas that it absorbs depends on its temperature. It is cooled by drawing out some liquid helium from the main

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Figure 3.14: Condensation procedure, adapted from [41]. The picture shows condensation’s working principle using a top loading sample holder. Even this is not our case, conceptually the procedure is the same.

bath, while a local heater is fitted near the sorb so that its temperature can be precisely controlled during the operation. The 1K pot is used to condense the3_He_{gas; it is fed from the liquid helium bath thanks to the capillar shown}

in figure 3.13 and a needle valve which allows us to regulate how much liquid helium is drew. When preparing for condensation, the sorb is warmed above 40 K, so that it won’t absorb 3_{He: it will instead desorb any residual} 3_He

. We cool down the 1K pot pumping 4_He _{out of it using a scroll pump, so}

that 3_He _{condenses and falls down to cool the sample and} 3_He _{pot to the}

temperature of the 1 K pot. When most of the gas has condensed, the 1K pot needle valve is closed so as 4_He _{from main bath is not needed anymore.}

Now the3_He_{pot is full of liquid} 3_He_{at approximately 1.2 K (figure 3.14.a).}

We now cool down the sorb, so that it reduces vapor pressure above the liquid 3_He _{in the pot and consequently sample’s temperature drops. As the}

limit pressure is approached, the temperature of the liquid3_He_{and thus our}

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temperature of the system.3_.

An important feature of this system is that sample’s temperature can be controlled by adjusting the temperature of the sorb. In fact, setting it between 10K and 40K (and thus not working in the total absorption or total desorption regime) makes it possible to control the pressure of the3_He_vapor

and thus the temperature of the liquid 3_{He. In order to obtain this control}

in a precise way, we use a temperature controller (Intelligent Temperature Control 503 -ITC 503- by Ofxord Instruments, provided together with the whole described setup) which monitors temperatures where sample, sorption pump and 1K pot are located; moreover, ITC503 is linked to local heaters near these areas. This allows us to achieve great and stable control on our system’s working temperature, allowing us to carry on long measurements using higher sample temperatures.

3.3.2 Measurement devices

The following paragraph is dedicated to the description of devices for transport measurements and of their features we exploited to perform our experiment. The setup consisted on different equipments for voltage and current supply and for detection. In particular, we used the following instru-ments:

• Yokogawa 7651 Programmable DC Source:

– High accuracy output: ±0.01% of setting ±200µV (10V range, 90 days, at 23±5◦_C);

±0.02% of setting ±100nA (1mA range, 90 days, at 23±5◦_C)

– High resolution output: 100nV DC (10mV range) – High response speed: 10ms/±0.1%

– Low noise: 15µVP−P(1V range, DC to 10Hz)

• HP 4142B Modular DC Source/Monitor:

3_{Note that in the operations described no heat is supplied directly to the liquid}3_He as this would evaporate it too quickly.

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– Measurement Range: 10A, 1000V – Resolution: 20fA, 4µV

– Speed: 4ms

– Accuracy: V-0.05%; I-0.2% • Keithley 2400 SourceMeter:

– Source Voltage from 5µV to 210 V; measure voltage from 1µV to 211V

– Source current from 50pA to 1.05A; measure current from 10pA to 1.055A

– Measure resistance from 100µΩ to 211MΩ – Maximum source power is 22W

– Possibility to perform 2 and 4-wire measurements • Agilent 34410A Multimeter:

– DC Voltage:

Measurement method: Continuously integrating multi–slope IV A/D converter;

Ranges: 0.1 V, 1 V, 10 V, 100 V, 1000 V;

Accuracy (% of reading + % of range): 0.0030 ± 0.0030 (0.1 V), 0.0020 ± 0.0006 (1 V), 0.0015 ± 0.0004 (10 V), 0.0020 ± 0.0006 (100 V), 0.0020 ± 0.0006 (1000 V);

Input resistance: selectable 10 MΩ or ≥10 GΩ for 0.1 V, 1 V, 10 V Ranges. Fixed 10 MΩ ± 1% for 100 V, 1000 V Ranges;

