Electrical and thermal measurements on superconducting proximized structures

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U

NIVERSITÀ DI

P

ISA

DIPARTIMENTO DI FISICA

TESI DILAUREA MAGISTRALE

Electrical and thermal measurements on

superconducting proximized structures

Candidato Relatore

Vittorio Buccheri Dott. Francesco Giazotto

Relatore interno

Prof. Stefano Roddaro

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Introduction

The great advances in nanotechnology of the last decades have made it possi-ble to exploit the exotic effects of quantum mechanics and to develop a new family of electronic devices (single electron transistors, electron pumps, qubits, etc.). In this context, a deeper understanding of the fundamental laws which rule the transport at the nanoscale is essential. Superconducting weak links [1] represent ideal candidates to this purpose, thanks to their unique properties, as the Josephson effect.

A weak link is a hybrid junction where superconductors (S) are connected with different types of materials, such as insulators (I) and normal metals (N). The unique phenomenology ruling such structures, which is called weak super-conductivity, is related to the macroscopic phase-coherence of the superconduct-ing pseudo-wave function and to the energy gap ∆ which characterizes the su-perconducting quasiparticles density of states [2]. For our purposes, we can dis-tinguish between two kinds of junctions: the tunnel junctions, which are ruled by tunneling processes, and the proximized junctions, which are ruled by An-dreev reflection [3]. In the former case, the metallic elements of the junction are separated by an insulator, in the latter one, normal metal and superconductors are directly in contact. Such junctions have several applications. They can be used, for instance, as sensing elements on more complex devices, especially in thermometry, refrigeration and detection [4]. Particularly, a magnetic field sen-sor is obtained if a superconducting loop is interrupted by two junctions. Such structure is called Superconducting Quantum Interference Device (SQUID) [2, 5] .

Proximized weak links are the main topic of this thesis. We focus on alu-minium/copper/aluminium junctions, where aluminium is used as the super-conductor and copper as the normal metal. Our goal is to give an experimental characterization of both electrical [3, 6, 7] and thermal [8, 9] properties of these structures.

The electrical characterization is performed on SNS SQUIDs [5] with differ-ent lengths for the normal metal wire which constitutes the junction. In partic-ular, we measure how the critical current of the SQUID depends on the main parameters which can be controlled experimentally, namely the applied

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mag-netic field, the temperature and the N-junction length.

On the other hand, the thermal measurements are performed on single SNS junctions. By defining source and drain the left and the right S-electrode of the junction, respectively, the experiment consists in warming up the source and in measuring how the drain temperature changes with respect to the source temperature. Then, by solving a balance equation for the system, we obtain the thermal conductance of the SNS junction. Both the source’s heater and the S-electrodes thermometers are realized with SISIS tunnel junctions. Particularly, one crucial issue for the success of the experiment is the proper calibration of such thermometers.

The samples are fabricated by multi-angle shadow mask evaporation and the measurements are performed in a 3_He-4_{He dilution refrigerator with base}

temperature of 25 mK.

In both experiments (electrical and thermal) a comparison with the theory is carried out. In particular, the diffusive transport in such structures is ruled by the Usadel equations [3], which are numerically solved by means of the library Usadel1 [8]. The limits where an analytic solution is available are discussed as well [6, 7, 9].

This thesis work has been realized at the NEST laboratories under the super-vision of Francesco Giazotto. The main fabrication steps, in particular the elec-tron beam lithography, the shadow mask-evaporation and the scanning elecelec-tron microscopy have been performed by Giuliano Timossi.

The thesis is organized as follows:

Chapter 1:Basic notions about superconductivity are introduced in the frame-work of the Bardeen-Cooper-Schrieffer (BCS) theory.

Chapter 2:Weak superconductivity is discussed. The first part of the chapter focuses on both the electrical and the thermal characteristics of tunnel junctions. Then, in the second part, the Proximity effect is presented and the SNS junctions are discussed. Numerical solutions and analytic limits are presented.

Chapter 3:The notion of the quasi-equilibrium regime is introduced.

Chapter 4: This chapter presents the devices fabrication processes. Details about the shadow mask evaporation and the other nano-fabrication techniques are given. Also, the main geometrical dimensions of the devices are reported.

Chapter 5:An introduction to the3_He-4_{He dilution refrigerators is given. In}

particular, the main characteristics of a Triton 200 cryostat are presented. Also, the main electrical components of the experimental apparatus are introduced.

Chapter 6:This chapter focuses on the electrical experiment. The main hall-marks of the SQUIDs are given at the beginning of the chapter. Furthermore

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Introduction vii the measurements and the data analysis are described. A comparison with the theory presented in chapter 2 is performed.

Chapter 7: This chapter focus on the thermal experiment. The first part of this chapter is devoted to the auxiliary components of the thermal devices, namely the cooling fins, the heaters and the thermometers. Particularly, the thermometers calibrations are discussed in detail. The measurements and the data analysis are described and, also in this case, the results are checked by us-ing the thermal theory discussed in chapters 2 and 3.

Chapter 8: The results and the conclusions of both the electrical and the thermal experiment are summarized. Finally, future perspectives of this work are discussed.

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Chapter 1 Superconductivity basics

This chapter presents some fundamental aspects of superconductivity and gives the necessary tools for understanding the topics treated afterwards. The goal here is to underline two fundamental features of superconductors: i) the macro-scopic long-range phase coherence of the superconducting pseudo-wave func-tion and ii) the presence of a temperature dependent energy gap in the supercon-ductors electron density of states (DOS). We will focus on the Bardeen-Cooper-Schrieffer (BCS) theory, because it gives an exhaustive description to the main features of superconductors.

1.1 Some phenomenological facts

Figure 1.1: Output of an ideal V-I measurements for an aluminium wire. (a) At T = 300 K we observe the ohmic linear dependence of V versus I, V = RI. (b) At T = 100 mK the current flows into the device without appreciable voltage drop, until the critical value Icis reached. For

I > Ic, the system reproduces again the ohmic behaviour.

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To introduce superconductivity we show an example of how this phenomenon arises experimentally. Let us imagine to measure the V-I characteristic of an alu-minium wire with section 500 × 500 nm2 _{and length 2 µm. We inject a current}

into the wire, measuring the voltage drop across it. If we repeat the measure-ments at two different temperatures, T ∼ 300 K and T = 100 mK, the results should be quite similar to the patterns showed in Fig 1.1 (a) and (b), respectively. Something happens in our system and modifies abruptly its transport be-haviour when it is cooled down to cryogenic temperatures. In fact, at room temperature, the device follows the Ohm’s law, V = RI (Fig. 1.1 (a)), whereas at T = 100 mK (Fig. 1.1 (b)) we observe the usual ohmic behaviour only for currents larger than a critical value Ic ∼ 19 µA, highlighted by the red disk in

the figure. By contrast, for I < Ic, we measure a current flowing through the

de-vice without building up any voltage drop: the dede-vice behaves as like a perfect conductor, i.e. it has no resistance.

This observation gives us two information. First: there is a critical value of temperature Tc, below which the system behaviour changes abruptly.

Sec-ond: such change is limited by a critical value, which is related with Ic. A very

similar phenomenology was found by Heike Kamerlingh Onnes in 1911 while measuring the temperature dependence of the resistance of a pure sample of mercury [10] and it was called superconductivity. It was shown that its nature is intrinsically quantum mechanical [11], and that the superconducting state is the stable thermodynamical state for certain metals below a threshold temperature which depens on the metals themselves.

An additional feature of the superconducting state is the Meissner-Ochsenfeld effect [11, 12], namely that the magnetic field can’t penetrate in the bulk of the superconductor. When a normal metal is cooled below its critical temperature Tc, magnetic field originally present inside the metal is expelled. More precisely,

the magnetic field penetrates in the superconductor but it decays exponentially over a characteristic length λL ∼ 500 nm [13]. In particular, when a

supercon-ductor is placed in an external magnetic field Be, a supercurrent Isis induced in

the superconductor in order to screen Be. We point out that Is, and in general

the supercurrent, circulates only in the region where the magnetic field can pen-etrate. As a consequence of the Meissner-Ochsenfeld effect, superconductors are characterized by a further critical quantity value for the superconductors, namely the magnetic field Bc, which breaks the superconducting state causing

the system to switch to the normal state. Again, we argue that the superconduct-ing state is limited by a critical value of energy, which it is related, in this case, with the critical field Bc.

