U
NIVERSITÀ DI
P
ISA
PH. D. THESIS IN CONDENSEDMATTERPHYSICS
Coherent thermodynamics and transport properties
of mesoscopic Josephson junctions
Author Advisor
Francesco Vischi Dott. Francesco Giazotto
Introduction
In the history of superconductivity, the discovery of the Josephson effect [1, 2] and the related technological development of superconducting junctions consti-tute an important breakthrough concerning the knowledge of the fundamental physics of superconductivity, the development of scientific research and the ap-plications.
From the fundamental point of view, the Josephson effect is the manifesta-tion of a macroscopic coherent quantum system, where the electric characteris-tics can be predicted within a heuristic quantum-tunneling model [1]. The ex-perimental confirmation of the Josephson effect was also unexpected by a part of the community, including Bardeen, that was doubtful about a model based on the propagation of correlations through an insulating barrier [3, 4].
From the research point of view, scientists suddenly started to investigate the presence of Josephson coupling in hybrid junctions, where superconducting parts are in contact with normal metal parts. The scientist’s efforts were enthu-siastic, and in about 15 years the concept of Josephson coupling was extended to superconducting junctions made of different materials and architectures [5]. While many experiments were performed, very sophisticated theoretical tools were developed in order to explain the propagation of superconducting correla-tions in normal metal parts and to predict the non-linear electric characteristics of superconducting junctions [6–12]. In this way, the heuristic model by Joseph-son has been complemented by innovative and detailed microscopical descrip-tions.
Finally, from the applications point of view, superconducting junctions opened a Pandora’s box of applications. On the side of pure electronics where thermal properties are not invoked, superconducting junctions have been exploited for realizing sensitive magnetometers [13–17], computing sys-tems [15, 18–20], high frequency digital-signal-processing [15], quantum optics and quantum computing [15, 21–25]. On the side of electro-thermal applica-tions, superconducting junctions show strongly interlaced thermal and elec-tronic properties, exploited for realizing experimentally solid-state refrigerators [26–29], thermometers [27, 30–32], photodetectors [33–37], heat valves [38–41] and heat-based electronics and computation [42–45].
In this thesis, we focus on a few applications that belong to the wide world of superconducting junctions physics. In detail, we study the coherent thermal
and electric properties of three different kinds of junctions. The main focus is dedicated to the theoretical study of coherent thermodynamic properties in Su-perconductor/Normal metal/Superconductor (SNS) junctions. The main result is that an SNS junction can be considered as a thermodynamic system where superconducting phase, supercurrent, temperature, and entropy are interlaced thermodynamic variables (see Pub. V), with application to solid-state cooling and thermal machines (see Pubs. II, IV).
The second system we study is based on graphene/insulator/superconduc-tor junctions. Graphene is a new material where the reduced dimensionality results in promising electric and thermal properties. Among them, the reduced electron-phonon thermal coupling compared to other materials constitutes an advantage for the refrigeration of the graphene electronic system. For this rea-son, we theoretically investigate the tunneling refrigeration (also called Peltier effect [29]) in this kind of junction and its application to bolometry, showing the positive effect of the electron refrigeration on the bolometric characteristics (see Pub. I).
Finally, we study experimentally the behavior of a superconducting interfer-ometer based on fully-metallic constriction-junctions, which are gated through a local electric field provided by gate electrodes. The effect of the electrostatic field on superconducting constrictions is a recently demonstrated new phenomenon and is not yet fully understood [46, 47]. The experiments show an interesting modulation of the interferometric characteristics depending on the gate volt-ages at the two junctions (see Pub. III).
The thesis is organized as follows. Chapter 1 introduces the physics of hy-brid systems, with a rapid overview of the theoretical treatment and experi-mental results. Particular attention is dedicated to SNS junctions, which is the subject of the following two chapters. Indeed, chapter 2 studies the connec-tion between the current-phase-relaconnec-tion in an SNS juncconnec-tion and its equilibrium thermal properties. These results are fully exploited in chapter 3 to build up a thermodynamic description of an SNS junction and apply it to solid state machines. Chapter 4 is focused on the cooling and bolometric characteristics of a superconductor/insulator/graphene system, where the cooling is accom-plished by electron tunneling under a specific voltage bias of the junction. Base temperature, thermal relaxation time, responsivity, and noise equivalent power are the discussed parameters of merit. Finally, chapter 5 is dedicated to the experimental investigation of a superconducting interferometer based on fully-metallic constriction-junctions controlled by field-effect.
List of publications
This thesis consists of an overview of the following publications, exclud-ing Pubs. VI, VII, which concern a topic treated durexclud-ing the master degree. Throughout the thesis, they will be referred to by their Roman numerals. The list is in chronological order from the most recent.
I F. Vischi, M. Carrega, A. Braggio, F. Paolucci, F. Bianco, S. Roddaro and F. Giazotto. Electron cooling with graphene-insulator-superconductor tunnel junctions and applications to fast bolometry.
arXiv:1906.10988 [cond-mat.mes-hall]
II F. Vischi, M. Carrega, A. Braggio, P. Virtanen and F. Giazotto. Thermodynamics of a phase-driven proximity Josephson junction. Entropy 21, 1005 (2019)
III F. Paolucci, F. Vischi, G. De Simoni, C. Guarcello, P. Solinas and F. Giazotto. Field-Effect Controllable Metallic Josephson Interferometer.
Nano Letters 9, 3238 (2019)
IV F. Vischi, M. Carrega, P. Virtanen, E. Strambini, A. Braggio and F. Giazotto. Thermodynamic cycles in Josephson junctions.
Scientific Reports 9, 3238 (2019)
V P. Virtanen, F. Vischi, E. Strambini, M. Carrega and F. Giazotto.
Quasiparticle entropy in superconductor/normal metal/superconductor proximity junctions in the diffusive limit.
Physical Review B 96, 245311 (2017)
VI F. Vischi, M. Carrega, E. Strambini, S. D’Ambrosio, F. S. Bergeret, Yu. V. Nazarov, F. Giazotto.
Coherent transport properties of a three-terminal hybrid superconducting interferometer. Physical Review B 95, 054504 (2017).
