Thermal computation and heat harvesting in hybrid superconducting tunnel junctions

PH. D. THESIS INPHYSICS

Thermal computation and heat harvesting in hybrid

superconducting tunnel junctions

Author Advisor

Giampiero Marchegiani Dr. Francesco Giazotto

Science never cheered up anyone. The truth about the human situa-tion is just too awful.

This thesis consists of an overview of the following publications. Throughout the thesis they will be referred to by their Roman numerals.

I G. Marchegiani, P. Virtanen, and F. Giazotto. On-Chip Cooling by Heating with Superconducting Tunnel Junctions. EPL 124, 48005 (2018).

II F. Paolucci, G. Marchegiani, E. Strambini, and F. Giazotto. Phase-Tunable Thermal Logic: Computation with Heat. Phys. Rev. Applied 10, 024003 (2018)

III F. Paolucci, G. Marchegiani, E. Strambini, and F. Giazotto. Phase-tunable temperature amplifier. EPL 118, 68004 (2017)

IV N.Ligato, G. Marchegiani, P. Virtanen, E. Strambini, and F. Giazotto. High operating temperature in V-based superconducting quantum interference proximity transistors. Sci. Rep. 7, 8810 (2017)

V G. Marchegiani, P. Virtanen, F. Giazotto, and M. Campisi. Self Oscillating Josephson Quantum Heat Engine. Phys. Rev. Applied 6, 054014 (2016).

Throughout the thesis the following acronyms will be used: • AC: alternating current.

• BCS: Bardeen Cooper Schrieffer. • COP: coefficient of performance • DC: direct current.

• DOS: density of states.

• EBL: electron beam lithography. • EM: electromagnetic.

• FI: thin ferromagnetic insulator. • GL: Ginzburg and Landau. • I: thin insulating film. • IPA: isopropanol.

• MIBK: methyl isobutyl ketone. • N: normal metal.

• PMMA: polymethyl metacrylate.

• RCSJ: resistively and capacitively shunted junction. • S: superconductor.

• S’: superconducting proximity film. • SEM: scanning electron microscope.

• SQUIPT: superconducting quantum interference proximity transistor. • TE: thermoelectric.

List of publications v

List of acronyms vii

Contents ix

Introduction xi

1 Charge and heat transport in superconducting tunnel junctions 1

1.1 History and basic properties of conventional superconductors . . 1

1.2 BCS superconductors in a uniform exchange field . . . 2

1.3 Quasiparticle transport in hybrid tunnel junctions . . . 7

1.3.1 N1IN2 junctions . . . 9

1.3.2 NIS junctions . . . 9

1.3.3 NFISmjunctions . . . 11

1.4 Coherence effects in superconducting weak links . . . 16

1.4.1 Electrical transport in S1IS2junctions . . . 16

1.4.2 Resistively and capacitively shunted junction model (RCSJ model) . . . 19

1.4.3 Proximity effect in a SNS weak link and SQUIPT . . . 20

1.5 Quasiequilibrium regime in mesoscopic circuits . . . 24

1.5.1 Electron-electron relaxation rate . . . 26

1.5.2 Electron-phonon relaxation rate . . . 27

1.5.3 Energy balance equations . . . 28

2 Heat recycle in hybrid superconducting-normal junctions 29 2.1 Self oscillating quantum heat engine . . . 29

2.1.1 Model and numerical computation . . . 30

2.1.2 Mean Power estimation in sinusoidal regime and main limitations of the engine . . . 33

2.2.1 Design and modeling . . . 36

2.2.2 Cooling power and efficiency . . . 39

2.2.3 Steady state electronic temperature of the normal island and discussion . . . 42

3 Temperature Amplification and Thermal Logic 45 3.1 Temperature Amplifier . . . 45

3.1.1 Design and mathematical model . . . 46

3.1.2 Amplification factor . . . 48

3.2 Thermal Logic: Computation with heat . . . 50

3.2.1 Modeling . . . 51

3.2.2 Negation Logic Gate . . . 54

3.2.3 Conjunction Logic Gate . . . 55

3.2.4 Disjunction Logic Gate . . . 57

3.2.5 Operation speed and discussion . . . 59

4 High operating temperature in vanadium-based SQUIPT 61 4.1 Design and fabrication . . . 62

4.2 Base temperature characterization and theoretical modeling . . . 64

4.3 Impact of the bath temperature . . . 67

Appendices 75 Appendix A Tunneling formalism 77 A.1 Charge transport . . . 79

A.2 Heat transport . . . 82

A.3 SIS junction . . . 84

A.4 NFISmjunction . . . 86

Appendix B Heat current in a N1IN2junction 89

Appendix C Fluxoid quantization for a single-junction loop 91

Appendix D Mean power estimation in sinusoidal regime 93

Appendix E Modeling of the nanowire DOS in the V-SQUIPT 95

Bibliography 99

The second principle of thermodynamics states that it is impossible to realize a transformation whose only result is to convert heat into work. For this rea-son, heat is commonly seen as the most degraded form of energy. Furthermore, it often represents the main source of noise for many physical processes, lead-ing to decoherence and, more generally, to a degradation of the information processing. So, it is important to control this quantity for the technological ad-vancement of low temperature nanocircuits, which are mainly proposed in the field of quantum technologies. In fact, it is a well know fact that the massive heat production occuring at the nanoscale sets a fundamental limit to the com-putational power of nano-transistors. In this respect, it is not surprising that the investigation of the thermal transport at the nanoscale is attracting a con-tinuously growing interest. Over the last 15 years, the old heat-disorder epit-ome has been challenged by a series of theoretical models and experimental evidences. In particular, a model for a thermal rectifier, proposed in 2002 by Ter-raneo, Peyrard and Casati [1], paved the way for the control of heat currents in phononic systems. The goal of Phononics [2] is to achieve a precise control over the thermal transport in phononic systems, similar to the one obtain in electronic with electric currents. Device like thermal diodes, transistor and memories has been proposed and investigated, thus opening the possibility of a computational platform based on heat (heat logic). In a similar fashion, superconducting tunnel junctions offer a perfect platform for the control of thermal currents, since the single-particle excitations are the main carriers of both charge and heat. Two features of the superconductors are particularly relevant in this respect: i) the presence of the superconducting gap, which acts as a energy filter and ii) the macroscopic phase coherence of the condensate wave function [3]. Concerning the latter, the phase modulation of the thermal current in tunnel junction be-tween two superconductors has been experimentally realized in 2012 [4], giving birth to the rising field of coherent caloritronics (from the latin word, Calor) [3, 5]. In this field, structure as thermal diode, transistor and memory have been real-ized or proposed.

electronic-caloritronic analogy, upon the inclusion of ferromagnetic barriers in proximity contact with the superconductors. This contact induces an effective exchange field in the superconductor and breaks the spin symmetry in the quasiparticle density of states [6]. As a consequence, a strong thermoelectric effect occurs in hybrid normal-superconducting tunnel junctions in the presence of a spin-filtering mechanism. This effect has been predicted very recently [7] and timely experimentally verified [8, 9]. Interestingly, the thermoelectric effect opens new possibilities for such structures. Our goal is to explore the rich phenomenology of effects correlated to this conversion. In particular we will describe proposals for thermal machines, like a heat engines which provides work in a contact-less way (Pub.V) and a refrigerator supported by the thermal energy provided to the thermoelectric element (Pub.I). In both cases, the schemes benefits from additional nonlinear effects of the superconducting junction, for instance the AC Josephson effect and the quasiparticle filtering due to the presence of the energy gap. Moreover, we design a new model of a temperature transistor based on the thermoelectric effect and we discuss how to implements all the fundamental thermal logic gates (Pubs. III,II). We remark that all the proposals in this work are only thermally biased, namely they don’t require any external input. They thus acts as energy harvester and may be relevant for a series of applications where on-chip operations are preferred.

The thesis is organized as follows.