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SECTION 1.2. BCS 3 with a junction of different materials, for instance a superconductor/insulator/-superconductor or a superconductor/insulator/-superconductor/normal metal/superconductor/insulator/-superconductor sandwich, its V-I patterns will be very similar to the ones of Fig 1.1. However, the micro-scopic mechanisms which rule such structures are more complex than a bulk superconductor. We will discuss about this systems in the second chapter.

1.2 BCS

The first widely successful microscopic explanation of superconductivity was presented by Bardeen, Cooper and Schrieffer (BCS) in a landmark paper in 1957 [14]. Here we describe the main results of the BCS theory which are relevant for the contents of this thesis.

1.2.1 Cooper pairs

As anticipated, the phenomenology described in the last section suggests that superconductivity is a thermodynamic phase of the electron gas inside the metal. In 1956 Cooper showed [15] that the ground state of a Fermi electron gas is un-stable if a weak attractive interaction between electrons pairs is switched on. In particular, he found that, when such attractive interaction affects the system, a configuration in which electrons are coupled in the so called Cooper pairs has an energy lower than the normal one ( i.e. with unpaired electrons). Hence the electrons condense in such pairs that minimize the energy and this results in the superconducting ground state [11].

The microscopic origin of such attractive interaction is a phonon-mediated process which affects the electrons near the Fermi surface and which becomes relevant at low temperature, as proposed a few years before the Cooper’s paper by Frölich [16]. A rough sketch of such electron-phonon interaction is given in Fig 1.2 (a). The transit of one electron (yellow point e1) among the ions (grey

points) causes a local deformation of the lattice which corresponds to a local surplus of positive charge. At low temperature the time needed to the lattice returns to its equilibrium position is higher than the transit time of the electron. Thus, the local deformation persists and, since its positive charge, it attracts another electron (yellow disk e2). As a consequence, the two electrons interact

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1.2.2 Macroscopic superconducting wave function

We point out that the individual pairs strongly overlap: the BCS coherence length ξ0represents an average distance in real space between the two electrons

of the Cooper pair, and we typically observe ξ0 ∼ 1000Å, whereas the average

distance between the centres of mass of the pairs is ∼ 10Å. Hence, there are million Cooper pairs within the coherence length [13].

A single wave function can be associated with a macroscopic number of electrons which are assumed to condensate in the same quantum state. In this sense, the superconducting state can be regarded as a macroscopic quantum state which can be described as a whole by a macroscopic complex pseudo-wave function of the form:

Ψ = ρ1/2ejφs _(1.1)

where ρ is the Cooper pair density and φS is the common phase shared by all

the particles in the condensate [2].

1.2.3 Energy gap

A Cooper pair can be broken by a certain amount of energy related to the pair binding energy, which depends cooperatively on the presence of all the other pairs [13]. In the BCS theory, a very important parameter related with the break-ing energy of the pairs is introduced: the energy gap ∆.

If we define λ the coupling constant of the electron-phonon interaction which is responsible of the Cooper pairing, the energy gap ∆(T ) is determined by the following equation [11, 13]: 1 λ = Z ~ωD 0 d 1 p2 _{+ ∆}2_{(T )}tanh p2_{+ ∆}2_{(T )} 2kBT , (1.2)

where kB is the Boltzmann constant and ωD is the frequency threshold for the

coupling interaction. In the weak coupling limit considered here, it holds kBT

~ωD EF, where EF is the Fermi energy. Near the critical temperature Tc,

equation (1.2) can be simplified in ∆(T ) ∆0 ≈ 1.74 1 − T Tc 1/2 , (1.3)

where ∆0 is the gap at zero temperature. The quantities ∆0 and Tc are related

by the following expression:

∆0 ' 1.764kBTc. (1.4)

Fig.1.2 (b) presents the solution of the self-consistent equation (1.2): we observe that for temperature lower than 0.4Tcthe gap is roughly constant.

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SECTION 1.2. BCS 5

Figure 1.2: (a) Qualitative picture of the electron-phonon process which causes the attrac-tive interaction between electrons: the transit of the electron e1 (right yellow point) affects the

ions configuration (grey points) causing a local accumulation of positive charge which attracts another electron e2(left yellow point). (b) Temperature dependence of the energy gap. It is

pos-sible to distinguish a region (T < 0.4Tc) in which ∆(T ) is approximately constant and another

region (T > 0.4Tc) where the gap monotonically decreases until it vanishes for T = Tc.

1.2.4 Quasiparticles and density of state

The presence of an energy gap in the spectrum of the elementary excitations, i.e. the quasiparticles, is the main hallmark of BCS description of superconductivity and it reflects the binding energy of the Cooper pairs. Here quasiparticles are excitations produced by the Cooper pairs breaking: for each broken pair two quasiparticles are excited. In the BCS description, it is proved [11, 13] that the quasiparticle excitations energy spectrum in a superconductor is E =√2_{+ ∆}2_,

where = ~k2_{/2m − E}

F, as depicted in Fig. 1.3 (a). By knowing E() we can

de-rive the quasiparticles density of state (DOS) NSof an ideal BCS superconductor

(Fig. 1.3 (b)): NS(E, T ) = |E| pE2_{− ∆}2_{(T )}Θ E 2_{− ∆}2 (T ) (1.5)

where Θ is the Heaviside step function. Note that NS(E, T ), as expressed in

(1.5), is normalized respect to NN, the normal state DOS.

Thus, in contrast with the normal metal behaviour in which all energy exci-tations are possible, in a superconductor there are no exciexci-tations in the energy interval [EF, EF + ∆], which means there are no quasiparticle states in the

en-ergy interval [EF − ∆, EF + ∆]. In fact, in this range the system doesn’t have a

sufficient energy to broke the Cooper pairs and thus to excite quasiparticles. For the same reason a gap of width 2∆ is opened around the Fermi energy in the DOS of the quasiparticles.

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Figure 1.3: (a) Normal metal (orange) and superconductor (blue) quasiparticles excitations spectrum near the Fermi energy. Note that in the normal case, as well as E > 0, the spectrum shows always available excitations states, whereas in the superconductor spectrum there are no excitations states for energy lower that the superconducting energy gap ∆0. (b) Normal

(orange) and superconducting (blue) DOS near the Fermi energy. A gap is opened around EF

in superconducting case and we can imagine that all the Cooper pairs are located at E = EF

(dotted pink line). A minimum energy equal to 2∆ is required to split a Cooper pair and excite two quasiparticles.

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Chapter 2 Weak Superconductivity

In the previous chapter we discussed the properties of the bulk superconduc-tors. Different phenomena appear in more complex structures, where supercon-ductors are connected by links of different materials, for instance normal metals or insulators. For example, in 1962 Brian David Josephson predicted [17] that a non dissipative current (supercurrent) can flow through two superconductors separated by a thin layer of insulator, thanks to a tunnelling process involving the Cooper pairs. This effect was experimentally proved by Anderson [18] in 1963 and it was called Josephson effect.

It soon became clear that such effect can be observed not only in the tun-nel-type junctions described by Josephson, but also in other kind of links where tunnelling processes are not involved [1]. The common hallmark of such links is that they can support some superconducting features, but they present criti-cal values lower than a bulk superconductor. For this reason we refer to such structures as weak links and to their phenomenology as weak superconductivity.

In the first two sections of this chapter we summarize those aspects that are common to all types of junctions. Then, in the third section, we focus on the tunnel junctions and finally, in the fourth one, we introduce the proximized junc-tions, namely those junctions where normal metal and superconductor are di-rectly interfaced. As we will see, in the last case, the microscopic explanation of the weak superconducting behaviour resides in the so-called Proximity effect. In both the tunnel and the proximized case, our aim is to give a description of the electrical and thermal properties of the junctions.