VII E. Strambini, S. D’Ambrosio, F. Vischi, F. S. Bergeret, Y. V. Nazarov and F. Giazotto. The ω-SQUIPT as a tool to phase-engineer Josephson topological materials.
Nature Nanotechnology 11, 1055 (2016).
List of acronyms
Throughout the thesis the following acronyms will be used: ABS Andreev bound states
BCS Bardeen-Schrieffer-Cooper COP Coefficient of performance CBC Continuity boundary conditions CPR Current-phase relation
DoS Density of states
G Graphene
KO Kulik-Omel’yanchuk
I Insulator / insulating
LNDoS Local normalized density of states
N Normal metal / normal
NEP Noise equivalent power RBC Rigid boundary conditions
S Superconductor / superconducting SEM Scanning electron microscope
List of symbols
Throughout the thesis, the following symbols will be used: a Junction cross-section parameter
C Heat capacity
∆0 Superconductivity gap at zero temperature
˜
∆ Induced mini-gap
∆ Pair potential
Particle energy from the Fermi energy EF Fermi level energy from the band bottom
εTh Thouless energy
E Josephson energy stored in the junction φ Pair potential phase
ϕ Phase difference across the junction
Φ Magnetic flux threading the superconducting ring ˆ
gR Momentum averaged Green function
I Supercurrent
Ic Critical supercurrent
I Dissipative electric current IS Switching current
` Junction length normalized to the coherence length LN Weak link length
N Quasi-particle LNDoS
N0,j DoS at Fermi energy per spin of the j part
NBCS Normalized BCS DoS
Pj Heat current in channel j
RN Weak link resistance
Rt Tunnel resistance in a GIS junction
R Responsivity
S Total entropy SBCS BCS entropy
δS Entropy difference from ϕ = 0
T Temperature
τj j-th Pauli Matrix
τth Thermal relaxation time
Contents
Introduction iii
List of publications v
List of acronyms vii
List of symbols ix
Contents xi
1 Introduction to hybrid systems and proximity effect 1
1.1 Proximity effect and its applications . . . 1
1.2 The Usadel equations . . . 6
1.3 SNS junction . . . 11
1.4 Discussion . . . 20
2 Quasi-particle entropy in an SNS junction in diffusive limit 23 2.1 Thermodynamic properties of an SNS junction . . . 24
2.2 Inverse proximity effect . . . 28
2.3 Analytical results for the inverse proximity . . . 32
2.4 Discussion . . . 35
3 Thermodynamics of a phase-driven SNS junction 37 3.1 Model . . . 37
3.2 Thermodynamic processes . . . 45
3.3 Thermodynamic cycles . . . 52
3.4 Discussion . . . 69
4 Refrigeration and bolometry in a SIGIS junctions 71 4.1 Model . . . 72
4.2 Base temperature . . . 80
4.3 Thermal response dynamics . . . 83
4.4 Biased SIGIS as bolometer . . . 86
5 Field-effect controllable Josephson interferometer 95
5.1 Field-effect in superconducting systems . . . 95
5.2 Experimental setup . . . 100 5.3 Experimental results . . . 104 5.4 Discussion . . . 113 Summary 117 Bibliography 119 Acknowledgements 137
Chapter 1
Introduction to hybrid systems and
proximity effect
This chapter is an introduction to the physics of hybrid superconducting sys-tems at thermal equilibrium and in the diffusive (dirty) limit. Its target is to set the physical and mathematical framework necessary for the next chapters.
Section 1.1 presents the hybrid superconducting systems and the proximity effect. A general picture of the Andreev reflection, the microscopical mecha-nism underlying proximity effect, is given in order to grasp the physical be-havior without going in the complex details of the Andreev theory. Section 1.2 overviews the mathematical methods of the quasi-classical theory of supercon-ductivity, focusing specifically on the case of diffusive transport at thermal equi-librium. These methods constitute the tools to obtain quantitative results, and they are practically applied in section 1.3 to the case of an SNS junction, within specific boundary conditions that simplify the treatment and yield closed-form results, showing the practical usage of the mathematical machinery.
1.1
Proximity effect and its applications
Superconducting hybrid systems are circuits constituted by Superconducting (S) parts in electric contact with non-superconducting parts, which often consist of Normal (N) metal [8, 27, 48–50], but also semiconductors [51, 52], graphene [30, 33, 53–56], and topological insulator [57–60] are involved.
The interest in hybrid systems is guided by applicative and theoretical rea-sons. Among the applications, the most investigated are
• Caloritronics. Management of heat currents is an important point for the realization of solid-state mesoscale thermodynamic machines. Hy-brid systems have interesting equilibrium and transport thermal proper-ties, which can be exploited for thermal routing or heat-based computa-tion [27, 38–40, 43, 44, 61–65]. Figure 1.1(a) shows a particular example, i.e.,
a heat rectificator diode based on a NINISIN hybrid system (adapted from Ref. [66]). Within the topic of caloritronics, this thesis studies the thermal properties of an SNS junction and its application as a solid-state thermal machine in chapters 2 and 3.
• Refrigerators. Realization of efficient micro-scale refrigerators is an im-portant industrial target, since it would allow the realization of cooling systems free from mechanical and gaseous parts, implying portability and lower production/maintenance costs. A particular kind of micro-scale re-frigerator is based on SINIS junctions [26–29, 67]. Figure 1.1(b) shows an interesting architecture based on a cascade of SINIS refrigerators made of Al and Cu (adapted from Ref. [28]).
• Photo-detection. Proximized systems enable strong transduction from thermal quantities to electrical quantities, hence allowing bolometric or calorimetric detection schemes [34]. Several different architectures have been explored, employing SINIS junctions [36], SNS [37], graphene [33,68]. As example, figure 1.1(c) shows the sketch of a proximized-graphene pho-todetector, with related Scanning Electron Microscope (SEM) micrograph in panel (d) (adapted from Ref. [33]). Here, a photonic stream heats the graphene flake, altering the supercurrent transport between the two super-conducting leads, realizing the transduction. Within the topic of photo-detection, we explore a hybrid photo-detector in chapter 4.