Chapter 1 provides the bases of electronic thermal transport in supercon-ducting tunnel circuits. It starts with a short review of the history of conven-tional superconductors, in the framework of the Bardeen-Cooper-Schrieffer the-ory (BCS thethe-ory). In the second section, we expose the main properties of a BCS superconductor in an uniform exchange field. In the following section we discuss the electric and energy transport properties of hybrid tunnel junc-tions due to quasiparticle tunneling. It is shown that a strong thermoelectric effect arises in a tunnel junction between a normal metal and a superconduc-tor with spin-splitted density of states, as long as the transport is spin polar-ized. Then, we consider coherence effects in superconducting junctions. Af-ter discussing both the microscopic and an effective model for the total current in a junction between two superconductors, we move to proximity effects in normal-superconductor clean junctions. We describe the operation of the super-conducting quantum proximity device transistor, which basically acts as a valve for the quasiparticle transport. At the end of the chapter, we discuss the condi-tions for the quasiequilibrium regime and the predominant thermal relaxation mechanisms in hybrid circuits. In Chapter 2 we present two proposals: the first

for a quantum heat engine which delivers power in a contactless way, by ex-ploiting the combination of a thermoelectric effect and the AC Josephson effect, the second regards a refrigerator for the electronic temperature powered by the heating of the thermoelectric element. In Chapter 3 we present our design for the temperature transistor and we discuss both the operation in the amplifica-tion regime, both the implementaamplifica-tion of all the three main logic gates for unitary gain. Chapter 4 is devoted to the electrical characterization of a superconduct-ing quantum interference proximity device with a superconductsuperconduct-ing rsuperconduct-ing made of a aluminum-vanadium bilayer. Finally, in the Conclusions, we resume the main results of the thesis work, discussing their possible applications.

Charge and heat transport in

superconducting tunnel junctions

1.1

History and basic properties of conventional

su-perconductors

The discovery of superconductivity dates back to 1911 [10], soon after the lique-faction of helium. Superconductors display two main properties when cooled down to cryogenic temperatures: i) a vanishingly small resistance, ii) the gener-ation of screening currents in response to an applied magnetic field smaller than a threshold value (called the critical magnetic field), with total expulsion of the field in the bulk. The latter effect was discovered by Meissner and Ochsen-feld in 1933 [11] and led to the phenomenological model by F. and H. Lon-don in 1935 [12]. They expressed the length for magnetic field penetration λL = pm/(µ0nse2) (experimental values range typically between 50 and 500

nm) in terms of the density of "superconducting electrons" ns, of mass m and

charge −e, which can flow with no dissipation and form a different fluid with respect to ordinary electrons (here µ0is the vacuum permeability). Thus, they

re-alized that the superconduncting state can be described as a macroscopic quantum state. A decisive step forward was taken by Ginzburg and Landau (GL) later in 1950 [13], with the introduction of a position-dependent complex pseudowave function Ψ(x) = |Ψ(x)|eiϕ(x)_{, with |Ψ(x)|}2 _{= n}

s(x). Through the minimization of

an appropriate energy density functional written in terms of the order parame-ter Ψ(x), they derived the two equations (GL equations)

1 2m∗[−i~∇ − e ∗ A(x)]2Ψ(x) + β|Ψ(x)|2Ψ(x) = −α(T )Ψ(x) (1.1) J(x) = e ∗ m∗|Ψ(x)| 2 (~∇ϕ(x) − e∗A(x)) (1.2)

where ~ is the reduced Planck constant and α(T ), β are phenomenological pa-rameters (T is the temperature). The second equation gives the electric den-sity current J for particles with effective mass m∗ and charge e∗, whereas the first is a stationary nonlinear Schrodinger-like equation (here A is the magnetic vector potential). They were able to give a phenomenological description of the normal-superconducting transition and to consider non-homogeneous sit-uations. The macroscopic collection of particles share the same phase ϕ and energy E, thus evolves in time coherently i~Ψ0(x, t) = EΨ(x, t)(here the prime mark denotes the first derivative with respect to time). This is not the case for a normal conductor, where the current is carried by quasiparticle excitations be-having similarly to free electrons. Due to the Fermi distribution, all the carriers have different energy, and hence a different (incoherent) time evolution. Con-versely, it is well known that a collection of bosons can share the same quantum state: this is known as Bose-Einstein condensation [14]. In fact, the origin of superconductivity, according to the Bardeen-Cooper-Schrieffer (BCS) theory, is due to the instability of the Fermi surface under the formation of bound states between electrons, called Cooper pairs. This happens whenever an effective at-tractive interaction, of arbitrary strength, is established between the electrons. In the BCS model, this attractive interaction is mediated by a weak electron-phonon interaction, characterized by a coupling constant −g < 0. The inte-raction is spin-singlet and characterized by a non-zero value of the correlation function ∆ = gP

khc−k↓ck↑i, which is proportional to the order parameter of the

GL theory Ψ ∝ ∆, as shown by Gorkov in 1959 [15] (here hi denotes the thermal average and ckσ is the annihilation operator for an electron with momentum k

and spin σ). The class of materials which are well described by the BCS model-ing is typically denoted as "conventional" or "classical" superconductors, which typically have low critical temperature <20 K. In this thesis we will always refers to them, since we are interested in dirty thin films [16] made through evapora-tion or sputtering of elements like aluminum (Al), vanadium (V), and niobium (Nb), which fall in the previous class. We will not discuss high temperature superconductors, or exotic superconductors with different kind of pairing.

1.2

BCS superconductors in the presence of a

uni-form exchange field

As already stated in the previous section, the expulsion of the magnetic field in the bulk (Meissner-Ochsenfeld effect) is a fundamental feature of the super-conducting state. By increasing the magnetic field, the energy associated with

0.5 1.0 0 0.5 1.0 

[

]

[

]

[

]

[

]

Figure 1.1: Two main features of BCS superconductors in a effective exchange field. a. Tem-perature dependence of the BCS order parameter for some values of heff ≤ 0.5∆0, obtained from Eq. (1.3). In the inset the phase diagram in the heff-T plane is dis-played either for a second order or a first order transition. b. Spin-splitting (red and blue lines) of the normalized quasiparticle DOS in an ideal BCS supercon-ductor (black lines). The dashed line (olive) expresses the subgap rounding in the presence of a phenomenological Dynes parameter Γ = 10−4∆0at T = 0.01 Tc0for heff = 0.

the screening currents increases up to a value, known as critical magnetic field, where the superconducting state is no longer energetically convenient. In ad-dition to this effect, called orbital depairing, the magnetic field couples to the spin of the electrons in the superconductor, by shifting the energy of the two spin bands (paramagnetic effect). As mentioned before, in a conventional su-perconductor each pair is composed of electrons with opposite spin, therefore the condensation energy of the superconducting state decreases. The orbital de-pairing is dominant either for bulk samples or for thin layers in a out-of-plane magnetic field. Conversely, the paramagnetic depairing is important for thin films when the magnetic field is applied in-plane, because the field penetrates uniformly the film, thus the screening effects are strongly suppressed. More-over, the paramagnetic effect matters for superconductors in close proximity with ferromagnetic elements, due to the effective exchange interaction of the spin of the conducting electrons in the superconductor and the localized mag-netic moments of the ferromagmag-netic element. The perturbation caused by this interaction decays over the superconducting coherence length ξs [17]. In this

section and in the rest of this thesis, we neglect the orbital depairing and con-sider a superconductor in an uniform effective exchange field acting only on the electron spin. This approximation holds well for films of thickness smaller than: i) the magnetic penetration length, when discussing the action of an external

magnetic field; ii) the superconducting coherence length ξs for a

superconduct-ing layer in proximity with a ferromagnetic material [17, 18]. The inclusion of a uniform exchange field in the BCS hamiltonian is straightforward and it was done back in the sixties [19, 20], yet it introduces new important features.