For practical reasons we will use the notation S, I, and N to indicate a su-perconductor, an insulator and a normal junction element, respectively. For in-stance a SN junction consists in a sandwich between a superconductor and a normal metal, and so on.

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2.1 Lumped junctions

As we have seen in Sec. 1.1, the screening supercurrent, which flows in response to an applied magnetic field, is confined to the surface of the superconductor. Its value, in fact, decays exponentially inward, with characteristic length λL =

(m∗/µ0nse∗2)1/2, where m∗ and e∗ are the effective values of the mass and of the

charge of the superconducting electrons, respectively, ns is their spacial density

and µ0 is the vacuum permeability. Usually λLis called London penetration depth,

and for common metals λL ∼ 100 nm [13]. All the junctions considered in

this work have a thickness smaller that λL, so that we can consider spatially

homogeneous the supercurrent and the phase in the single junction element. Junctions of this kind are called lumped junctions [19].

2.2 Dynes factor

In real junctions, as the ones considered in this work, it is customary to introduce a phenomenological parameter γ in order to account for non-ideal aspects of the superconductor [20]. Thus we need to modify the DOS expression given in Eq. (1.5) to: NS(E, T ) = Re ( E + iγ∆0 p(E + iγ∆0)2− ∆2(T ) ) , (2.1)

where ∆0 is the energy gap at T = 0 and γ accounts for all sort of

mecha-nisms that can create available quasiparticles sub-gap states. As the γ-factor increases, more sub-gap states become available and consequently the DOS as-sumes a more smeared shape (see Fig.2.1). Typical values of γ are in the range 10−4÷ 10−2

[21–23] .

2.3 Tunnel junctions

Let us imagine two metal electrodes sandwiched by an insulator layer which is thick enough to forbid classical electron transfer from one electrode to the other one, but it is still thin to make it possible quantum tunnelling. If we refer to the electrodes as 1 and 2, we can write the tunnelling current that flows from 1 to 2, in the case of a voltage-biased junction, as follows [2]:

I1→2 =

1 eRt

Z +∞ −∞

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SECTION 2.3. Tunnel junctions 9

Figure 2.1: Phenomenological density of states (DOS) for a superconducting electrode of a Josephson junction. The red line shows the modified DOS with γ = 10−4. By contrast, the blue dotted line shows the DOS of an ideal BCS superconductor as in Fig. 1.3 (b).

where e is the elementary charge, Rtis the tunnel barrier resistance, Ni(E,T) is

the normalized DOS Eq. (2.1) of the i-th electrode, and f0(E, Ti) =

1 ekB TiE _{+ 1}

(2.3) is the Fermi-Dirac distribution. Also we set the electrochemical potentials µ1 =

eV and µ2 = 0. See Fig. 2.2 for a scheme of the system we want to describe: such

kind of picture about junctions is called semiconductor model.

The meaning of Eq. (2.2) is that the tunnel current flowing from 1 to 2 is proportional, for each energy, i) to the tunnel resistance Rt, ii) to the number of

electrons available in 1, N1(E − eV, T1)f0(E − eV, T1), and iii) to the number of

states available in 2, N2(E, T2)(1 − f0(E, T2)). By following the same argument,

we can write also the I2→1 term, and hence the total current flowing from 1 to 2

is It= I1→2− I2→1, namely: It(V, T1, T2) = 1 eRt Z +∞ −∞

dEN1(E − eV, T1)N2(E, T2)× (2.4)

×[f0(E − eV, T1) − f0(E, T2)].

If both electrodes are normal metals (NIN junction), we can use the approxi-mation Ni(E, Ti) ∼ 1[2] and Eq. (2.4) becomes very similar to the well known

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Figure 2.2: Scheme of a tunnelling process occurring in a voltage biased tunnel junction be-tween two N electrodes. The voltage bias shifts the chemical potentials with respect to each other (here we set µ2= 2) and thus stimulates the tunnelling process.

But we are interest in those cases where one or both electrodes are super-conductors. Thus in the following we present two particular examples of such junctions: the SIN junction and the SIS one. We focus on these two structures because we have used them in the thermal experiment (see Chapter 7 ).

2.3.1 SIN junctions

In Fig. 2.3 (a) we observe the semiconductor model of a SIN junction in the absence of a voltage bias. Let try to apply the argument of the last section in this new configuration. As mentioned, the N-electrode DOS is roughly constant, NN ∼ 1, whereas for the superconductor we can use the smeared density of

state described in Eq. (2.1). Thus we specify the expression of Eq. (2.4) for the SIN junction as follows [2, 4]:

I_qpSIN(V, TS, TN) =

1 eRt

Z +∞ −∞

dENS(E − eV, TS)[f0(E − eV, TS) − f0(E, TN)], (2.5)

where now Rtis the resistance of junction in the normal state.

Fig. 2.3 (b) shows the voltage dependence of ISIN

qp : we note that at low

tem-perature the voltage across the junction can increase without appreciable growth of the current, as long as the critical value V = ∆/e is reached. Then, at the crit-ical point, the I-V pattern switches and the curves tends to a linear shape whose slope is related with the normal resistance as 1/Rt. As the temperature increase,

such switch becomes less marked, until the limit case T = Tc, for which the

junction acts as an usual resistor. By looking again at the (a)-side of Fig. 2.3, it is possible to explain the I-V patterns in terms of electrodes DOS shift. Let suppose to set both the N-electrode chemical potential and the temperature to zero. By Increasing the voltage drop, there is an upper shift of the S-electrode DOS: until the lower band edge of the S-DOS doesn’t reach µN, no quasiparticle current is

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Figure 2.3: (a) Semiconductor model of a SIN junction for V = 0. The thermally excited quasiparticles in the upper band of the S-electrode can tunnel in the N-electrode, as indicated by the red arrows. As V > 0, the band levels shift with respect to each other and the tunnelling processes increase until the critical value V = ∆/e is reached, for which the lower band edge of the S-electrode matches with the Fermi level of the N-electrode. (b) Quasiparticle current versus the voltage bias, at different bath temperature. Curves refer to the case TS =TN, Tc = 1.5K,

γ = 5 × 10−3and Rt= 1kΩ.

quasiparticles can tunnel from S to N, thus the junction assumes a resistive be-haviour. Such argument holds also in the case T > 0 but we have to take into account the quasiparticles thermally excited in the upper band of S, which can tunnel in N, also if eV < ∆0 (see the red arrows in the figure). Obviously, as the

temperature increases, the thermal effect outweighs and the switching becomes less sharp.

The temperature dependence of the I-V characteristic of the SIN junction is very important for thermometry application. In fact, often it is exploited to mea-sure the electronic temperature on the N-electrodes of the junction [4, 24]. For the same goal can be exploited the SIS junction [25]. In particular, to perform our thermal measurements, we used a SISIS junction whose I-V thermal depen-dence has been exploited to obtain the temperature of the central S-island. See Chapter 7 for further details.