• Quantum computing. Recently, hybrid semiconducting/superconduct-ing systems have been under attention for quantum computsemiconducting/superconduct-ing and topo-logical quantum computing. An example is the gatemon, consisting of a proximized epitaxial-grown semiconductor nanowire, provided with an external gate that tunes the nanowire Fermi level [52, 69, 70]. A gatemon is shown in figure 1.1(e), with magnification on the junction in panel (f) (adapted from Ref. [69]).
• Superconducting FET transistors. Another important topic is the realiza-tion of fast and low-dissiparealiza-tion Field-Effect Transistors (FET). The appli-cations of these devices are very extended, in particular to classical very high-speed computing. The field-effect on Josephson junction is treated in detail in chapter 5.
• Magnetometers. Magnetometry is one of the most successful applications of superconductivity. Many devices (especially in medicine) have mag-netometers based on Superconducting QUantum Interferometer Devices (SQUIDs), consisting practically of one superconducting ring interrupted by two tunnel junctions. Recently, a different magnetometer architec-ture has been realized, consisting of a superconducting ring interrupted by a normal metal wire junction [17, 71–73], as shown in figure 1.1(g,h)
1.1. Proximity effect and its applications 3 (adapted from Ref. [17]). This junction is in tunnel contact with a super-conducting probe, which can reveal electrically the phase-dependent vari-ation of quasi-particle density of states in the wire, caused by the magnetic flux, realizing then the magnetometry.
Under the theoretical point of view, the interest concerns the transport of superconducting coherent properties into the normal parts, where the electron-electron coupling is absent. Since the early experiments in the sixties, it has been realized that an N metal in electric contact with an S metal acquires superconducting-like properties. This alteration of an N metal and the related set of phenomena have been dubbed as proximity effect [50]. The phenomenol-ogy of proximity effect includes magnetic screening [74], gapped density of states [72, 75, 76], and altered heat capacity [77–79]. Another strong evidence comes from measurements of the conductance of NS junctions: given the inter-face conductance GN N of the contact in the normal state (i.e., above the critical
temperature of S), the conductance GN S is doubled in the superconducting state
if the NS interface is transparent. On the contrary, the conductance GN S is
sup-pressed in case of low transparency, i.e. in the NIS limit [48, 80, 81]. Finally, an important manifestation of the proximity effect concerns the disappearance of the N resistance in an SNS junction, related to the possibility of supercurrent transport [5].
The Andreev reflection
The microscopical mechanism underlying the proximity effect is the Andreev reflection [10, 48, 49, 82], sketched pictorially in figure 1.2. Let us consider an electron propagating from the N metal toward a clean NS interface. If its energy (measured from the Fermi energy) is below the superconducting gap ∆, it cannot propagate in the S part due to the absence of available states below the gap. Instead, it couples to another electron in the S part, forming a Cooper pair. As a result of the charge and energy conservation in this scattering process, a hole is retro-reflected in the N metal, i.e., reflected in the same direction of the incident electron.
The retro-reflected hole has two properties in particular. First, it has a mo-mentum mismatch with the incident electron. Indeed, for energy conservation, the latter has energy EF + and momentum ∼ kF + kF/EF while the hole has
energy EF − and momentum ∼ kF − kF/EF. Hence, a momentum
(wave-vector) mismatch δk ∼ 2kF/EF exists between the incident electron and the
retroflected hole [48, 49, 83].
The second important property is that the Andreev reflection is a coherent process. After scattered on the SN surface where the superconducting pair po-tential is ∆eiφ, the phase-difference between the incident electron and the
Figure 1.1. Examples of devices based on hybrid systems. More details about the presented devices are in the text. (a) Thermal diode, from Ref. [66]. (b) Refrigerator based on SINISs in cascade, from Ref. [28]. (c,d) Graphene-based bolometer, from Ref. [33]. (e,f) Gatemon quantum bit, adapted from Ref. [69]. (g,h) Magnetometer, from Ref. [17].
1.1. Proximity effect and its applications 5
Figure 1.2. Andreev reflection mechanism. (a)Three frames reporting the scattering of an electron on an SN clean interface. When an electron with energy below the superconducting gap ∆ scatters with the interface, it couples with another electron forming a Cooper pair. As a result of energy and charge conservation, a hole is reflected back in the normal metal in the same direction of the incident electron. This particular kind of reflection is called retro-reflection.
(b)Cooper pair transport in an SNS junction through Andreev bound states. The electron and hole states are coupled through the coherent scattering provided by the Andreev reflection. (c) Characteristic lengths in a diffusive-transport system: the mean free path lmfp, the Andreev pair
coherence length L, the phase-breaking length Lφ.
correlations rise in the N metal, realizing hence the proximity effect in the N metal. Furthermore, if these coupled electron-hole states can be picked up by a second interface, the transport of supercurrent through the SNS junction is allowed, as sketched in figure 1.2(b). These electron-hole correlated states trans-porting supercurrent are dubbed Andreev Bound States (ABS), while we call the electron-hole couple in the N metal Andreev pair. The N part of an SNS junction is dubbed weak link [5].
For simplicity, figure 1.2(b) depicts Andreev bound states in the ballistic case, where the mean free path lmfp is larger than the N weak link. However, in this
thesis, we focus on diffusive systems, where the transport is mediated by sev-eral elastic scattering, as depicted in figure 1.2(c). In diffusive transport, differ-ent characteristic lengths play a role. One of them is the mean free path lmfp,
characterizing on average the distance between the scattering centers, that we assume to scatter the quasi-particles elastically in the metal.
Another important characteristic length is the Andreev pair coherence length L. Let us consider the electron and the hole in an Andreev pair: after a scattered
trajectory of total length d, they have cumulated a phase difference δφ = d δk, being δk the momentum mismatch. When δφ = π after a distance dπ = π/δk,
the shift between the electron and hole is of the order of the Fermi wavelength, and the diffusion trajectory of electron and hole will take different paths: the Andreev pair is hence broken. In order to translate the diffused trajectory length dπ in an effective distance L from the interface, we consider the relation L ∼
√
Dt ∼ pDd/vF. The final result is the coherence length of the Andreev pair
L =
r ~D
. (1.1)
We can note that Andreev pair coherence length diverges for → 0, i.e., for an electron-hole couple close to the Fermi energy. However, an Andreev pair can be broken as well by other processes, like spin-flipping or inelastic scattering, that introduce a phase breaking length Lφas an upper limit to an Andreev pair
diffusion.