The first regards the superconducting-normal transition, which depends on the effective exchange field heff induced in the superconductor. Since we

con-sider here a single superconductor and we neglect the orbital effect, we can fix the gauge in order to have a real-valued order parameter ∆. The self consistency equation for ∆(heff, T )reads [19]:

1 λ = Z ~ωD 0 d 1 2p2 _{+ ∆}2_(h eff, T ) X σ=±1 tanh p 2_{+ ∆}2_(h eff, T ) − σheff 2kBT ! , (1.3)

where kBis the Boltzmann constant,  is the energy of an electron in the

conduc-tion band of the normal metal measured with respect to the chemical potential, ωD is the frequency cutoff for the electron-phonon coupling and σ is the spin

label (we will use the notations σ = ±1 or σ =↑, ↓ in an interchangeable way). In the derivation of this expression, the density of states (DOS) in the normal state νNis assumed to be constant and equal to ν0 = νN(EF), where EFis the Fermi

en-ergy. In fact, for low temperature superconductors kBT  ~ωD  EF, thus the

variation of νNon this small energy scale is totally negligible and λ = gν0. In the

absence of the field heff = 0, the order parameter ∆(0, T ) decreases

monotoni-cally with T and becomes zero when T approach the critical value Tc0, and a

sec-ond order phase transition to the normal state occurs (black curve in Fig. 1.1a). In the weak coupling limit λ  1 considered here, the zero temperature value of the order parameter ∆0 = ∆(0, 0)is proportional to the critical temperature,

namely

∆0 =

π

eγkBTc0' 1.764kBTc0, (1.4)

where γ is the Euler constant. If we assume a second order phase transition for arbitrary heff, by setting ∆(heff, T ) = 0in Eq. (1.3) and solving for T = Tc(heff), we

get the red curve in the inset of Fig. 1.1a, which is multi-valued in heff. However

the solution corresponding to the reentrant segment does not correctly minimize the free energy [19]. If the Eq. (1.3) is solved consistently with a zero free energy difference between the normal and the condensed state, one gets the black curve in the inset of Fig. 1.1a (the two curves coincide for heff . 0.6 ∆0). Thus the

tran-sition is of second order for heff . 0.6 ∆0and of first order, i.e. the order

param-eter remains finite at the transition, for larger fields [19, 20]. Note that the stable solution for the critical temperature decreases monotonically with heff and it is

zero at heff = ∆0/

√

2 ∼ 0.707 ∆0. This value, known as Chandrasekhar-Clogston

analyzed in this thesis, we will always consider exchange fields smaller than 0.5 ∆0, where the transition is of second order. The temperature dependence of

∆(heff, T ), obtained by solving self-consistently Eq. (1.3), is displayed in Fig. 1.1a

for some values of heff ≤ 0.5 ∆0. At T = 0, the order parameter is equal to ∆0

independently of heff and it decreases monotonically by raising the temperature

and it is zero at T = Tc(heff), as already discussed. To conclude this

discus-sion, we remark that the non-homogeneous Fulde-Ferrell-Larkin-Ovchinokov state [23, 24] is not considered here, since it is strongly suppressed and hardly detectable in dirty thin films as the ones we are interested in.

A second feature regards the distribution of elementary excitations (quasi-particle excitations). In the absence of an exchange field heff = 0, the

quasi-particle spectrum is spin-degenerate, with energy E = p2_{+ ∆}2_{(0, T ). The}

quasiparticle DOS for single spin band is νBCS(E, T ) = ν0|<[G(E, T )]| = ν0      < " E + iΓ p(E + iΓ)2_{− ∆}2_{(0, T )} #     , (1.5)

where G(E, T ) = (E + iΓ)/p(E + iΓ)2_{− ∆}2_{(0, T ), with Γ = 0}+_{, is the normal}

Green’s function for an ideal BCS superconductor [25]. As discussed above, the DOS in the normal state is set to νN = ν0, hence it is convenient to consider a

normalized dimensionless DOS, namely NBCS(E, T ) = νBCS(E, T )/ν0, which is

displayed at T = 0 in Fig. 1.1b (black solid curve). Note that the quasiparticle DOS has a gap of size 2∆ around the Fermi level. These missing states are shifted on the edge of the gap region, where peaks appear. For this reason the absolute value of the order parameter is often called energy gap. However, we remark that the correspondence between the order parameter and the gap in the DOS is peculiar of the BCS theory. It is no longer valid for heff 6= 0 and more

generally in the presence of the proximity effect, discussed in the next section. When an exchange field is present, the spin-degeneracy is removed and both the quasiparticle spectrum Eσ = p2 + ∆2(heff, T ) − σheff and the normalized

DOS Nσ(E, T, heff) = NBCS(E + σheff, T )are spin-splitted by a Zeeman effect [in

the latter expression ∆ = ∆(heff, T )]. The DOS for the spin up and spin down

are the blue and the red curves shown in Fig. 1.1b, respectively. Note that the effective gap in the DOS in now reduced to 2(∆0 − heff). This spin splitting

plays a crucial role in the thesis, because it breaks electron-hole symmetry in the DOS for each spin band. When combined with a spin-filtering mechanism, it produces a strong thermoelectric effect in hybrid superconducting structures, as discussed in the next section.

The spectral properties of the superconducting DOS can be investigated by measuring the charge current due to quasiparticle tunneling flowing through a

junction made between a normal metal N and a superconductor S by means of a thin insulating layer I (NIS junction), as we will argue in the next sec-tion. However, in real junctions, the conductance for energy smaller than ∆ is larger than expected. This effect can be included by considering a finite value for Γ in Eq. (1.5), as first modeled by Dynes and Garno [26, 27]. De-spite some very recent claims of a microscopic interpretation [28], this param-eter is still phenomenological at the present time. In particular, it accounts for all kinds of mechanisms (for instance, environmental photon-assisted tunnel-ing [29]) that can create a quasiparticle lifetime broadentunnel-ing and a finite amount of available effective sub-gap states. Typical values for realistic tunnel junctions are Γ = 10−5 − 10−2_∆

0 [4, 29, 30]. In the following we will always implicitly

assume a finite Γ parameter in the expression of the superconducting DOS, by making the formal replacement NBCS(E, T ) → NS(E, T ) (see the dashed

olive curve in Fig. 1.1b for heff = 0). Furthermore, to make the notation less

cumbersome, we will drop the exchange field dependence of the DOS of the spin-splitted superconductor. To keep track of this dependence, we will replace Nσ(E, T, heff) → NσSm(E, T ), where Smstands for magnetized superconductor.

In the discussion of this section, we neglected spin-relaxation phenomena as the spin-flip and the spin-orbit scattering [6, 18, 31], which modifies qualita-tively the picture. The former is typically related to the presence of magnetic impurities and has a strong effect on the superconducting state: for large val-ues produces gapless superconductivity (and the transition becomes of second order at each exchange field heff). The latter does not affect directly the

super-conducting state for heff = 0, but it is detrimental for the effects which we exploit

in our proposals at finite exchange field because i) it rounds the superconduct-ing DOS, and ii) for a large couplsuperconduct-ing it can suppress the spin-splittsuperconduct-ing in the DOS. However, the simpler description of this section is still relevant for this work. In fact, the superconducting structures discussed in our proposals are made of aluminum (Al), which is characterized by ν0/V ' 1.6 × 1047J−1m−3 (V

is the volume), EF ' 11.7 eV [32] and a small spin-orbit coupling [33, 34], due to

its low atomic number. In fact, the spin-splitting of the DOS in Al has been first observed a long time ago [35] and it is reasonably described by the model of this section. The size of the energy gap in Al depends on the film thickness [36–40] and varies from ∆0 = 180 µeV for a bulk sample [41] up to more than 450 µeV

(in a 2-nm-thick film, as shown for instance in Ref. 42), corresponding to critical temperatures ranging from 1.2 K to 3.0 K. Used in combination with magnetic insulator like europium sulfide (EuS) [43], Al displays a strong exchange field heff ∼ 100 µeV [44]. The assumption about uniform magnetization holds well in

range 30 − 60 nm [45].

1.3

Quasiparticle transport in hybrid normal-

super-conducting tunnel junctions

The investigation of the charge transport in tunnel junctions provides a power-ful method to investigate the spectral properties of mesoscopic systems. A tun-nel junction is a connection between two conducting electrodes realized through a thin insulating barrier (∼ nm), where particles can flow thanks to the quantum tunnelling but classical transport is forbidden, due to the high energy barrier. In the following we consider two electrodes, named left (L) and right (R), either in thermal equilibrium or quasiequilibrium [46], where the electronic distributions f (E − µL,R, TL,R)are expressed by Fermi-Dirac functions:

f (E − µL,R, TL,R) =

1

exp[(E − µL,R)/(kBTL,R)] + 1

, (1.6)

which define the electronic temperature TL,R and the chemical potential µL,R of

the electrodes. The validity of this description involving Fermi-Dirac distribu-tion funcdistribu-tions will be discussed in Sect. 1.5. When a voltage V is applied be-tween L and R, the chemical potentials shift with respect to each other µL− µR =

−eV , where −e is the electron charge. Since the transport properties depend only on this difference, in this thesis we follow the standard notation to mea-sure the energy with respect to the chemical potential of the right electrode, namely we set µL = −eV and µR = 0. In this section we only consider a direct

current (DC) voltage bias. The effects of an alternating current (AC) bias are briefly discussed in Sec. 1.4.2 and in the next chapter.