2.3.2 SIS junctions

In contrast to the SIN junction, in the SIS case we have to consider also the tunnelling of the Cooper pairs. In fact, as discussed in the first chapter, we can describe the electrons Fermi sea in each superconducting electrode with a macroscopic wave function, Ψ1 = ρ1/21 eiφ1 and Ψ2 = ρ1/22 eiφ2, where ρiis the

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representation of a junction where two superconducting electrodes are sand-wiched by a thin insulator layer. If the I-layer thickness is thin enough, the tails of Ψ1 and Ψ2 overlap, as depicted in the figure, and thus there is a finite

proba-bility that the pairs can tunnel from one superconductor to the other [2]. If we

Figure 2.4: Scheme of a SIS junction. The insulating barrier is large enough to forbid classical electron transfer from one S electrode to the other one. However, it is also thin enough to guar-antee a finite overlap (brown area) between the superconducting wave function tails, and thus the pairs tunnelling is possible.

consider the more general case, in which superconducting electrodes lay at dif-ferent temperature and have difdif-ferent values of energy gap and Dynes factor, (T1, ∆1, γ1) and (T2, ∆2, γ2), the critical current Ic is given by the generalized

Ambegaokar-Baratoff formula [26]: Ic(T1, T2) = 1 2eRt Z ∞ −∞

dE{[1 − 2f0(E, T1)]Re[F (E, T1)]Im[F (E, T2)]

+[1 − 2f0(E, T2)]Re[F (E, T2)]Im[F (E, T1)]

, (2.6)

where Rtis the resistance of the junction in the normal state, e is the elementary

charge, F (E, T1,2) = ∆1,2(T1,2) q (E + iγ1,2∆0(1,2))2− ∆21,2(T1,2) , (2.7)

and f0 is the Fermi-Dirac distribution. We used the notation ∆0i to indicate the

zero temperature gap energy of the i-th superconductor. Here we assume the same chemical potential for the electrodes, µ1 = µ2 = 0. By defining δ=∆01/∆02,

in Fig.2.5 (a) we plot Eq. (2.6) in the equilibrium case (T1=T2) both for δ=1

(dot-ted green line) and δ=0.8 (dashed pink line). The latter case is plot(dot-ted also for a non-equilibrium configuration, in particular for T2 = 0.05T1(blue line). We have

γ=5×10−3 for both superconductors, Tc1= 1.25 K, Tc2= 1.5 K, and Rt= 10 kΩ. In

this work we don’t consider the supercurrent contribution at V 6= 0, which is the a.c. Josephson effect.

On the other hand, the quasiparticle current term Iqp depends on the

volt-age drop applied across the junction. Setting µ1 = eV and µ2 = 0, Iqp can be

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Figure 2.5: (a) Generalized Ambegaokar-Baratoff formula for δ=1 (dotted green line) and δ=0.8 (dashed pink line) in the thermal equilibrium case T1= T2= T. The blue line corresponds

to a thermally biased junction, with T2 = 0.05T1and for δ = 0.8. (b) Quasiparticle current in a

SIS junction for different values of Tbathand for Tc1 = 1.25K, Tc2= 1.5K. The red arrows mark

the matching peaks. In both figures γ1= γ2= 5 × 10−3and Rt= 10 kΩ.

Iqp(V, T1, T2) =

1 eRn

Z +∞

−∞

dEN1(E − eV, T1)N2(E, T2)[f0(E − eV, T1) − f0(E, T2)],

(2.8) where Ni(E, Ti) is the Si-electrode DOS, as written in Eq. (2.1). In Fig. 2.5 (b),

Iqp is plotted at different bath temperatures (now we are assuming a thermal

equilibrium case, T1 = T2 = Tbath), with the same parameters (Tci, γi, Rt) used for

the the (a)-panel. The red arrows indicate the so-called matching peaks, namely a current enhancement corresponding to the voltage value (∆1 − ∆2)/e.

To summarize the electrical features of the SIS junction, in Fig.2.6 we show the semiconductor model of the SIS junction at three different voltage bias con-ditions. In the zero bias case (Fig. 2.6 (a)) the electrochemical potential of S1

matches with the one of S2, so we have the supercurrent due to the Cooper pairs

tunnelling, as described by Eq. (2.6). As soon as a voltage bias appears across the junction, the dissipative current due to the quasiparticles starts flowing through the I-barrier. Again, by increasing the voltage V , we cause a shift of the elec-trochemical potential that facilitates the quasiparticle flow. When the voltage bias reaches the value V = (∆2 − ∆1)/ewe have a match between the edge of

the upper S-electrodes bands (Fig. 2.6 (b)) which reflects the current enhance-ment of the matching peak in the I-V characteristics. Such peaks are due to the fact that the thermally excited quasiparticles in the upper band of S1 find

sud-denly a large number of available states in the upper band of S2. Finally, when

V = (∆1 + ∆2)/e, the filled band of one S-electrode matches with the empty

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that brings the junction in the normal state.

Note that we use the notation normal state to indicate the I-V range where the junctions don’t show a weak behaviour, i.e. where the junction resistance is Rt. We point out that the normal state of the junction doesn’t imply the normal

state of the superconducting electrodes. Also, we refer to the junction configu-ration V = 0 and V 6= 0 as zero-voltage state and voltage state (or resistive state), respectively.

Figure 2.6: Scheme of the semiconductor model for a SIS junction. (a) Zero-voltage state: µ1

matches with µ2(we set µ2= 0), leading to a supercurrent due to the Cooper pairs tunnelling.

No quasiparticle current is observed. (b) At µ1= ∆2− ∆1we observe the matching peaks due

to the perfect match between the upper bands of the superconductors. (c) When µ1reaches the

value ∆1+ ∆2the junction is totally open to the quasiparticles flow and the V-I characteristics

of the junction assume an ohmic pattern. Adapted from [24].

2.3.3 Heat transport in tunnel junctions

Equations (2.5) and (2.8) can be easily modified to obtain the heat current J flowing through a SIN and a SIS junction respectively. We focus only on the configurations exploited in our measurements.

Firstly, we consider the voltage unbiased case for the SIN junction, which is described by: J_qpSIN(TS, TN) = 1 e2_R t Z +∞ −∞

dEENS(E, TS)[f0(E, TS) − f0(E, TN)]. (2.9)

Thus, JSIN

qp (TS, TN)represents the power flowing from S to N: if TS >TN, JqpSIN >

0 and the S-electrodes is cooled, vice versa if TS < TN, JqpSIN < 0 and the

S-electrodes is heated.

On the other hand, for the voltage biased SIS junction, we find:

JS1IS2 qp (V, T1, T2) = 1 e2_R t Z +∞ −∞

dE(E − eV )N1(E − eV, T1)N∈(E, T2)

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SECTION 2.4. SNS junctions 15

Figure 2.7: (a) Heat power transmitted from S to N by a SIN junction, as a function of the S-electrode temperature. We set TN = 0.5Tc, Tc = 1.5K, γ = 5 × 10−3and Rt= 1kΩ (b) Heat

power extracted from the S1electrode in a voltage biased S1IS2junction. Parameters used are:

T1= T2= 0.45K, Tc1 = 1.5K, Tc2= 0.5K, γ1= γ2= 5 × 10−3and Rt= 1kΩ.

Also in this case, the formula computes the power that flows from S1 to S2. For

the SIS junction we are considering only the contribution due to the quasiparti-cle current since the supercurrent of the Cooper pairs is dissipation-less and it carries no entropy.

In Fig. 2.7 the heat power flowing through the SIN and the SIS junction in the discussed configurations is plotted in panel (a) and (b), respectively. We observe in both cases a wide range in which S, or S1 are cooled. To explain this cooling

we refer again to the semiconductor models of figures 2.3 and 2.6: we note that before the switch to the normal state of the junction, the flowing current is made of the more energetic quasiparticles of S in the SIN and of S1 in the SIS. Since

the more energetic quasiparticles are also the hotter ones, by flowing through the I-barrier, they carry heat to the other electrode. As a consequence both the S-electrode of SIN and the S1-electrode of SIS are cooled.

2.4 SNS junctions

In contrast to the SIS junction, where the current is the result of a tunnelling process, in the SNS junctions the supercurrent is due to the superconducting correlations induced in the normal island N. In fact, when a superconductor is put in clean electrical contact with a normal metal, the latter, near the interface, exhibits superconducting features, in particular a mini gap opens in the normal metal DOS and a supercurrent can flow through a SNS junction. This induced "superconducting-like" behaviour is known as the proximity effect.

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2.4.1 The proximity effect

Figure 2.8: Schematic representation of the Andreev reflection in a ideal N-S interface. An electron with energy EF <E<EF + ∆which moves from the N to the S region doesn’t find

available states for the transmission. However, it can be reflected as a hole by involving a second electron, and thus adding a Cooper pair to the superconducting condensate of the S-electrode.

The microscopic origin of the proximity effect is a particular reflection pro-cess, the Andreev reflection, whose mechanism is schematically represented in Fig 2.8.