Let us now consider an SNS junction where the weak link part has length LN. This case defines an energy band for the transported Andreev pairs, since
only the ones with energy such that L < LNcan diffuse to both the interfaces.
This energy band, defined by L = LN, is the Thouless energy
εTh= ~D
L2 N
. (1.2)
The Andreev pairs formed by an electron-hole couple with individual energy || above εThare broken before diffusing from one interface to the other.
A particular case is when the junction is small enough and all the ABSs below the superconducting gap contribute to the supercurrent transport. In this case, the junction length has to be lower than a length scale defined by εTh = ∆0, i.e.,
when the Thouless energy is equivalent to the gap ∆0at T = 0. This scale is the
coherence length ξ0 = r ~D ∆0 . (1.3)
Given ξ0, a junction is considered in short or long regime according if LN ξ0
or LN ξ0respectively.
Experiments [5, 72, 75, 84, 85] show that the characteristic length ξ of these system is of the order of 0.1 µm, setting the physics of proximized systems in the mesoscopic domain.
1.2
The Usadel equations
In the previous section, we pictorially introduced the Andreev reflection mech-anism and the diffusive transport of Andreev pairs in normal metals. In this section, we focus on the related mathematical formalism for the proximity ef-fect in hybrid diffusive systems in thermal equilibrium. The passage from the Andreev reflection mechanism to the effective transport theory is not trivial and requires many steps in order to move from the cumbersome Gork’ov equations to the handier Usadel equations [6, 11, 12, 86]. We remark that we focus on
ther-1.2. The Usadel equations 7 mal equilibrium, which is a specific case of the broader quasi-classical theory of superconductivity, where also quasi-equilibrium cases can be treated within the Keldysh formalism [83, 86, 87].
The properties and observables of the system are described by the momentum-averaged retarded Green function ˆgR, that is a 2 × 2 matrix defined in
the electron-hole (Nambu) space and function of the energy (from the Fermi energy) and the position r:
ˆ gR(r, ) = gR(r, ) fR(r, ) ˜ fR(r, ) −gR(r, ) . (1.4)
The off-diagonal components fR, ˜fR, called anomalous function or pair amplitude,
quantify the electron-hole correlations [6, 86]. They are zero in a normal non-proximized metal, and different from zero in a superconducting metal or a prox-imized metal.
The Green function ˆgRsatisfies the Usadel equations [11]
~D∇(ˆgR(r, )∇ˆgR(r, )) = [−iˆτ3+ ˆ∆(r), ˆgR(r, )] , (1.5)
where D = vFlmfp/3is the electron diffusion constant (that we assume
position-independent), ˆτj is the j-th Pauli matrix. The term ˆ∆is
ˆ ∆(r) = 0 ∆(r) ∆∗(r) 0 , (1.6)
where ∆(r) is the pair potential, a complex function ∆ = |∆(r)|eiφ(r) of the
po-sition r. The pair potential has to be calculated self-consistently with ˆgR(, r)
through ∆(r) = λ 4i Z +Ec −Ec tanh 2T [fR(r, ) − fA(r, )]d , (1.7) where λ is the interaction strength and Ecis a energy cut-off necessary to
nor-malize the logarithmic divergence of the point-like interaction model of the elec-tron attraction [83], fAis the anomalous component of the advanced Green
func-tion ˆgA ˆ gA(r, ) = gA(r, ) fA(r, ) ˜ fA(r, ) −gA(r, ) . (1.8)
which can be obtained from the retarded Green function as gA = −ˆτ
3(ˆgR)†τˆ3. It
correlation (fR6= 0) but null pair potential, since λ = 0. Finally, equation (1.5) is
complemented with the pseudo-normalization (ˆgR)2 = I.
From the Green function ˆgR, it is possible to extract the spectral quantities, functions of energy that are related to the macroscopical thermodynamic quan-tities through statistical integrals, involving a spectral quantity multiplied with a distribution function [86, 88]. In this thesis, we focus in particular on two ther-modynamic variables, that are the entropy and the supercurrent, given by sta-tistical integrals on two spectral quantities respectively: the quasi-particle Local Normalized Density of States (LNDoS) and the spectral supercurrent.
The LNDoS N (r, ) is the quasi-particle Density of States (DoS) per spin at position r and energy , normalized to the Fermi energy DoS per spin N0. Two
particular cases of LNDoS involve the S and N bulk metals: in the former, the LNDoS has the BCS form
NBCS() = Re || √ 2− ∆2 , (1.9)
while in the N metal it is
N () = 1 . (1.10)
Although the bulk DoSs of N and S are position-independent, the LNDoS de-pends in general on both energy and position r. The LNDoS can be extracted from ˆgRwith the prescription
N (, r) = RegR(, r) . (1.11)
The spectral supercurrent density jsp, which quantifies the energy-density of
the ABSs, can be extracted from ˆgRwith the prescription [16, 83, 87]
jsp(, r) =
1
4Tr ˆτ3 ˆg
R∇ˆgR− ˆgA∇ˆgA
. (1.12)
The supercurrent density jscan be calculate from the spectral supercurrent
den-sity jspthrough the statistical integral [16, 83, 87]
js(r) = σ −2e Z ∞ −∞ jsp(, r) tanh 2T . (1.13)
Two particular simple solutions of the Usadel equations are the N and S bulk Green function. The homogeneous bulk solutions are given by imposing the derivative term in the left-hand side of equation (1.5). For a N metal, it is
1.2. The Usadel equations 9 where we imposed ∆ = 0 in equation (1.7). The solution is
ˆ gNR() = 1 0 0 −1 . (1.15)
As expected, this solution yields the normal metal DoS N = 1 and its anomalous off-diagonal components (the electron-hole correlations) are zero.