For the purpose of this thesis, we consider a spin-conserving and elastic tun-nelling process, but we retain the possibility of a spin-dependent transport, in order to include also magnetic junctions. Note that, by assuming equal spin populations in Eq. (1.6), we neglect any kind of spin accumulation mechanism. The rigorous treatment for the tunneling transport is presented in Appendix A. In this section we focus only on the quasiparticle contribution, that can be de-rived by means of a phenomenological model, first proposed by Giaever and Megerle in 1961 [47], which deals with normal electrons rather than quasipar-ticles. In this respect, we remark that, in this work, the word "electron" and its derivatives are sometimes used for convenience to indicate both the carriers in a normal metal and the quasiparticle excitations in a superconductor. According to the Fermi Golden rule, the flow of electrons with energy E and spin σ from

the L to the R electrode for unit time is [25] IσL→R(E) =

2π ~

|Tσ(E)|2f (E + eV, TL)νσL(E + eV, TL)[1 − f (E, TR)]νσR(E, TR)

(1.7) Here, Tσ(E) is the probability amplitude for a transmission between states of

equal energy E and same spin σ and νσL,R(E, TL,R) is the spin DOS in the left,

right electrode. Hence, f (E + eV, TL)νσL(E + eV, TL)is the number of available

electrons on the L electrode and [1−f (E, TR)]νσR(E, TR)the number of available

states in the R electrode. An analogous expression can be written for IσR→L(E)

by exchanging L ↔ R, E ↔ E + eV : the net flow is IσL→R(E) − IσR→L(E)and

the total particle current is obtained by integrating over E and summing over σ. The electrons have charge −e and energy E which must be included in Eq. (1.7) when computing the correspondent current. The quasiparticle charge reads (the symbol I in the subscript stands for insulator)

I_LIRqp (V, TL, TR) = X σ Gσ e Z ∞ −∞

dENσL(E + eV, TL)NσR(E, TR)

× [f (E, TR) − f (E + eV, TL)] (1.8)

and the energy current flowing out of the left electrode is ˙ Qqp_LIR(V, TL, TR) = X σ Gσ e2 Z ∞ −∞

dE(E + eV )NσL(E + eV, TL)NσR(E, TR)

× [f (E + eV, TL) − f (E, TR)]. (1.9)

Here NσL,R(E, TL,R)are the spin DOS in the L, R electrode normalized in terms

of the DOS at the Fermi level νσL,R(EF), and Gσ = 2π|Tσ|2νσL(EF)νσR(EF)e2/~

is the spin σ conductance in the normal state. In the derivation of these expres-sions, the energy dependence of the transmission probability Tσ(E) has been

neglected, since we consider cryogenic temperatures kBT  EF and low

volt-ages eV  EF. In our notation, the conventional direction for the charge and

the heat current follows the order of the symbols in the subscript: for instance for the subscript LIR the current is positive if it flows from L to R, negative otherwise. As a consequence, we have I_RILqp = −I_LIRqp . On the other hand, due to the energy conservation, the heat current flowing out of the R electrode is

˙

Qqp_RIL = ˙W − ˙Qqp_LIR, where ˙W = −I_LIRqp V is the power associated with the work done by the junction. In the remainder of this section, we first consider the el-ementary case of a tunnel junction between two normal metals (N1IN2), then

we focus on the hybrid normal-superconductor junction either in the absence (NIS) or in the presence (NFISm) of a magnetic barrier. As mentioned before, the

subscript "m" is used here and in the following to remark that the DOS in the su-perconductor is spin-splitted by the proximity of the ferromagnetic insulator FI.

Both the hybrid junctions show peculiar transport properties, as the NIS cool-ing, which has been exploited in Pub. I. However, only in the magnetic junction a thermoelectric (TE) effect is generated, thanks to the breaking of electron-hole symmetry in the tunneling transport.

1.3.1

N

IN

junctions

As previously stated, for eV  EF, the DOS of a N electrode can be set to its

value at the Fermi level, namely the rescaled DOS reads NN(E) = 1. In this

case the transport is not spin-polarized, hence G↑ = G↓ = GT/2(here and below

GT = G↑ + G↓ is the total conductance in the normal state) and we can rewrite

the charge current Eq. (1.8) as: IN1IN2(V ) =

1 eRT

Z ∞

−∞

dE[f (E, TN2) − f (E + eV, TN1)] =

V RT

, (1.10)

since the integral R_−∞∞ dE[f (E, TN2) − f (E + eV, TN1)] = µN2 − µN1 = eV

irre-spective of TN1,N2 [25]. Thus, the current is proportional to the applied voltage,

as in a standard Ohmic resistor, with resistance RT = G−1T . The heat current of

Eq. (1.9) reduces to the following expression: ˙ QN1IN2(V, TN1, TN2) = 1 e2_R T Z ∞ −∞

dE(E + eV )[f (E + eV, TN1) − f (E, TN2)], (1.11)

which can be evaluated explicitly (see Appendix B) ˙ QN1IN2(V, TN1, TN2) = L0 2R_T(T 2 N1 − T 2 N2) − V2 2R_T, (1.12)

where L0 = π2kB2/(3e2) is the Lorenz number. Upon setting V = 0, i.e. by

considering a pure temperature bias, we recover the well-known Wiedemann-Franz law [48]. As a consequence, the ratio between the thermal conductance K = d ˙QN1IN2/dT and the electric conductance GTis proportional to the

temper-ature, namely K/(GTT ) = L0. This result reflects the fact that we are analyzing

the transport between two normal metals, where the charge carriers and the energy carriers are the same kind of particles.

1.3.2

NIS junctions

The quasiparticle charge current in a NIS junction is given by the following ex-pression: INIS(V, TN, TS) = 1 eRT Z ∞ −∞

-1

0

1

-1

0

1

I

[



/(

eR

)]

V [



/e]

0.0

0.5

1.0

1.5

-3

0

3

6

T / T

V [



/e]

Q

[



/(

e

R

)]

x10

Figure 1.2: Electric and energy transport in a voltage biased NIS junction. a. Normalized current-voltage characteristics of a NIS junction for different values of TN = TS= T. b. Normalized energy current flowing out of the N electrode of the NISm junction vs. V for the same values of TN = TS = T. In both panels, we set Γ = 10−4∆0.

Figure 1.2a shows the INIS-V characteristics for some values of TN = TS = T.

Note that the current is odd in V , namely INIS(V, TN, TS) = −INIS(−V, TN, TS),

due to the electron-hole symmetry of the superconducting DOS NS(E, TS) =

NS(−E, TS). The current is strongly suppressed for V < ∆0/e at low

tempera-tures T  Tc0, due to the gap in the DOS of the S electrode, leading to a

dif-ferential conductance G(V < ∆0/e) ' ΓGT/∆0. Upon raising the temperature,

the INIS-V characteristic gets increasingly more rounded and linear, due to the

broadening of the Fermi distributions and the suppression of the order param-eter. When T → Tc0, the current approaches the Ohmic behaviour of the NIN

junction.

Let’s consider now the energy transfer in a NIS junction. The energy re-moved from the N electrode in the unit time is given by the following formula:

˙ QNIS(V, TN, TS) = 1 e2_R T Z ∞ −∞

dE(E + eV )NS(E, TS)[f (E + eV, TN) − f (E, TS)].