Let us consider an ideal SN interface: if an electron in the N-side with energy EF < E < EF + ∆and wave vector ke moves towards the S electrode (see the

yellow disk in Fig 2.8), it will be neither transmitted nor normally reflected. Indeed, transmission is forbidden because energy must be conserved in the pro-cess, and the electron can not find any available state in the S-side at the energy E < EF + ∆. Moreover, since the interface is ideal, the electron can not be

nor-mally reflected, because there is no barrier which could absorb the momentum difference of the reflection. However, thanks to an additional electron involve-ment, it can be converted in a Cooper pair in the S-side of the interface. In order to make it happen, the wave vector of the second electron must have the same magnitude and the opposite direction with respect to ke. Furthermore, since

the momentum has to be conserved, a hole (see the green disk in Fig 2.8) with wave vector kh = kewhich moves in the opposite direction with respect to the

first electron, has to be produced. In conclusion the first electron has been retro-reflected as a hole and a Cooper pairs has been created in the S-electrode.

If we imagine a SNS junction instead the simple NS interface, following the argument just used, the hole will be in turn retro-reflected as an electron at the opposite interface, causing the destruction of a Cooper pair in the left S-electrode. Thus it will be a cycle that in fact transfers the pair correlated electrons from one superconductor to the other one [3]. As a consequence, the N-link is

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SECTION 2.4. SNS junctions 17 capable to support a supercurrent.

For the sake of completeness, if the incident electron has energy |E| > ∆, quasi-particle states are available in the S-electrode and also normal specular reflection occurs with a certain probability. For the systems here considered, even if we are dealing with real SN interfaces, the Andreev reflection is still a dominant process with respect to the normal transmission or reflection [27, 28].

We emphasis that the Andreev reflection affects not only the N-link but also the S-electrodes. In fact as well as a superconducting behaviour is induced in the normal region, the superconducting features of the S-electrodes become weaker. This effect is noted as inverse proximity. Nevertheless, as we will discuss in the next section, the dominance of such effects depends strongly on the dimensions of the system considered. In our case, we analyse junctions whose S-electrodes dimensions are several times larger than the N-link, so we will neglect the in-verse proximity effect.

2.4.2 Electrical Transport

To model a transport phenomenon in a SNS junction we must take into account that the proximity effect modifies the N-DOS by inducing a spatially dependent mini-gap. Furthermore it depends strongly on the boundary conditions, namely on the BCS gap ∆ of the S-electrodes and on the length of the N-link. Due to the geometrical dimensions of our system, in the following we adopt a 1-D approx-imation, which reduces the spatial DOS dependence to the coordinate x along the longitudinal direction of the junction. A general approach to study such systems involves the solution of the Usadel equations. Usadel equations are dif-ferential equations obtained in the so called quasi-classical approximation, and they are written in terms of the Green’s functions of the system. Such formalism is really complicate and an exhaustive description of it goes beyond the purpose of this work. A more detailed description can be found in [3, 29, 30].

To analyse the systems relevant for this work, we can focus on the so called θ − parametrization, and the Usadel equations reads:

~D∂x2θ = −2iE sinh(θ) +

~D 2 (∂xχ)

2_{sinh(2θ) + 2iRe{∆} cosh θ}

(2.11a) ~D∂xjE = −2iIm{∆} sinh θ, (2.11b)

where D is the diffusivity and we have defined jE = − sinh2θ∂xχ. The complex

x-dependent superconducting order parameter ∆eiφ_{is obtained self-consistently}

from:

∆eiφ= λ 4

Z

dERe{sinh(θ)eiχ} tanh

E 2kBT

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Here φ and λ are the superconducting phase and the strength of attractive in-teraction (see sections 1.2.2 and 1.2.3 ). The functions θ(x, E) and χ(x, E) are complex valued and depend on the spacial coordinate x and the energy E. In particular they are defined so that in a superconductor θ = artanh(∆/E) and χ = φ, whereas in a normal metal θ = 0 and χ is not defined.

After the equations (2.11) have been solved, we can use the functions θ(x, E) and χ(x, E) to calculate the quantities of interest for our analysis. In particular, we can write the proximized N-link DOS N , normalized respect to its normal value, as

N (x, E) = Re{cosh θ(x, E)}, (2.13)

and, for our purpose, the supercurrent ISN S(φ) =

1 2eRN

Z +∞

−∞

dEjS(E, φ) tanh

E 2kBT

, (2.14)

where RN is the resistance of the N-link. The maximum of ISN S(φ)with respect

to φ at a certain temperature T is the critical current that flows through the SNS junction for that temperature, therefore we can obtain the function Ic(T )by

max-imizing Eq (2.14) for different values of T . In the following two sub-sections we define two known analytic limits in which Eq. (2.14) is simplified.

2.4.3 Characteristic lengths

In dealing with mesoscopic systems, it is important to distinguish between the different characteristic lengths which are involved in the transport phenomena. For example, in a junction system, we have the length of the junction LN, but

also leand lφ, that are the electron elastic mean free path and the single particle

de-phasing length, respectively. The SNS junctions treated in this work are in the so called diffusive limit, namely these three characteristic lengths are in the order relations le< LN < lφ. In addiction, the proximity effect introduces a

char-acteristic length ξN Swhich represents the depth for pairs correlation penetration

in the normal island [1]:

ξN S =

r ~DN

∆ , (2.15)

where DN = levF/3is the diffusion coefficient of the N region and vF is the Fermi

velocity.

Thus we can define two new limits: the junction is long for LN >> ξN S and

short for LN << ξN S. In the short limit we can consider all the N-link

proxi-mized, namely the pairs coherence is transmitted along all the N-link length. In this case the junction supports a supercurrent following the same principles

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SECTION 2.4. SNS junctions 19 which rule the superconductors. By contrast, in the long junction limit, only the portions of the normal wire near the SN interface are proximized. Nevertheless, also in this case a supercurrent can flow through the junction. Its "vehicle” is no more the Cooper pairs long-range coherence, but the intrinsic capacity of the N wire to preserve the pairing correlation induced by the S-electrodes at the inter-face, also where the normal metal is no more proximized. In this case we can roughly consider ∆g ∼ 3.1Eth[28], where Eth is the so-called Thouless energy.

2.4.4 Thouless energy

Sometimes it is useful to define long and short regime by using an energetic ar-gument. With this aim we introduce the Thouless energy ET h = ~DN/L2N which

represents the characteristic energy scale for a diffusive system. We have to con-front ET h with energy gap. In this case long junction means ∆ >> ET h and short

junction means ∆ <<ET h. The limit defines the energy value which sets the

cutoff for the supercurrent which can flow through the junction at low temper-ature [29]: Ic(T = 0) ∝    ∆ if ∆ << ET h (short junction) ET h if ∆ >> ET h (long junction) (2.16)

The argument is the same of the last section. In the short limit, the junction is totally proximized, transport phenomena are ruled prevalently by supercon-ducting properties and consequently the gap ∆ sets the upper limit of the su-percurrent at the low temperature (as in the tunnel junctions). On the other side, the diffusive transport is the mechanism providing the supercurrent in the long junctions, thus the Thouless energy is the low temperature scale for this limit.

2.4.5 Short limit

In the short limit, Eq. (2.14) is simplified in [Heikkila]:

IS = 2πkBT eRN cos(φ/2) ∞ X n=0 ∆ p∆2_cos2_{(φ/2) + ω}2 n (2.17) arctan ∆ sin(φ/2) p∆2_cos2_{(φ/2) + ω}2 n ! ,

where ωn = πkBT (2n + 1). Also in this case, to find Ic(T )equation (2.17) has to

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2.4.6 Long limit

In the long limit, and in particular in the regime of high temperature, kBT > Eth

we can write directly the expression for the critical temperature [6, 7]:

Ic(T ) = 64πkBT eRN ∞ X n=0 q 2ωn ET h∆ 2_{(T )exp}n₋q2ωn ET h o ωn+ Ωn+p2(Ω2n+ ωnΩn) 2 (2.18)

where Ωn(T ) =p∆2(T ) + ωn2(T )and again ωn(T ) = πkBT (2n + 1).