For a S metal, the bulk Green function ˆgR
BCS is given by imposing a
homoge-neous pair potential ∆ = ∆eiφand a null space derivative in equation (1.5). The
solution is ˆ gBCSR () = 1 p1 − ∆2/2 1 i∆eiφ/ i∆e−iφ/ −1 , (1.16)
where ∆ has to be calculated with the self-consistency equation (1.7). Substitut-ing the anomalous component fR, fAin (1.7), after some algebra we recover the
self-consistency equation of the BCS theory 1 = λ Z Ec 0 tanh p ξ2+ ∆2(T ) 2T ! 1 pξ2+ ∆2(T )dξ , (1.17)
where ξ2 = 2 + ∆2. The microscopical parameters λ, E
c can be erased
con-sidering that in the weak coupling limit λ → 0 it holds ∆0 = 2Ece−1/λ, where
∆0 = ∆(T = 0) is the gap at zero temperature. Moreover, by imposing
∆(Tc) = 0, we obtain that the critical temperature is Tc ≈ 1.14Ece−1/λ,
equiv-alent to the well known relation
∆0 ≈ 1.764Tc . (1.18)
As expected, the quasi-particle DoS extracted from (1.16) returns the BCS form NBCS = Re
n
||/p2− ∆2(T )o.
Parametrization
The Usadel equations can be attacked with proper parametrizations that take into account the pseudo-normalization condition (ˆgR)2 = I. The θ-parametrization and the Riccati θ-parametrization are the most used.
The θ parametrization [16, 83, 86, 88, 89] maps ˆgR in two functions θ(r, ) and
χ(r, )such that
ˆ
gR(r, ) =
cosh(θ) sinh(θ) exp(iχ) − sinh(θ) exp(−iχ) − cosh(θ)
The Usadel equations are hence transformed in ~D∇2θ = −2iE sinh θ + ~D
2 (∇χ)
2
sinh(2θ) + 2i|∆| cos(φ − χ) cosh θ (1.20)
~D∇ · jE = −2i|∆| sin(φ − χ) sinh θ , (1.21)
where jE = −(∇χ) sinh2θ.
The quasi-particle LNDoS is
N (, r) = Re {cosh[θ(, r)]} , (1.22)
while the spectral supercurrent is
jsp(, r) = Im {jE} . (1.23)
The θ-parametrization is suitable for an analytical approach to the solution. On the opposite, for numerical calculation, the θ-parametrization is an inconve-nient choice, since 1) the hyperbolic functions are unbounded, 2) they are 2πi-periodic and can yield spurious solutions, 3) if |θ| is small, the function χ can have steep variations. For numerical calculations it is better to employ the Ric-cati parametrization [83,87,90,91], based on two coherence functions a(r, ), b(r, ) such that ˆ gR(r, ) = 1 1 − ab 1 + ab 2a −2b −(1 + ab) (1.24)
where the Usadel equations reads ~D∇2a + 2b(∇a)2 1 − ab = −2ia + i∆ ∗ a2+ i∆ (1.25) ~D∇2b − 2a(∇b)2 1 − ab = −2ib + i∆ ∗ b2− i∆ . (1.26) The LNDoS is N (, r) = Re 1 + ab 1 − ab . (1.27)
and the spectral current density is jsp = 2Re
b∇a − a∇b (1 − ab)2
. (1.28)
The Riccati parametrization has been used for the numerical calculations in this work. The latter have been performed with the Python open source library Us-adel1 [89].
1.3. SNS junction 11
Figure 1.3. Sketch of an SNS junction.The normal metal weak link, LNlong, is in clean electric
contact with the superconducting leads. Aj, N0,jare respectively the cross-section and the DoS
at Fermi energy per spin of the j metal. The pair potential ∆ has a phase drop ϕ = φR− φL
across the junction.
1.3
SNS junction
An SNS junction, sketched in figure 1.3, consists of two S leads in electric contact with an N weak link. The S leads have superconducting gap at zero temperature ∆0 and critical temperature Tc. The weak link and the S leads have respectively
cross-sections AN and AS, and DoS per spin at Fermi energy N0,N, N0,S. The
conductivities of the two parts are σj = 2e2DN0,j. The weak link is long LNwith
resistance RN = LN/ANσN.
In order to simplify the treatment of the Usadel equations, we consider sys-tems that can be treated within the quasi-1 dimensional approximation, i.e., when AS, AN ξ20.
For simplicity, we define here two adimensional parameters ` = LN ξ0 (1.29) a = σSAS σNAN , (1.30)
that will be used later.
Rigid boundaries conditions
Here, we introduce a model of junction based on Rigid Boundaries Conditions (RBC), an approach that allows obtaining predictions in good agreement with the experiments within a simplified mathematical treatment. This model is based on the assumption that the superconducting leads are much larger than the normal metal weak link, and hence the two leads can be considered as bulk superconductors with the role of “reservoirs of coherence” [88]. Hence, the Green functions in the leads are not part of the calculations but constitute fixed boundary conditions, while the Green function has to be determined only in the N weak link.
Figure 1.4. SNS junction in the rigid boundaries model.The Green function ˆgRsatisfies
equa-tion (1.32), where ˜xis the position normalized to the length LN. The boundary conditions are
imposed at the SN surfaces, where ˆgR is imposed to be equal to the bulk expression ˆgR BCS in
equation (1.16).
The RBC model is sketched in figure 1.4. The Usadel equations reduce to ~D∂x(ˆgR(x, )∂xgˆR(x, )) = [−iˆτ3, ˆgR(x, )] , (1.31)
where we set ∆ = 0 since the coupling strength is λ = 0 in the N weak link. This equation is usually rewritten normalizing the lengths to the junction length LN:
∂˜x(ˆgR(˜x, )∂˜xˆgR(˜x, )) = −i εTh ˆ τ3, ˆgR(˜x, ) (1.32) where εTh is the Thouless energy introduced in equation (1.2). For simplicity,
we set ˜x = 0in the middle of the weak link. The RBC at the interfaces are
ˆ
gR(˜x = −1/2, ) = ˆgBCSR (, φ = −ϕ/2) (1.33)
ˆ
gR(˜x = 1/2, ) = ˆgBCSR (, φ = ϕ/2) , (1.34)
where ˆgR
BCS is the bulk Green function in equation (1.16) and ϕ is the phase
dif-ference across the junction. Importantly, the parameter a does not appear in the boundary conditions of the RBC model.