(1.14) Due to the electron-hole symmetry of NS(E, T ), ˙QNIS(V, TN, TS)is an even

func-tion of the voltage bias V . Figure 1.2b displays the behavior of ˙QNIS as a

func-tion of V > 0 for the same values of TN = TS = T of Fig. 1.2a. Interestingly,

˙

QNIS is non-monotonic both in the temperature and in the applied voltage. For

and increasing for voltages approximately smaller than ∆(0, T )/e. After reach-ing a maximum, ˙QNIS decreases and becomes negative at higher voltages, thus

yielding heating of the N electrode. This heat removal is due to the presence of the superconducting energy gap, which provides an energy filter allowing the transmission of only hot quasiparticles of energy E > ∆(0, T ). This explains why there is an optimal temperature for refrigeration, T ' 0.45 Tc0 [46]. At

low temperatures T  Tc0 only a vanishing small number of quasiparticles is

available for refrigeration, whereas for T → Tc0 the suppression of the order

parameter destroys the effect. In fact, in a NIN junction a voltage bias always produces heating [i.e, Joule heating ˙QN1IN2(V, T, T ) = −V

2_/(2R

T)]. This

refrig-eration mechanism, known as NIS cooling, was first demonstrated in 1994 [49]. Since then, it has been extensively studied both from a theoretical and an ex-perimental point of view [46, 50], in order to increase the cooling power (for the state of the art, see Refs. 51, 52). In this direction it is more effective to connect two NIS junctions back to back and realize a SINIS cooler [53], which yields double cooling power. The latter is exploited in the proposal of Pub. I to cool down a normal metal island thanks to the heat injected in a thermoelectric ele-ment, as discussed in Sec. 2.2. On the other hand, it is not possible to cool down the superconducting electrode of a NIS junction by voltage biasing, namely the heat current from the S electrode ˙QSIN = − ˙QNIS − INISV is always negative for

TN = TS (see inset of Fig. 1.3b). It is worth mentioning that a temperature

bi-ased (TS 6= TN) NIS junction shows remarkable properties even at zero voltage

bias. In fact, | ˙QNIS(0, TN, TS)|(and so | ˙QSIN|) is not symmetric under the exchange

TN↔ TS, due to the temperature dependence of ∆. As a consequence, the

mag-nitude of the heat current flowing from the hot to the cold electrode of the NIS junction depends on which electrode (either N or S) is heated up. Thus the NIS junction behaves as a intrinsic thermal diode [54, 55].

1.3.3

NFIS

junctions

Here we consider a tunnel junction which is only slightly different from the previous one, namely the insulator is replaced with a ferromagnetic insulator. Interestingly, the physical properties of the junction change quite dramatically. First, since the insulator is magnetic, the height of the energy barrier for tunnel-ing depends on the spin, thus G↑ 6= G↓. With no loss of generality, we can set the

FI magnetization along the z axis and introduce a parameter P = (G↑− G↓)/GT

to describe the degree of polarization of the barrier. Furthermore, due to the proximity of the FI, the quasiparticle DOS in the superconductor is Zeeman-splitted by the exchange interaction heff, as discussed in the previous section.

-1

0

1

-3

0

3

6

9

V [



/e]

Q

[



/(

e

R

)]

x10

-2

-1

0

1

2

-1.0

-0.5

0.0

0.5

1.0

I

[



/(

eR

)]

V [



/e]

Figure 1.3: Electric and energy transport in a voltage biased NFISmjunction. a.Normalized current-voltage characteristics of a NFISmjunction for different values of heffboth at intermediate P = 0.3 (dashed) and full P = 1 (solid) spin polarizations. Here we set TN = TS = 0.01 Tc0. The NIS characteristic at the same temperature is shown for a comparison (black). b. Normalized energy current flowing out of the N electrode of the NFISmjunction vs. V for the same values of heff at TN = TS= 0.45 Tc0. Here we set P = 0.9. In the inset we plot the corresponding normalized energy current flowing out of the S electrode of the NFISmjunction. In both panels, we set Γ = 10−4_∆

0.

The quasiparticle charge currents now reads [7, 56] INFISm(V, TN, TS) = X σ Gσ e Z +∞ −∞

dENσSm(E, TS)[f (E, TS) − f (E + eV, TN)]

(1.15)

= 1

eRT

Z +∞

−∞

dE[N+(E, TS) + P N−(E, TS)][f (E, TS) − f (E + eV, TN)]

(1.16) where in the second expression N±(E, TS) = [N↑Sm(E, TS) ± N↓Sm(E, TS)]/2and

the dependence on P is made explicit. Note that both a finite polarization P 6= 0 and a spin splitting heff 6= 0 [hence N−(E, TS) 6= 0] are required to break the

electron-hole symmetry of the quasiparticle transport across the barrier. This concept is important for the TE effect discussed below. Let’s first consider a zero thermal gradient TS = TN= T.

Figure 1.3a displays the INFISm-V characteristics for some values of heff for

P = 0.3(dashed) and P = 1 (solid) at T = 0.01 Tc0. Note that the characteristics

charac-teristic is displayed in black for a comparison). At intermediate polarization, the contribution from the two spin bands increases the subgap current. However, for a totally polarized system, the characteristics are simply shifted of a quantity heff/ewith respect to the NIS characteristic. Note that in the latter case the

con-ductance is exactly one half with respect to the unpolarized situation, though, because for P = 1 we have G↓ = 0, hence the total conductance in the normal

state is GT(P = 1) = G↑ = GT(P = 0)/2. Before moving to the TE effect, we

discuss the thermal transport in a voltage-biased NFISmjunction. The

quasipar-ticle heat current flowing out of the N element, i.e. the cooling power of the N element, reads ˙ QNFISm(V, TN, TS) = 1 e2_R T Z +∞ −∞

dE(E + eV )[N+(E, TS) + P N−(E, TS)]×

× [f (E + eV, TN) − f (E, TS)]. (1.17)

Figure 1.3b shows the voltage dependence of ˙QNFISm for the same values of

heff of Fig. 1.3a for P = 0.9 at T = 0.45 Tc0, where the NIS cooling mechanism

is optimized. The electron-hole asymmetry produces either heating ( ˙QNFISm < 0

for V > 0) or cooling ( ˙QNFISm > 0for V < 0) of the N electrode at small absolute

voltage |V |. As a consequence, unlike the ordinary NIS cooling mechanism, we can have also refrigeration of the S electrode. The corresponding cooling power of the S electrode ˙QSmFIN = − ˙QNFISm − INFISmV is displayed in the inset

of Fig. 1.3b. This electron temperature cooling mechanism occurring in NFISm

element has been only very recently investigated [57] and it may be relevant for applications where the refrigeration of superconductor is required. For smaller values of P , the picture is qualitatively similar. The degree of asymmetry of the curves is lowered and hence the cooling (for V < 0) and the heating (for V > 0) power of the N island at small absolute voltage are reduced, as well as the maximum cooling power of the N island for V < 0.

Let’s now include a thermal gradient across the junction TS 6= TN. Even at

V = 0 the second term in the square brackets of the quasiparticle current of Eq. (1.16) is different from zero [since N−(E, TS) 6= N−(−E, TS)], hence a

ther-moelectric effect occurs. This effect has been first predicted only a few years ago [7] and it is has already been proved experimentally [8, 9]. In these works, a slightly different structure has been investigated, namely a ferromagnetic con-ductor tunnel coupled to a superconcon-ductor with spin-splitted DOS. Despite that, the origin of the TE effect is still the same, namely the breaking of the electron-hole symmetry of the tunneling transport, which in the latter comes from the different DOS of the two spin bands in the ferromagnetic element [18].

0.0 0.1 0.2 0.3 T_N=T_hot -0.6 -0.3 0.0 0.3 0.6 -0.3 -0.2 -0.1 0.0 V [₀/e] I NFISm [  0 /( eR T )] Thot / Tc0= 1.00 0.75 0.60 0.45 0.30 0.15 T_S=T hot b FI S_m N a 1-P N_ 0.0 0.3 0.6 0.9 -0.6 -0.4 -0.2 0.0 0.2 0.1 0.3 T_S=T_hot T_hot [T_c0] I NFISm [  0 /eR T ] V [ 0 /e ] 0.3 0.6 0.9 0.6 0.4 0.2 0.0 -0.2 -0.1 -0.3 T_N=T hot T_hot [T_c0] Shortcircuit (V=0) Open circuit (I=0) Load (R_L=2R_T) c N_ ↓ N ↓ N