2.4.7 Usadel1

Unfortunately, as we will discuss in the following chapters, none of the junc-tions considered here is in the short limit. Most of them they are long juncjunc-tions, with ∆/ET h ∼ 20, and two of them are in the intermediate limit, ∆/ET h ∼ 1.

Furthermore, among the long ones, only about one we have data in the right temperature range, namely for kBT > ET h, to apply Eq (2.18). Thus we have

to solve the Usadel system (2.11) and obtain Ic(T ) as described above. For this

goal we used the open source software Usadel1 [8] which has been implemented with the aim of studying proximized systems.

Firstly the 1-D geometry has to be imposed: in the SNS junction case, we have one wire (the normal link) and two superconducting nodes (the electrodes). Then, the software requires as input parameters the diffusivity of the normal metal link, the N-link length and the critical temperature of the superconduct-ing electrodes. Settsuperconduct-ing this parameters, Usadel1 returns Ic(T ) in the required

range of temperature.

In Fig. 2.9 (a) and (b) is plotted the critical current of a SNS junction as func-tion of temperature for the short and the long limit, respectively. In both case we plotted the numerical solution obtained with Usadel1 (blue lines) and the sim-plified result of the limits (yellow paths). As expected, solutions for the short junction matches quite well, whereas in the long limit, the match holds only at high temperatures.

2.4.8 Thermal transport

The heat current across a SNS junction is carried by electron(hole)-like quasipar-ticles which can be transmitted via the SN interface. For the thermal modelling of such system we follow the description presented in a recent paper, Ref. [9].

Let us consider a SNS junction of N-link length LN, where each S-electrode

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SECTION 2.4. SNS junctions 21

Figure 2.9:Temperature dependence of the critical current of a SNS junction. (a) Short junction limit: for Rn = 0.6 Ω, LN = 50nm. (b) Long junction limit for Rn = 8 Ω, LN = 1 µm. In both

plots the numerical solutions of Usadel equation (blue line) and the analytic limits solution (orange pattern) are shown.

respectively. In the inset of Fig. 2.10 we depict a scheme of the junction: we suppose the S-electrodes lay at different temperatures, TLand TR, which leads

to a heat flow through the junction. Furthermore we take into account a pos-sible phase difference, φ = φL − φR, between the superconductors. We’re not

considering the voltage biased case, so we set µL = µR = 0. The finite width

of the junction leads to a number Nt of transport channels. Such system can be

solved by following the scattering matrix approach [31, 32], which gives us the transmission probability Tnof the n-th channel:

Tn= 2DnξLξR

DnξLξR+ (2 − Dn)(E2− |∆L∆R| cos φ)

((2 − Dn)ξLξR+ Dn(E2− |∆L∆R| cos φ))2

, (2.19)

where ξi = pE2− |∆i|2 and 0 ≤ Dn ≤ 1 is the transmission eigenvalue of the

n-th channel of the N-link. To obtain the total heat current JSN S we have to

in-tegrate each Tnover the energy and to sum the contributions of the Nttransport

channels. Thus we have: JSN S = 2 h Nt X n=1 Z ∞ |∆|

dEETn(E)(f0(E, TL) − f0(E, TR)), (2.20)

where h is the Planck’s constant, f0(E, Ti)with i = L, R is the Fermi-Dirac

distri-bution of the left and right electrode, respectively. Note that we have integrated from the energy starting value |∆| = max{∆L, ∆R}, in fact below this value no

quasi-particles can flow through the junction since no states are available in the S-electrodes. From equations (2.19) and (2.20) we can derive the heat conduc-tance of a SNS junction, κSN S = ∂JSN S/∂δT

δT =0, in the case of Nt channels,

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κSN S = 1 2h Nt X n=1 1 kBT2 Z ∞ |∆| dE E 2 cosh2(E/2kBT ) Tn(E) δT =0. (2.21)

But we are interest in a SNS where the normal metal link is a diffusive metal, i.e. the number of channels Nt is very large. In this limit the value of the

transmission probability Dnhas been found to be statistically distributed by the

Dorokhov distribution [33]: ρ(Dn) = Ntle 2LNDn √ 1 − Dn , (2.22)

where le is the electron mean free path. Thus, the average heat conductance is

given by: hκSN Si = Z 1 0 dDnκ (n) SN Sρ(Dn), (2.23)

where κ(n)SN S is the single-channel contribution which we can replace with Eq

(2.21): hκSN Si = 1 2hkBT2 Z ∞ |∆| dE E 2 cosh2(E/2kBT ) Z 1 0 dDnρ(Dn)Tn(E) δT =0. (2.24)

Here we use an important result [31]: R₀1dDnTn(E)ρ(Dn)is energy and

phase-independent and simply equals Ntle/LN. It follows a simple analytical

expres-Figure 2.10: Average normalized heat conductance in the diffusive limit. Inset: scheme of thermally biased SNS junction.

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SECTION 2.4. SNS junctions 23 sion for the normalized average heat conductance:

hκSN Si hκNi = 2 − 6 π2 " |∆| kBT 2 + |∆| kBT ln f (|∆|) − 2Li2(−e |∆| kB T₎ # , (2.25)

where hκNi = κ0Ntle/L is the average heat conductance for the junction in the

normal state, κ0 = π2kB2T /3h, and Li2is the dilogarithmic function, defined as

Li2(z) = − Z z 0 ln(1 − v) v dv, (2.26) with z ∈ C.

Eq. (2.25) shows the important result that hκSN Si in a diffusive SNS

junc-tion is phase-independent. Furthermore, superconductivity features affect such equation just via the larger energy gap |∆| of the superconducting electrodes. The shape of Eq. (2.25) is plotted in 2.10. Note that the argument that leads to Eq. (2.25) is strictly correct only in the short junction limit.

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Chapter 3 Thermal model: basic elements

In the last chapters we have discussed the properties of the superconducting junctions disregarding the environment where they operate. In real systems, they are components of a mesoscopic device which in turn resides inside a cryo-genic experimental apparatus. This fact has a crucial impact on the thermal behaviour of the system, as discussed in this chapter.

3.1 Quasi-equilibrium regime

In our experiments, the devices are set in a cryogenic dilution refrigerator (see Chapter 5), which is characterized by a base temperature Tbath. By using the

discussed Josephson junctions properties, it is possible to manage the electronic temperature Teof a generic electrode leading to Te6= Tbath.

In order to give a more insightful view on this point, we can describe the generic component of the device (either N or S) by means of the thermal model displayed in Fig. 3.1 (a). It is composed of two main thermal subsystems: the electron gas (red box) and the ions lattice (yellow box). Each subsystem has internal relaxation process with characteristic rate γe−e (for the electrons) and

γph−ph (for the phonons), and they interact with a rate Γe−ph. Moreover, the

phonons of the component interact with the phonons of the substrate (blue box), with a rate Γph−ph.

The temperature of each subsystem is well defined as long as the characteris-tic relaxation rates internal to them are larger than the characterischaracteris-tic rate due to the reciprocal interactions among the subsystems (γ >> Γ). This hypothesis al-low us to apply the Fermi-Dirac statistic for the electrons and the Bose-Einstein statistic for the phonons, and to define separately their temperatures, Te, Tphonon

and Tbath respectively. For our purposes, we can consider the device phonons

thermalised to the substrate phonons, i.e. Tphonon = Tbath[34]. When the

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ature of each subsystem is well defined, we refer to the full equilibrium regime if all these temperatures are equals, and to the quasi-equilibrium regime if they are not. On the other hand, if the hypothesis γ >> Γ doesn’t hold, no temperature can be defined and the system is in the non-equilibrium regime. The systems we will study are always in a quasi-equilibrium regime. Furthermore we can con-sider the substrate phonons as a thermal reservoir whose temperature Tbath can

be set externally and remains constant.