Let us discuss some numerical results, in order to grasp the underlying mechanisms of the proximity effect. Figure 1.5 describes many features of the LNDoS at ϕ = 0 and ∆(T ) = ∆0. The top row of figure 1.5 reports the LNDoS
N as a function of energy and position ˜x. An induced mini-gap ˜∆appears in the weak link. The width of the induced mini-gap depends on the weak link length `, as reported in the bottom row of figure 1.5. Consistently with the picture of the Andreev reflection, the ABSs are placed in an energy bandwidth upper-limited roughly by the Thouless energy εTh/∆0 = 1/`2. Above this energy, the
Andreev pairs are broken and constitute quasi-particle excitations, giving rise to the dependency of the mini-gap on ` in figure 1.5(c,d). In particular, numerical simulations show that in the long limit ˜∆ ∼ 3.1εTh.
1.3. SNS junction 13
Figure 1.5. Induced mini-gap in an SNS junction at ϕ = 0. (a)LNDoS N versus position ˜xand energy for length ` = 2 (εTh = 0.25∆0). At the SN interfaces (˜x = ±0.5) the DoS has the BCS
form. In the middle of the junction the induced mini-gap ˜∆has width ∼ 0.5∆0. (b) Cuts from
panel (a) for chosen positions ˜xin legend. (c) DoS in the middle of the junction as a function of the junction length. For ` → 0, the mini-gap tends to the S lead gap. (d) Cuts from panel (c) for chosen junction lengths ` in legend.
Let us consider now the evolution of the mini-gap ˜∆versus the phase, stud-ied in figure 1.6, for a simulation with ` = 0.5. Interestingly, the quasi-particle DoS is phase-dependent: the induced mini-gap shrinks till complete closure at φ = π. Figure 1.6(a,b) reports the evolution of LNDoS in the center of the junc-tion as a funcjunc-tion of the phase ϕ. It can be noticed that ˜∆is fully open at ϕ = 0 and shrinks till closure at ϕ = π, where the LNDoS is metal-like. The evolution of the LNDoS N (˜x, )versus the phase over the whole junction is reported in the four frames in figure 1.6(c). We can note that the DoS at the SN interfaces (˜x = ±1/2) is not affected by ϕ since it is set equal to the bulk DoS by the rigid boundary equations (1.34).
The supercurrent is given by the statistical integral I(ϕ, T ) = 1 2eRN Z ∞ −∞ ˜ jsp() tanh 2T d , (1.35)
Figure 1.6. Induced mini-gap versus ϕ. (a)LNDoS N in ˜x = 0(middle of the junction) versus energy and phase ϕ. The length is ` = 2 (εTh = 0.25∆0). (b) Cuts from panel (a) for the
chosen phases ϕ in legend. (c) Frames representing the evolution of the LNDoS N (˜x, )versus the phase, for ϕ values written on top of the panels.
˜ jsp() = 1 4Tr ˆτ3(ˆg R∂ ˜ xˆgR− ˆgA∂˜xˆgA) . (1.36)
Note that ˜jsp has not the same physical dimensions of jsp, since it has been
ob-tained by spatial derivation with the normalized coordinate ˜x. Moreover, we dropped the spatial dependence of ˜jspsince the spectral current is conserved in
the weak link when ∆ = 0.
The relation I(ϕ, T ), linking the supercurrent I, the phase ϕ, and the temper-ature T is dubbed Current Phase Relation (CPR), and depends on the physical and geometrical factors of the junction. References [5, 84] give a review of different CPRs [5, 84].
Figure 1.7(a,b) shows the spectral supercurrent ˜jsp(, ϕ), that is an odd
func-tion of and ϕ. From ˜jsp, the supercurrent in figure 1.7(c) can be obtained.
Closed-form results in short junction limit
In the short junction limit, it is possible to obtain a closed-form solution. This limit is given by ` 1 or, equivalently, by εTh → ∞, implying that the Usadel
1.3. SNS junction 15
Figure 1.7. Spectral supercurrent in an SNS junction,with length LN= 2ξ0. (a) Color plot of
˜
jsp as function of and ϕ. We can note that the spectral supercurrent is energetically confined
between the mini-gap and the gap. (b) Related cuts from panel (a) for the chosen ϕ in legend.
(c)Supercurrent I obtained from ˜jspthrough the statistical integral (1.35).
equations (1.32) reduce to
∂x˜(ˆgR(˜x, )∂ ˆgR(˜x, )) = 0 . (1.37)
The solution, expressed within the θ-parametrization defined in equation (1.19), is [73, 88]
θ = arccosh [α(ε, ϕ, T ) cosh [2˜x arccosh [β(ε, ϕ, T )]]] (1.38) χ = − arctan [γ(ε, ϕ, T ) tanh [2˜x arcosh [β(ε, ϕ, T )]]] , (1.39) where α = s ε2 ε2− ∆2(T )cos2(ϕ/2) (1.40) β = s ε2− ∆2(T )cos2(ϕ/2) ε2− ∆2(T ) (1.41)
γ = pε
2− ∆2(T )cos2(ϕ/2)
∆(T )cos(ϕ/2) (1.42)
The LNDoS is, according prescription (1.22),
N (, ˜x) = Re {α(ε, ϕ, T ) cosh [2˜x arccosh [β(ε, ϕ, T )]]} . (1.43) This form of N is studied in figure 1.8. Each column corresponds to a different value of ϕ, reported in the box on the top. The two rows show N as 3D surface and contour plot respectively. The LNDoS in the middle of the junction ˜x = 0 has a particular simple form
N (, ˜x = 0) = Re ( p2− ∆2(T ) cos2(ϕ/2) ) , (1.44)
that is the same form of the BCS DoS with a reduced gap provided by the mini-gap ˜∆(ϕ, T ) = ∆(T )| cos(ϕ/2)|.