Figure 1.4: Thermoelectric effect in a NFISmjunction. a. Origin of the thermoelectric (TE) current in the semiconductor model. The spin current in the up band at high energy it is not fully compensated by the spin down current, due to the spin-filtering by the magnetized barrier. b. Normalized current vs normalized voltage for Tcold = min(TS, TN) = 0.01 Tc0 and different values of Thot = max(TS, TN), either for TS= Thot(top) or TN= Thot(bottom). The dashed lines are useful to vi-sualized the Seebeck and the short-circuit operation, given by the interceptions of the characteristics with the horizontal and the vertical line, respectively. The solid black line set the operation of the TE element when it is connected to a load resistor characterized by RL = 2RT. c. Normalized current (red) and normalized voltage (blue) vs Thotfor TS= Thot, TN = 0.01Tc0(left panel) and TN= Thot, TS= 0.01Tc0 (right panel). The solid lines are the current at V = 0 and the thermo-voltage at open circuit. The corresponding dashed lines are the thermo-current and the thermo-voltage for a finite load of resistance RL= 2RT. In all the panels, we set heff = 0.4∆0, P = 0.9 and Γ = 10−4∆0.

of the TE in the NFISm element, as displayed in Fig. 1.4a. For simplicity, we

consider V = 0 and TS = 0. Due to the spin splitting of the DOS in the S

electrode, the hot quasiparticle in the N electrode can flow only in the spin-up band (spin-upper black line). For the same reason, only the quasiparticle of the S electrode in the spin-down band can fill the corresponding holes in N (lower black line). However, due to the magnetization of the barrier, the latter current is reduced by a factor 1−P , thus a finite charge current is generated. As mentioned before, we remark that it is the combination of the spin-splitted DOS in the S electrode and the spin filtering that produces the TE effect. In the absence of the latter, only a thermospin current is generated [18].

Figure 1.4b displays selected portions of the INFISm-V characteristics for TN6=

TS, in order to focus on the TE effect. We set the exchange field to heff = 0.4 ∆0

and the polarization to P = 0.9. In the top panel we fix TN = 0.01 Tc0 and we

consider different values of TS > TN. Conversely, in the bottom panel, TS =

0.01 Tc0 and TN > TS runs over the same values of TS in the top panel. In each

panel, we describes three operation modes [56] for the TE element: i) thermo-current at zero voltage (short-circuit), identified by the interception of the curves with the vertical dashed lines, ii) thermo-voltage at open circuit, given by the crossing with the horizontal dashed lines (Seebeck regime), iii) general close circuit operation, given by the interception with a generic load line (solid black curve). These quantities are plotted against Thot = max(TS, TN) for TS > TN

(left panel) and TN > TS (right panel) in Fig. 1.4c. Let’s consider the first case

(solid red lines). The sign of the thermo-current (positive for TS = Thot and

negative for TN = Thot) depends on the electron-nature of the charge carriers

and on our convention on the current sign (positive if it flows from N to S). Note that the absolute thermo-current increases monotonically with TNfor TN= Thot,

whereas is not monotonic in TS for TS = Thot, due to the critical temperature

Tc(heff = 0.4 ∆0) ∼ 0.87 Tc0 of the Sm electrode (see Fig. 1.1). In fact, no TE

occurs when the Smelectrode is in the normal state. A similar discussion holds

for the thermo-voltage at open circuit (solid blue lines), which is positive for TN = Thot and negative for TS = Thot, in order to balance the diffusive charge

current in the structure. Again, the absolute thermo-voltage is monotonic in TN

for TN = Thot and non-monotonic in TS for TS = Thot, for the aforementioned

reason. In the generic close circuit operation (dashed lines) both the current and the voltage are finite. In particular, due to the monotony of the INFISm-V

characteristics, both the charge current and the voltage are smaller in absolute value than the thermo-current in the short-circuit case and the thermo-voltage in the Seebeck regime, respectively. The features of the TE in the NFISmelement

To conclude this discussion, we remark that high values of spin-polarization were reported for ferromagnetic barriers bases on europium chalcogenides [58], with values up to ∼ 85% for EuS and larger than ∼ 97% in europium selenide (EuSe) [18, 59].

1.4

Coherence effects in superconducting weak links

In the previous section we investigated the quasiparticle transport in hybrid normal-superconducting tunnel junction. When two superconductors are cou-pled through a weak connection, for instance a thin insulating barrier (SIS junc-tion) or a normal metal wire (SNS juncjunc-tion), an additional transport mechanism arises. In these situations, as first predicted by Josephson in 1962 [60] and sub-sequently verified [61] for a SIS junction, the two superconducting condensates interact and a dissipationless current can flow, even in the absence of a voltage (or a temperature) bias. In this section we first consider a SIS junction and de-scribe the coherent transport both in the DC and the AC regimes. In the latter case, we present a standard effective model, called resistively and capacitively shunted junction model (RCSJ model) [25, 41], for the description of the dynam-ics. Then we consider a SNS weak link, where coherent effects are induced by the proximity effect. This structure is the fundamental element in a supercon-ducting proximity interference transistor (SQUIPT) [62], which acts a valve for quasiparticle transport, as shown in the end of the section.

1.4.1

Electrical transport in S

IS

junctions

Let’s consider two tunnel-coupled superconducting electrodes S1 and S2, with

generic energy gaps ∆0,S1 and ∆0,S2 (corresponding to the critical temperatures

Tc0,S1 and Tc0,S2). The total charge current through the junction for a DC voltage

bias VJis [25] (see also Appendix A)

IS1IS2(VJ, TS1, TS2, ϕJ) = I qp

S1IS2(VJ, TS1, TS2) + I ph

S1IS2(VJ, TS1, TS2, ϕJ). (1.18)

where ϕJ = ϕS1 − ϕS2 is the difference between the phases of the two

super-conducting condensates ϕS1, ϕS2. The first term on the right side of Eq. (1.18) is

the quasiparticle contribution of Eq. (1.8), with the proper substitutions of the superconducting DOS. Figure 1.5a shows the I_Sqp_1IS2-V characteristics for some values of TS1 = TS2 = T. The curves display some similarities with the ones of

the NIS junction. In particular, at low temperatures T  Tc0,S1, Tc0,S2, the current

is strongly suppressed for |VJ| < (∆0,S1 + ∆0,S2)/e and the behavior is Ohmic

-1

0

1

-1

0

1

T / T

I

[



/(

eR

)]

V [



/e]

0,5

1,0

0

0,5

1,0

1,5

I

[



/(

2

eR

)

]

T

[

T

]



/

=

Figure 1.5: Electric transport in a S1IS2 junction. a. Normalized quasiparticle current-voltage characteristics for some values of TS1= TS2= Tand for ∆0,S2/∆0,S1 = 0.5. The peaks which appear at intermediate temperatures are due to the match-ing of the BCS smatch-ingularities in the DOS. b. Normalized Josephson critical cur-rent Ic(T ) vs. T for different values of ∆0,S2/∆0,S1. In both panels, we set ΓS1/∆0,S1 = ΓS2/∆0,S2 = 10

−4_.

peaks centered in VJ = ±|∆S1(0, T ) − ∆S2(0, T )|/e appear at intermediate

tem-peratures, due to the matching of the BCS singularities in the quasiparticle DOS. This creates a region |VJ| & |∆S1(0, T ) − ∆S2(0, T )|/eof negative differential

con-ductance [41].

The second term on the right side of Eq. (1.18), first predicted by Josephson, gives the phase coherent contribution to the charge current, namely

I_Sph

1IS2(VJ, TS1, TS2, ϕJ) = IJ(VJ, TS1, TS2) sinϕJ+ Iint(VJ, TS1, TS2)cosϕJ. (1.19)

where the time evolution of the phase difference is given by dϕJ

dt = 2e

~VJ. (1.20)

In the first term on the right side of Eq. (1.19) the amplitude is IJ(VJ, TS1, TS2) = 1 πeRT p.v. Z Z ∞ −∞ dEdE0MS1(E, TS1)MS2(E 0_{, T} S2) E − E0_{− eV} J ×[f (E, TS1) − f (E 0 , TS2)], (1.21)

where MS1,S2(, TS1,S2) = |<[FS1,S2(E, TS1,S2)]| are the Cooper pair BCS DOSs

and FS1,S2(E, TS1,S2) = ∆S1,S2(0, TS1,S2)/

q

(E + iΓS1,S2) 2_{− ∆}2

anomalous Green’s functions in the superconductors [25]. Above p.v. denotes the principal value of the integral. By inspection, one can see that IJis different

from zero even in the absence of a voltage or a temperature bias, hence a non-dissipative current (proportional to the sine of the phase difference) can flow in a SIS junction, due to the tunneling of Cooper pairs (DC Josephson effect). In particular, for VJ= 0and TS1 = TS2 = T, we have

Ic(T ) = IJ(0, T, T ) = 1 2πeRT     Z ∞ −∞ dE=FS1(E, T)F ∗ S2(E, T) [1 − 2f (E, T )]     . (1.22) This quantity gives the maximum current which a SIS junction can support at VJ= 0, and it is called Josephson critical current.