Input Bias Current: ≤50 pA at 25 ◦_C

Input Terminals: copper alloy;

DC CMRR: 140 dB for 1 kΩ unbalance in LO lead. ±500 VDC maximum;

– DC Current:

Ranges: 100 µA, 1 mA, 10 mA, 100 mA, 1 A, 3 A;

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≤0.03 V Burden Voltage), 0.007 ± 0.006 (1 mA, ≤0.3 V Burden Voltage), 0.007 ± 0.020 (10 mA, ≤0.03 V Burden Voltage), 0.010 ± 0.004 (100 mA, ≤0.3 V Burden Voltage), 0.050 ± 0.006 (1 A, ≤0.8 V Burden Voltage), 0.100 ± 0.020 (3 A, ≤2 V Burden Voltage); Shunt Resistor: 0.1Ω for 1 A, 3 A, 2 Ω for 10 mA, 100 mA, 200 Ω for 100 µA, 1 mA;

Input Protection: Externally accessible 3 A, 250 V fuse; • SR830 DSP Lock-In Amplifier:

– Signal Channel:

Single-ended (A) or differential (A-B) Voltage inputs;

Signal filter: 60 (50) Hz and 120(100) Hz notch filters (Q=4); Gain accuracy: ±1% from 20◦_{C to 30}◦_{C (notch filters off), ±0.2}

% typical;

Input noise: 6 nV/√Hz at 1 kHz (typical); Sensitivity from 2 nV to 1 V;

– Internal Oscillator (Sine wave): 1 mHz to 102 kHz Frequency;

25 ppm +30 µHz frequency accuracy;

Output Amplitude: 4 mVrms to 5 Vrms (into a high impedance load) with 2 mV resolution;

Amplitude Resolution: 1 %;

– 4 BNC digital to analog DC outputs (AUX) ±10.5 V full scale, 1 mV resolution. 10 mA max output current;

• DL 1211 Current Preamplifier:

– Sensitivity: 10−3 _{to 10}−11 _{A/V with nine decade current gain}

ranges;

– Gain Multiplier: output gain multiplier with 3 steps: X0.1, X1.0 and X10;

– Gain Accuracy: ±2% of reading for 10−3 _{to 10}−9 _{A/V ranges;}

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– Input Offset Current: Less than 0.5x10−13 _{A @ 50} ◦_C;

– 4 outputs (BNC): 2 600Ω;

Lo-Z output (to 25mA, 50Ω); Unity gain output;

– Output Level: 22Vpp into 1KΩ load. (Lo-Z output);

– Output stability: output voltage offset stability better than 0.003% per ◦_C

How all these devices have been used relatively to our measurements will be described in chapter 4, where we will discuss about data we obtained.

Thermoelectric Effects in Nanowire Quantum Dots

Universit`

a degli Studi di Pisa

Thermoelectric Effects in Nanowire Quantum Dots

Contents

Abbreviations

Introduction

Chapter 1

State of the Art

1.1

Semiconductor nanowires

1.2

Quantum dots

1.3

Thermoelectrics in nano-sized systems

Chapter 2

Theoretical Models and

Concepts

2.1

Transport in quantum dots: the constant

interaction model

2.1.1

Current-voltage relation in the Coulomb

block-ade regime at a fixed V

configuration

2.1.2

Coulomb blockade: fixed V

2.1.3

Coulomb diamonds

2.2

Thermoelectrics

2.2.1

Thermoelectric effects in bulk materials

2.2.2

Boltzmann transport equation

2.2.3

Using Boltzmann equation for thermoelectrics

2.2.4

Thermoelectric effects in quantum dots

Chapter 3

Experimental Methods

3.1

Nanowire growth

3.2

Device fabrication

3.2.1

Ultraviolet lithography

3.2.2

Nanowire deposition, imaging and CAD

design-ing

3.2.3

Electron beam lithography

3.2.4

Device packaging

3.3

Transport measurements setup

3.3.1

Cryogenics

3.3.2

Measurement devices