The spectral supercurrent is
˜ jsp() = 0, 0 < < ∆(T )| cos(ϕ/2)| π∆(T ) cos(ϕ/2) √ 2−∆2(T ) cos2(ϕ/2), ∆(T )| cos(ϕ/2)| < < ∆(T ) 0, > ∆(T ) , (1.45)
defined in the interval 0 ≤ ϕ ≤ 2π with continuation given by 2π-periodicity in ϕ.
For negative energies, it holds ˜jsp(−) = −˜jsp(). According to the
prescrip-tion (1.35), the CPR for a short juncprescrip-tion is
I(ϕ, T ) = π∆(T ) eRN cosϕ 2 Z ∆(T ) |∆(T ) cos(ϕ/2)| 1 p2− ∆2(T ) cos2(ϕ/2)tanh 2T d , (1.46) defined in the interval 0 ≤ ϕ ≤ 2π with continuation is given by 2π-periodicity in ϕ. Expression (1.46) is dubbed Kulik-Omel’yanchuk (KO) current-phase rela-tion, after the authors that derived it [92].
Experimental demonstration
The agreement between the quasi-classical theory and the experiments has been proved positively in literature. Here, we focus on the experimental proofs of the induced mini-gap and the flowing supercurrent in an SNS junction.
1.3. SNS junction 17 Figure 1.8. Closed-form solution of quasi-particle LNDoS in the short junction limit. The four columns report the LNDoS N versus the normalized coor dinate ˜x and the ener gy , given by equations (1.39 ,1.40 ,1.41 ,1.42 ). The first row reports N ( ˜x, ) as 3D plots, while the second row contour plots. It is possible to appr eciate the evolution of the LNDoS fr om the center , given by equation (1.44 ), to the SN interfaces, given by the BCS form (1.9 ). The phase values ar e chosen accor ding the clarity of the plots. In this figur e, the gap is at zer o temperatur e ∆ 0 .
Figure 1.9. Spectral supercurrent in the short junction limit. (a)Color plot of ˜jspas function
of and ϕ. We can note that the spectral supercurrent is energetically confined between the mini-gap and the gap. (b) Related cuts from panel (a) for chosen ϕ in legend. (c) Supercurrent I, obtained from the spectral supercurrent in panel (a).
The alteration of the LNDoS in a N weak link has been demonstrated with tunneling experiments. Let us consider indeed a normal metal electrode, dubbed probe, in tunnel contact with the weak link over a point-like area around xpr. Let us consider moreover an equilibrium temperature T and a tunnel
resis-tance in the normal state Rt. When a voltage difference V is applied to this
junction, a normal electron dissipative current flows, given by the integral [6] I(V ) = 1
eRt
Z ∞
−∞
N (, x = xpr, ϕ)[f ( − eV, T ) − f (, T )]d . (1.47)
From this integral, the differential conductance G(V ) = dI /dV can be calcu-lated. At low temperature T → 0, the differential conductance is
G(V ) = 1 Rt N = eV ∆0 , x = xpr, ϕ , (1.48)
i.e., it is proportional to the LNDoS.
Figure 1.10 reports the results of tunneling experiments from References [72, 75]. In both devices, the weak link was phase-biased by a superconducting
1.3. SNS junction 19
Figure 1.10. Experimental evidence of the induced mini-gap.(a)SEM micrograph of a SQUIPT device, consisting of a superconducting aluminum ring interrupted by a normal-metal copper weak link. The latter is in tunnel contact with a normal-metal aluminum-manganese electrode.
(b)Color plot of the differential conductance versus the threading flux Φ and the voltage bias V. (c) Representative image showing the STM experiment on a device made of an Al ring inter-rupted by an Ag weak link. The STM tip allows measuring the differential conductance versus position and voltage bias. (d) Differential conductance versus the voltage bias for chosen posi-tions on the device. The gap is ∼ 200 eV wide on the ring (typical Al value), while the mini-gap in the center of the junction is ∼ 100 eV. Panels (a), (b) are adapted from Ref. [72]. Panels (c), (d) are adapted from Refs. [75, 76].
ring pierced by a magnetic flux Φ, through the quantization expression ϕ = 2πΦ/Φ0, and Φ0 = h/2e ≈ 2 × 10−15Wbis the flux quantum. In Ref. [72], the
probe electrode is nanofabricated in contact with the center of the weak link, as reported in figure 1.10(a). Figure 1.10(b) shows a color plot of the differential conductance as a function of the voltage bias V and flux Φ, showing the depen-dence of the mini-gap ˜∆on ϕ in the middle of the junction.
In Ref. [75], the probe is constituted by the tip of a Scanning Tunnel Micro-scope (STM), as sketched in figure 1.10(c). In this case, the tip can be spatially moved and the DoS can be probed over the device surface. As predicted by the theory, the LNDoS evolves from a BCS form on the bulk of the ring to a mini-gapped form in the weak link, see figure 1.10(d).
The findings of the quasi-classical theory concerning the quasi-particle DoS have been confirmed in other experiments, see Refs. [16, 17, 71, 76, 93–98] and Pubs. VI,VII.
Experimental results concerning the supercurrent in an SNS junction are reported in figure 1.11. Panel (a), adapted from Reference [99], displays the
Figure 1.11. Experimental measurements of supercurrent in SNS junctions. (a)Temperature dependence of the critical current Icversus the temperature T in Nb/Cu/Nb junctions. The dots
are the experimental measurements. The solid lines represent the theoretical prediction with a temperature-independent gap. The dashed lines represent the theoretical prediction with the BCS temperature-dependent gap. The fits are made with the Thouless energy εTh as the only
parameter. The three groups of curves correspond to different samples. Adapted from Ref. [99].
(b)Supercurrent versus phase in Nb/Ag/Nb junctions. In particular, the skewing of the CPR is reproduced. Adapted from Ref. [100].
measured critical current Ic(dotted curves) versus the temperature T for three
junctions of different length. The theoretical curves (dashed lines), are obtained through a fit with the Thouless energy as the only free parameter. Figure 1.11(b) shows the measured CPR of an SNS junction. In this case, the theory is able to reproduce the non-sinusoidal skewed CPR, typical of SNS junctions. Other suc-cesses of the quasi-classical theory in the treatment of the supercurrent transport in hybrid systems can be found in the reviews [5, 84].