Figure 1.5b displays the temperature evolution of Ic(T ) for some values of

∆0,S2/∆0,S1. The current is maximum at T = 0, where it approximately reads [25]

Ic(0) ' π~∆0,S1∆0,S2/[eRT(∆0,S1 + ∆0,S2)], it decreases monotonically with the

temperature, and it is zero for T = min(Tc0,S1, Tc0,S2). For ∆0,S1 = ∆0,S2 the

crit-ical current is given by the Ambegaokar-Baratoff expression [63] , i.e. Ic(T ) =

[π∆(0, T )/(2eRT)]tanh[∆(0, T )/2kBT ], which vanishes at Tc0,S1with a finite slope.

For any different ratio of the two energy gaps, the slope of Ic(T ) at

T = min(Tc0,S1, Tc0,S2)is infinite.

For VJ 6= 0, the phase evolves linearly in time, according to Eq. (1.20). As a

consequence, the current oscillates in time with frequency νJ = 2eVJ/h ∼ 0.48

GHz/µV×VJ (AC Josephson effect) and zero mean value [41, 60]. This

DC-to-AC conversion is used in Pub. V to design an engine that transmits power in a contactless way, as discussed in the next chapter.

The second term on the right side of Eq. (1.19) is given by the following expression Iint(VJ, TS1, TS2) = 1 eRT Z ∞ −∞

dEMS1(E + eV, TS1)MS2(E, TS2)

× [f (E, TS2) − f (E + eV, TS1)]. (1.23)

This term is zero at VJ = 0, hence it only gives a dissipative contribution to the

charge current. In the literature it is known as "quasiparticle-pair interference term", because it can be pictured as a quasiparticle tunneling process mediated by a concomitant destruction and creation of Cooper pairs on both sides of the SIS junction [25]. Despite several experiments demonstrating its existence in the ’70s, the cosine term was confirmed with good precision only recently [64].

It is worth noticing that, in a thermally biased SIS junction, the thermal coun-terpart of Iint is a fundamental ingredient of coherent caloritronics [3]. In

tem-perature biased SIS junction depends on the superconducting phase across the junction, namely ˙ QS1IS2(TS1, TS2, ϕJ) = ˙Q qp S1IS2(TS1, TS2) − ˙Q int S1IS2(TS1, TS2)cosϕJ. (1.24) where ˙Qqp_S

1IS2 is the quasiparticle contribution of Eq. (1.9) and ˙Q int

S1IS2 is the

am-plitude of the phase-coherent term. This fact was first experimentally proven almost 40 years later in 2012 [4], and it can be used to manage heat currents in hybrid superconducting systems [3, 66–68]. In this thesis we won’t discuss ther-mally biases Josephson Junction, hence this energy exchange mechanism is no further discussed in the following. However, for a matter of completeness, the expressions of ˙Qqp_S

1IS2, ˙Q int

S1IS2 are reported in Appendix A.

1.4.2

Resistively and capacitively shunted junction model (RCSJ

model)

Figure 1.6: RCSJ model for a SIS junction under DC current bias. a. Equivalent circuit for a SIS tunnel junction in the RCSJ model, composed of a resistor RJ, a capacitance CJ and a nonlinear element which provides Josephson supercurrent Icsin ϕJ b. Particle in a tilted washboard potential: for low biases Idc< Icthe phase is locked to the minimum of the potential (non-dissipative regime). At larger currents, i.e. for Idc > Ic, the phase evolves in time (voltage state). c. Voltage evolution in the overdamped limit for different values of the input current Idc> Ic.

In the last subsection, we discussed the conversion of a DC voltage into an AC current in a SIS junction, due to the time evolution Eq. (1.20) of the phase difference ϕJ. For a time dependent bias, the current evolution is more complex,

due to the nonlinear self-coupling of the junction [69]. For this reason, it is cus-tomary to discuss the dynamics by using an effective model, first introduced by Stewart and McCumber in 1968 [70, 71], known as resistively and capacitively

shunted model (RCSJ model). The SIS junction is replaced by a circuit composed of the parallel of a nonlinear element which provides Josephson supercurrent I_csin ϕ, a resistor RJ and a capacitance CJ (see Fig. 1.6a). As a consequence, the

total current flowing through the junction in the AC regime is the sum of the Josephson supercurrent, the dissipative contribution due to quasiparticle and the displacement current due to the capacitance of the junction, namely

IJ(t) = Icsin ϕ(t) + VJ(t) RJ + CJ dVJ(t) dt . (1.25)

Upon rewriting VJ in terms of ϕJ by using Eq. (1.20), one gets a second order

ordinary differential equation for ϕJ. In the literature, the simplest case involve

a DC current bias IJ(t) = Idc(we assume here Idc> 0with no loss of generality):

the evolution described by Eq. (1.25) can be visualized as the motion of a clas-sical particle (with position ϕJ) in a tilted washboard potential (see Fig. 1.6b).

For Idc < Ic, the particle "seats" on the local minimum and the phase is locked

ϕJ = arcsin(Idc/Ic), hence VJ = 0 and a dissipationless supercurrent flows in

the circuit. Conversely, for Idc > Ic, there is no longer a local minimum in the

potential and ϕJ evolves in time, thus the transport is dissipative VJ 6= 0. In

the overdamped limit CJ → 0 we can neglect the last term on the right side

of Eq. (1.25), and the differential equation can be solved explicitly [25, 72]. The voltage dynamics reads

VJ(t) = RJIc

(Idc/Ic)2− 1

Idc/Ic− cos(ωJt)

, (1.26)

where ωJ= 2πνJis the Josephson angular frequency corresponding to the mean

voltage ¯VJ= RJpIdc2 − Ic2. Figure 1.6c displays the voltage evolution of Eq. (1.26)

for different values of Idc > Ic. At low current biases, i.e. Idc & Ic, the voltage

evolution consists of a succession of peaks located at ωJt = 2πn (here n is an

integer number), where the denominator of Eq. (1.26) reads Idc/Ic− 1 ≈ 0, hence

the signal is characterized by the contribution of many harmonics of angular fre-quency nωJ. By raising the current, the voltage pulses broaden and for Idc  Ic

the evolution reduces to a harmonic oscillation VJ(t) = RJIdc+ RJIccos(ωJt)[as

can be seen by using the expansion 1/(1−x) ∼ 1+x valid for x → 0 in Eq. (1.26)]. This model is used in the next chapter to describe the time evolution of the heat engine.

1.4.3

Proximity effect in a SNS weak link and SQUIPT

As mentioned before, the Josephson effect was first predicted in a tunnel junc-tion between two superconducting electrodes. However, a non-dissipative cur-rent is also observed in other kinds of superconducting weak links, for instance

superconductors separated by a short normal metal wire (SNS junction), con-strictions, point contacts and so on. For our purposes, we focus only on the first case. Interestingly, the vanishing resistance of SNS contacts was first re-ported [73] and investigated [74, 75] before the original article of Josephson. Generally, when a superconductor is brought in good electric contact with a normal metal, superconducting correlations are induced in the normal metal wire [76]. This effect is known as the proximity effect. The origin of this effect is the Andreev reflection [77, 78], which is a possible scattering mechanism for an electron of a normal metal colliding at the NS interface. If the energy of the elec-tron (with respect to the chemical potential) is smaller than the superconducting energy gap, it cannot be transmitted. In particular, it can be normally reflected or can be retroreflected as a hole (Andreev reflection), adding a Cooper pair to the superconducting condensate. In a clean SNS junction, the double Andreev reflection at the boundaries give rise to Andreev bound states, which carry the supercurrent (see Fig. 1.7a).