For the sake of completeness, the thermal model of Fig 3.1 (a) should include the photon gas. However we will not treat this topic since the energy exchanges with the photon component are negligible in the systems studied in this work [35].

Figure 3.1: (a)Schematic thermal model of a mesoscopic device. (b) Heat power exchanged due to the electron-phonon interaction for a normal metal (orange line) and a superconductor (blue line).

3.2 Energy balance equations

The temperature we want to manage is the electronic one Te, which is a

vari-able of almost all the expressions discussed in the second chapter. On the other hand, Tbathis easily settable, since it corresponds to the temperature of the

cryo-genic apparatus which used. Thus, let’s set Tbath and imagine to perform some

electrical measurements, for instance to obtain the I-V characteristic Iqp(V )of a

SIS junction. Since Iqp depends on the electronic temperature of the

supercon-ducting electrode, we are able to extract from the electrical measures a thermal information, namely the temperature of the S-electrodes.

With this goal in mind, in the framework of the quasi-equilibrium regime, we have to impose a balance equation which accounts for all the heat current,

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SECTION 3.2. Energy balance equations 27 namely the total input heat current Jtot

in and the total output heat current Jouttot,

for each electrode. Thus we have to solve: J_intot − Jtot

out = 0. (3.1)

We put this equation equal to zero because we are interested in the stationary case. In particular, Eq. (3.1) accounts both for the heat flowing between the S-electrodes (see Sec. 2.3.3 ) and the heat flowing due to the electron-phonon coupling. In the following section we give an expression about such terms.

3.2.1 Electron-phonon channel

As mentioned above Tphonon = Tbath, thus we consider only the contribution due

to the interaction between the electrons and the phohons of the device.

Normal metal case

For a normal metal, the heat current JN

e−ph which flows from the electron

subsys-tem to the phonon one, due to the electron-phonon coupling, is [4]: J_e−phN = ΣV (T_eα− Tα

bath) (3.2)

where V is the volume of the metal island, Σ is a parameter depending on the materials and α is an integer number which depends on the disorders properties of the metal. In our case, copper Cu has been used as normal metal, so we have ΣCu = 3 × 109 Wm−3K−5 and αCu = 5[4]. The orange line in Fig. 3.1 (b) show

the behaviour of Eq. (3.2) for a copper element at Tbath = 50mK.

Superconductor case

By considering a S-electrode, the electron-phonon heat current is expressed by a more complex expression [36]:

J_e−phS (Te, Tbath) = − ΣV 96 ζ(5) k5 B Z ∞ −∞ dEE Z ∞ −∞ d2sgn() L(E, E + , Te) coth 2kBTbath [f(1)(E, Te) − f(1)(E + , Te)] −f(1)_{(E, T} e)f(1)(E + , Te) + 1 . (3.3)

where the function L(E(1)_{, E}(2)_{, T}

e) = NS(E(1), Te)NS(E(2), Te)[1−∆2(Te)/E(1)E(2)],

ζ(5) ' 1.04and f1_{(E, T ) = f}

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Al as superconductor, ΣAl = 2.4 × 109 Wm−3K−5 [4]. In Fig 3.1 (b) the blue

line shows the pattern of Eq. (3.3) for a superconducting aluminium film with Tc= 1.4K, γ = 10−4 and again Tbath = 50mK.

At low electronic temperatures, the superconductor is less coupled to the phonons with respect to the normal metal of several order of magnitude, due to its energy gap which suppress the energy exchange. Nevertheless, as the temperature increases, JS

e−phon become comparable with Je−phonN . In chapter 7 we

will apply topic treated here about our thermal measurements performed on a SNS junction. Further details will be given there.

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Chapter 4 Devices fabrication

In this chapter we describe the fabrication process of our device step by step. The Fabrication was performed at the NEST laboratories.

4.1 Shadow mask evaporation

The shadow mask evaporation is a common nano-fabrication technique [37]; in Fig. 4.1 an example of the process is schematically reported.

Figure 4.1: Shadow mask process.

The devices are fabricated in a silicon (Si) substrate where a layer of silicon oxide (SiO2) has been grown (Fig. 4.1 (a)). The first step consists in coating

the substrate with a resist mask composed of a bi-layer of two different poly-mers. More specifically, a ballast thicker layer is deposited onto the substrate and it is topped by a stencil thinner layer (Fig. 4.1(b)). The second step of the

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process is the Electron Beam Lithography (EBL) in which the sample is pat-terned by a highly focused beam of accelerated electrons. The electrons react locally with the polymers, changing their molecular structures and their solu-bility properties, so that it is possible to remove selectively the exposed zones. The stencil layer is mainly sensitive to the incident electrons, whereas the ballast layer is more sensitive to both the electron beam and to the secondary electrons backscattered from the substrate. As a result, the stencil layer is patterned ac-cording to a clear shape, whereas the ballast layer acac-cording to a less defined one (Fig. 4.1 (c)). Thus, the sample is developed, namely the exposed zone is removed by a suitable solvent (Fig. 4.1 (d)).

Figure 4.2: (a)Essential scheme of the evaporator. The sample can be transferred from the sample holder of the load chamber (right triangle) to the one of the main chamber (left triangle), and vice versa, along the direction of the double green arrow, to perform the oxide/metal lay-ers growth. In the bottom of the main chamber, the crucibles (coloured disks) can be selected by a mechanical system, whereas in the load chamber a valve controls the oxygenation of the environment. (b) Lateral section view of a typical double Metal/Insulator/Metal junction. (c) Top view of the entire sample: the replica of the structure due to the multiple-angle evaporation process is shown. The dotted area represents the device.

The next step is the multi-angle evaporation. The sample is loaded in a UHV (ultra-high-vacuum) chamber (minimum pressure ∼ 10−11 mbar) above some crucibles which contain the metals that have to be evaporated (Fig 4.2 (a)). A fo-cused and movable electron beam is generated by a tungsten filament through the thermionic effect. The beam hits the metal inside the crucibles and causes its evaporation; thus the evaporated atoms are deposited on the sample. In par-ticular, they deposit on the substrate by following the shadow pattern of the EBL process. The sample holder inside the evaporator can be tilted in order to create different metallic layers in different positions (Fig. 4.1 (e)). The evaporation rate

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SECTION 4.2. On-chip implementation 31 is monitored by a crystal, which has been previously calibrated. The vibrations of the crystal are related to the impact of the evaporated atoms. The last step is the lift-off process where the resist mask and the metal on top of it are removed by immersing the sample in acetone at 50◦C (Fig. 4.1 (f)).

As we have discussed in the second chapter, it is often necessary to alternate metal electrodes with insulator barriers. For this reason the evaporator setup makes it possible to move the sample holder in a separate chamber where the oxide layers are grown thanks to a oxygenated environment (see the load chamber in Fig, 4.2 (a)). Some further sketches of the shadow mask-evaporation are given in Fig. 4.2 (b) and (c), where a section and a top view of a junction are depicted, respectively. Note that the core of the sample is connected to big pads (panel (c)) which facilitate the implementation of the sample in a circuit, as explained afterwards.

At the end of the fabrication process, the sample is observed by using a SEM (Scanning Electron Microscope) to ensure the good result of the full process. In Fig 4.3 we show two examples of SEM images.

Figure 4.3: (a)Top view of a large area of the sample. Six groups are visible and each group contains three SNS-SQUID devices. The big pads discussed in the text are clearly visible. Inset: a zoom of the yellow circled area which contains one subgroup of devices. The red circled area displays one single device. (b) Detail of a single SNS SQUID. Also here we note the replica of the structure due to the multi-angle evaporation.

4.2 On-chip implementation

After the nano-fabrication process, samples have been tested by MiBots. A Mi-Bot is a robot which performs about micro-metrical movements around a station (see Fig. 4.4 (a)). Furthermore, it has a metallic tip which can be moved up and down (also in this case by a micrometer quantity) in order to perform electrical measurements. By placing a device in the MiBot station, which is monitored

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thanks to a microscope, it is possible to move finely both the Mibots and their tips, thus ensuring the electrical contact with the pads of the device, as shown in the panel (b) of Fig. 4.4. Thus the electrical connections of the fabricated samples can be tested.