Interestingly, the proximity effect modifies also the spectral properties of the normal metal wire, where a so-called minigap appears in the quasiparticle DOS [79]. In order to describe non-homogeneous problems involving coher-ence effects, a quasiclassical approach is extremely powerful [80–82]. In par-ticular, it consists in averaging out the oscillations on the Fermi wavelength (λF ∼ 0.5 nm is very short compared to the typical dimension of the films

inves-tigated) and consider a Boltzmann-like equation of motion, called Eilenberger equation [83]. In the diffusive limit, i.e. when the mean free path is much shorter then the length of the sample, the Eilenberger equation reduces to the Usadel equation [81, 84]. For the purpose of this discussion, we can focus on the ther-mal equilibrium situation and consider a one-dimensional Usadel equation,

~D ∂ ∂x  ˆ g∂ ˆg ∂x  + ( + iΓ)[ˆτz, ˆg] = 0 (1.27)

where x ∈ [−`/2, `/2] is the spatial coordinate of the nanowire (` is the length of the proximized wire). Here D is the diffusion coefficient, Γ is a phenomeno-logical parameter which accounts for inelastic scattering, τz is the third Pauli

matrix in the Nambu representation [85]. Moreover, ˆg is a matrix in this space with the form ˆg = gτz + ˆf, where ˆf the anomalous Green’s function that is

off-diagonal in Nambu space. In mesoscopic physics, there is a relevant energy scale, called Thouless energy ETh, which, for diffusive systems, can be written in

terms of the diffusion coefficient and the geometric length of the system, namely ETh = ~D/`2. In particular, in this system, it sets the scale for the short junction

limit, i.e. for ETh  ∆0, where the proximity effect is maximized and the spatial

F

I

[



/(

eR

)]

Q

[



/(

e

R

)]

F [F

]

Figure 1.7: Superconducting quantum interference proximity device as a quasiparticle valve a. Cooper pair transmission in a SNS weak link. The supercurrent is sup-ported through the creation of Andreev bound states, due to the double Andreev reflection at each NS interfaces. Here e and h represents electron and hole, respec-tively. b. Scheme of the SQUIPT: a voltage bias is applied between a supercon-ducting ring interrupted by a normal metal wire (red) and a normal probe (blue). The probe is tunnel coupled to the ring. The DOS in the wire depends on the phase bias across the wire ϕJ(see the text), which is controlled by the flux Φ of the ex-ternal magnetic field, that is applied out-of-plane. The black dashed line gives the integration contour path for the derivation of the fluxoid quantization Eq. (1.29). c. (Top) Flux modulation of the normalized current for TS0 = T_N = 0.1 T_c0for some values of the applied voltage. (Bottom) Flux modulation of the normalized heat current at zero voltage bias V = 0 for TN= 0.01 Tc0and some values of TS0 > T_N. In both panels, we set Γ = 10−4∆0.

for a perfectly clean contact [86]: Nwire(E, T, ϕJ, x) =     <  GϕJ(E, T ) cosh  2x ` cosh −1 G(E, T ) GϕJ(E, T )  .     (1.28)

Here GϕJ(E, T )is defined as G(E, T ) (see Sec. 1.2), upon the replacement

∆(0, T ) → g(T, ϕJ), where g(T, ϕJ) = ∆(0, T )| cos(ϕJ/2)|is known as minigap.

The proximity effect is the fundamental ingredient of a device called super-conducting quantum proximity transistor (SQUIPT) [62], schematically repre-sented in Fig. 1.7b. The SQUIPT is composed of a superconducting ring in-terrupted by a short normal wire (red) and a probing electrode (blue) tunnel coupled to the normal wire. In this geometry, the phase difference across the junction ϕJ is controlled by applying an external out-of-plane magnetic field,

along the dashed path in the superconductor in Fig. 1.7b, see Appendix C] ϕJ+

2π Φ0

(Φ + LRingIRing) = 2πn. (1.29)

Here LRing is the total (geometric + kinetic) inductance of the superconducting

ring, IRing is the circulating current in the ring, Φ is the flux of the external

mag-netic field, and n is an integer number. Typically, one can make the inductance term negligible, as we will discuss later in Ch. 4. In this case, the external flux dependence of the DOS is Nwire(E, T, Φ, x) = Nwire(E, T, 2πΦ/Φ0, x), which is

periodic in Φ with period Φ0. The quasiparticle current across the probe-wire

junction is (for convenience, we use the symbols "S’" and "pr" to denote the proximized normal metal wire and the probe, respectively)

IS0_Ipr(V, T_S0, T_pr, Φ) = 1 eRT Z ∞ −∞ dE 1 wpr Z x0+wpr/2 x0−wpr/2 dxNwire(E + eV, TS0, Φ, x)

×Npr(E, Tpr)[f (E, Tpr) − f (E + eV, TS0)],

(1.30) where Npr(E, Tpr) is the density of state in the probe, wpr is the width of the

probe and x0 is the position of the center of the probe with respect to the wire.

In the rest of this section, we consider a normal metal probe Npr(E, Tpr) = 1, of

negligible width, perfectly centered with respect to the wire (x0 = 0), hence

1 wpr

Z x0+wpr/2

x0−wpr/2

dxNwire(E+eV, TS0, Φ, x) ∼ N_wire(E+eV, T_S0, Φ, 0) = |<[G_ϕ

J(E, TS0)]|

which is a BCS-like DOS with energy gap equal to the minigap g(T, ϕJ), with

ϕJ = 2πΦ/Φ0. Nevertheless, the finite width of the probe and the position will

be considered in the analysis of Ch. 4.

Figure 1.7c displays the flux modulation of the quasiparticle transport across the probe-wire junction (we consider a normal metal probe). In the top panel the quasiparticle charge current

IS0_IN(V, T_S0, T_N, Φ) =

1 eRT

Z ∞

−∞

dENwire(E + eV, TS0, Φ, 0)

× [f (E, TN) − f (E + eV, TS0)] (1.31)

is displayed for some values of the applied voltage and TS0 = T_N = 0.1 T_c0 .

In the semi-period [0,Φ0/2], the current is minimum and strongly suppressed

at Φ = 0 (n is an integer value), where the induced minigap is maximum, and increases monotonically up to Φ0/2 (in Φ = Φ0/2, we have ϕJ = π and the

to the normal probe at zero voltage bias ˙ QS0_IN(T_S0, T_N, Φ) = 1 e2_R T Z ∞ −∞

dEENwire(E, TS0, Φ, 0)[f (E, T_S0)−f (E, T_N)]. (1.32)

for some values of TS0 > T_N. Again, in the semiperiod [0,Φ₀/2], the heat current

is minimum at Φ0. However, in this case the behavior is not monotonic with the

flux and the current is maximum for a flux slightly smaller than Φ0/2.

In summary, the SQUIPT behaves as a valve for quasiparticle transport, whose state (open/close) depends on the applied magnetic flux. For this reason, the SQUIPT is an essential component of the temperature transistor investigated in Pubs. III,II, as we will discuss in Ch. 3. Moreover, the experimental characteri-zation of a voltage biased SQUIPT will be presented in Ch. 4.

1.5

Quasiequilibrium regime in mesoscopic circuits

In the previous sections, we discussed the remarkable properties of the charge and energy transport in hybrid superconducting tunnel junctions. In doing so, we focused only on the electronic system of the electrodes, and we assumed that the description is valid independently on the voltage and the temperature potentials applied (we have only required eV, kBT  EF). Even putting aside

all the technological limitations, this picture alone is incomplete also from a the-oretical perspective. First, in real systems, the electronic system is never truly isolated. As a matter of fact, the superconducting state itself exists thanks to the interaction with the phonon system, which must be included in the discussion. Second, the application of voltage or temperature gradients consists of particle and energy injection, which may affect the equilibrium distribution of these two subsystems. Therefore it is crucial to consider the relevant energy relaxation mechanisms that occur in mesoscopic circuits at low temperatures.

For the purpose of this discussion, a generic N or S thin film can be pictured as shown in Fig. 1.8a. We identify three main different subsystems, namely the electronic (red) and the phonon (yellow) system of the film, and a generic sub-system (blue) which acts as a thermal bath for the phonons of the film. In a typical experiment, the latter is the phonon system of the substrate where the film are deposited, which is directly cooled down to Tbath by a refrigeration

ma-chine (for instance a dilution fridge). As discussed before, in this thesis we are mainly interested in the electronic subsystem. In order to apply a temperature gradient, power ˙Qin is injected in this subsystem, which can relax by means of

the electron-electron and the electron-phonon coupling (with relaxation rates γe−e and γe−ph, respectively). The latter induces a heat current ˙Qe−ph released