Figure 4.4: (a)A picture of the MiBots in their station, with the sample to be tested. (b) Top view of the same picture with the microscope: the Mibots metallic tips are in electrical contact with the pads of the device. (c) A picture of the samples glued on a dual-in-line sample holder, ready for the experiment. The wires which connect the gold pads of the sample holder to the pads of the device can be observed. They are aluminium wires welded by an ultrasonic bonder.

After the MiBots tests, the device is glued in a dual-in-line sample holder and, finally, electrical connections with the pins of the sample holder are made by an ultrasonic bonder by using aluminium wires (Fig 4.4 (c)). Now the sample is ready for the experiment.

4.3 Samples

SQUID

Two set of SQUID devices, set A and set B, have been fabricated in two different fabrication processes. In both cases, aluminium and copper have been chosen as the superconductor and the normal metal, respectively. A two-angle evapo-ration has been performed with no oxidation. For the set A, we present three devices, which have been named A330, A460 and A650. On the other hand, B260 and B960 are the devices of set B. In Fig. 4.5 the SEM pictures of these

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SECTION 4.3. Samples 33 devices are reported and the main geometrical characteristics of the SQUID are summarized in Tab. 4.1.

Figure 4.5:SEM pictures of the fabricated and measured SQUIDSs.

Figure 4.6: The picture displays the difference between the length of the junction LSN S,

marked by the green double arrow, and the geometrical effective length LG−ef f, which is half

of the length marked by the yellow double arrow and takes into account the overlap between the N-link and the S-electrodes.

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Device LSN S/w/h SAl sCu ASQU ID LG−ef f nm µm2 _nm A330 330/120/25 2 11 750 A460 460/120/25 2 11 900 A650 650/140/25 2 11 1065 B260 260/80/25 14 12 550 B960 960/150/25 8 12 1250

Table 4.1: Geometrical dimensions of the fabricated SQUIDs. LSN S/w/hdenote the length/

width/ thickness, of each normal metal link, respectively. SAl

sCu denote the ratio between the

normal metal section and the section of the SQUID arms close to the junction and ASQU ID

denote the SQUID area. Finally, LG−ef f is the geometrical effective length of the junction (see

Fig. 4.6). The samples marked with the same initial letter have been made in the same fabrication process and have been measured together. Note that the names of the devices just include information about the length of their N-links.

SNS junctions

The thermal measurements have been performed on the sample named G250. As showed in Fig. 4.7, G250 presents a complex structure. Each S-electrode of the SNS junction is equipped with five additional probes: four S-probes which contact the S-electrode from the bottom and a large N-electrode which is placed on the opposite side with respect to the N-link.

Figure 4.7: SEM pictures of the device used for the characterization of the thermal transport in the SNS junction.

All the additional probes are connected to the S-electrodes by I-barrier. Thus, the fabrication process consists in a four shadow-angles evaporation with

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oxi-SECTION 4.3. Samples 35 dation step. Also in this case aluminium and copper have been used as super-conductor and normal metal. The main geometrical characteristics of the device are summarized in Tab. 4.2.

Device LN/wN/hN LS/wS/hS wp/hp wP/hP LG−ef f

nm nm nm nm nm

G250 250/100/20 4000/450/100 80/50 150/50 900

Table 4.2: Geometrical characteristics of the device fabricated to perform the thermal mea-surements. LN/wN/hN and LS/wS/hS denote the length/ width/ thickness of the N-link and

of both the S-electrodes of the SNS junction, respectively. wp/hp and wP/hP are the width/

thickness of the thin and the thick S-probes, respectively, and LG−ef f is the geometrical

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Chapter 5 Cryogenic instruments

Measurements have been performed in a Triton 200 cryostat by Oxford Instru-ments [38], which is a3_He-4_{He dilution refrigerator system. In this chapter we}

briefly describe how this kind of cryostat works.

5.1

3

He-

4

He dilution refrigerator system

Figures 5.1 show the key properties of the3_He-4_{He mixture which are exploited}

in a dilution refrigerator system.

Figure 5.1: (a) Phase diagram for a 3_He-4_{He mixture. (b) Temperature dependence of the}

vapour pressure for3_{He and}4_{He. Adapted from [39].}

Figure 5.1 (a) displays the phase diagram of the mixture. Note that, below the temperature value T = 0.87 K, two different phases coexist on the border of the forbidden region. As the temperature drops, one of these phases becomes more and more rich of3_{He (see the yellow arrow), whereas the other one evolves}

to-wards the lower3_{He concentration (see the light blue arrow), until it reaches the}

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fixed valued 6.6% for temperatures lower than ∼ 0.1 K [39]. We call concentrated phase the former and diluted phase the latter [39].

A dilution cryostat works thanks to a thermodynamic cycle which exploits such two-phase separation. The thermodynamic cycle is schematically pictured in Fig.5.2 (a): a mixing chamber contains the mixture cooled below T = 0.87 K, the yellow colour denote the concentrated phase, the light blue denotes the di-luted phase and the yellow bubbles represent the 3_{He 6.6% impurities in the}

concentrated phase. As shown by the figure, the mixing chamber is connected with the still plate, so that the latter is partially filled by the dilute phase. Now it comes into play the second hallmark of the mixture: the vapour pressure of

3_{He is much bigger of the} 4_{He one (see Fig.5.1 (b)), thus if we pump over the}

still basically only3_{He will be extracted. As a consequence, some}3_{He atoms of}

the concentrated phase must pass to the dilute phase in order to maintain the equilibrium fraction value defined by the temperature (see Fig.5.1 (a)). The tran-sition from the concentrated to the dilute phase is an endothermic process, so it cools down the temperature of the mixture [39].

Figure 5.2: (a)Scheme of the core of a dilution refrigerator: 4_{He is painted in light blue,}3_He

is represented in yellow (full yellow for the concentrated phase, yellow bubbles for the diluted one).The heat exchanger box consists of a tangle of coil tubes which increase the thermal contact between the cold outcoming flow and the hot incoming one. (b) Scheme of a large view of a dilution cryostat: the extracted 3_{He follows the red line and reaches the traps, where it is}

purified and thrown back into the cryostat by the green line. The sample holder is in thermal contact with the mixing chamber so that its temperature is reduced during the cycle. All the core of the dilution refrigerator is shielded by different shells where a vacuum of the order of ∼ 10−6_mbar_{is pumped. The top plate is controlled externally.}

Figure 5.2 (b) gives a schematic representation of the full dilution refriger-ator setup. The red line takes the helium into the liquid nitrogen traps where

Electrical and thermal measurements on superconducting proximized structures

U

NIVERSITÀ DI

P

ISA

Electrical and thermal measurements on

superconducting proximized structures

Contents

Introduction

Chapter 1

Superconductivity basics

1.1

Some phenomenological facts

1.2

BCS

1.2.1

Cooper pairs

1.2.2

Macroscopic superconducting wave function

1.2.3

Energy gap

1.2.4

Quasiparticles and density of state

Chapter 2

Weak Superconductivity

2.1

Lumped junctions

2.2

Dynes factor

2.3

Tunnel junctions

2.3.1

SIN junctions

2.3.2

SIS junctions

2.3.3

Heat transport in tunnel junctions

2.4

SNS junctions

2.4.1

The proximity effect

2.4.2

Electrical Transport

2.4.3

Characteristic lengths

2.4.4

Thouless energy

2.4.5

Short limit

2.4.6

Long limit

2.4.7

Usadel1

2.4.8

Thermal transport

Chapter 3

Thermal model: basic elements

3.1

Quasi-equilibrium regime

3.2

Energy balance equations

3.2.1

Electron-phonon channel

Chapter 4

Devices fabrication

4.1

Shadow mask evaporation

4.2

On-chip implementation

4.3

Samples

SQUID

SNS junctions

Chapter 5

Cryogenic instruments

5.1

He-

He dilution refrigerator system