1.4
Discussion
In this chapter, we quickly summarized the main points of proximity effect and the related treatment within the quasi-classical theory of superconductiv-ity. We introduced the main applications of hybrid systems and the underly-ing microscopical mechanism of Andreev reflection, which determines the for-mation of Andreev bound states in SNS junctions. An overview of the mathe-matical methods for a diffusive hybrid system in thermal equilibrium has been also given, introducing the momentum-averaged Green function ˆgR, the Usadel
equations, and how the LNDoS N and the spectral supercurrent ˜jsp can be
ex-tracted from ˆgR. These methods have been applied to the specific case of an SNS
junction, showing the induced mini-gap ˜∆in the weak link and the emergence of a Current-Phase Relation I(ϕ, T ). In particular, closed-form solutions of the Usadel equations are discussed for the short junction limit LN/ξ0 1. Finally,
1.4. Discussion 21 we reported many experimental results that show the existence of the induced mini-gap and the skewed CPR in SNS junctions.
Chapter 2
Quasi-particle entropy in an SNS
junction in diffusive limit
In this chapter, we study the quasi-particle entropy in an SNS junction and its dependence on the phase difference across it. This point can be demonstrated from both a statistical and thermodynamic point of view. The statistical argu-ment relies on the fact that entropy is related to the population of quasi-particles, which density of states depends on the junction phase, as a consequence of the proximity effect. This microscopical mechanism is hence the backbone of en-tropy dependence on the junction phase drop.
From a thermodynamic point of view, the phase-dependence of entropy is implied by a requirement of thermodynamic consistency. Indeed, a Maxwell relation links the entropy state function with the junction current-phase relation, connecting hence these two phase-dependent quantities.
Both the statistical and thermodynamic arguments correspond to two meth-ods to calculate the phase-dependence of entropy in an SNS junction: the for-mer involves a statistical integral on the quasi-particle density of states, while the thermodynamic approach is based on the Maxwell relation applied to the current-phase relation. The two methods must yield the same result; however, we show in this chapter that particular boundary conditions in the microscopic model can yield thermodynamically inconsistent results.
Specifically, we demonstrate numerically that rigid boundary conditions in the Usadel equations return results that violate the Maxwell relation between current and entropy. Moreover, we show that this point can be cured within a different microscopic model, based on continuity boundary conditions.
The result of this discussion is to highlight the important role in the ther-modynamic properties played by the inverse proximity effect, i.e., the anti-proximization of the superconducting leads operated by the normal weak link. Moreover, the current/entropy Maxwell relation yields a powerful recipe to cal-culate the thermodynamic properties of a hybrid system.
mechanism in an SNS junction, showing that it can be calculated through a sta-tistical integral or from the Maxwell relation. The section ends by showing the thermodynamic inconsistency between these two methods when calculations are made within a model based on rigid boundary conditions. This issue is then solved in section 2.2, where we recover the thermodynamic consistency employ-ing a model with continuity boundaries, which includes the inverse proximity effect. The latter is quantitatively studied in section 2.3. Finally, we briefly show an application to the calculation of the heat capacity in the proximized junction.
2.1
Thermodynamic properties of an SNS junction
In the previous chapter, we have shown that, in the quasi-classical theory of su-perconductivity, the system is described by a Green function ˆgR(, r)depending on energy and position r. From ˆgRit is possible to extract spectral quantities
that allow obtaining macroscopic quantities through statistical integrals. For ex-ample, from the spectral supercurrent jsp, it is possible to obtain the supercurrent
density or the CPR of a junction.
In a similar fashion, in this section we exploit the LNDoS N (, r) to calculate the entropy density of a hybrid system. The entropy density is related to the LNDoS through the relation
S(ϕ, T, r) = −4N0,r
Z ∞
−∞
N (r, , ϕ)f (, T ) log(f (, T ))d , (2.1) where N0,ris the DoS at Fermi energy per spin of the material at position r and
f is the Fermi distribution f () = 1
e/T + 1 . (2.2)
The total entropy of a system is given by the integral over its volume V S(ϕ, T ) =
Z
V
S(ϕ, T, r)d3r .
(2.3) We remark that expression (2.1) is obtained from the sum-over-states S = = −(kB)P
R
i [filog fi+ (1 − fi) log(1 − fi)] through the f symmetry f () = 1 −
f (−)[101].
Two direct applications of equation (2.1) concern for example the limit-cases of bulk N and S metals. For a N metal, it is N = 1 and the R f () log f ()d = π2T /6, yielding the well known Sommerfeld expression
SN(T ) =
2π2
2.1. Thermodynamic properties of an SNS junction 25 It is worth to remark that this results holds in the N = 1 approximation (dubbed Cooper approximation), based on the fact that we consider temperatures low enough to neglect the energy dependence of the DoS [101].
For a superconductor, the entropy density is SBCS(T ) = −4N0,S Z ∞ −∞ Re ( || p2− ∆2(T ) ) f (, T ) log(f (, T ))d , (2.5)
obtained substituting N with NBCS in equation (1.9). At low temperatures T
∆0, SBCS(T ) has the exponentially suppressed form due to the presence of the
gap [7, 102] SBCS(T ) ≈ √ 2π r ∆0 T e −∆0/TV N 0,S∆0 . (2.6)
From the expression of the entropy density S(T, r), other thermodynamic quantities can be derived, for example the specific heat:
C(T ) = T∂S
∂T . (2.7)
The heat capacity of a system is given by the integral over its volume V C(ϕ, T ) =
Z
V
C(ϕ, T, r)(d)3r . (2.8)
The specific heat for a normal metal is CN(T ) =
2π2
3 N0,NT . (2.9)
For a superconductor, at low temperatures CBCS(T ) ≈ √ 2π ∆0 T 3/2 e−∆0/TN 0,S∆0 . (2.10)
Panels (a,b) of figure 2.1 show respectively the entropy and the specific heat of both N and S metals. Above the critical temperature, ∆(T ) = 0 is zero and the DoS recovers the normal metal form N = 1. We can note the well known second-order transition consisting in a continuous entropy and in a discontinu-ous specific heat.
In the case of an SNS junction, the LNDoS N is a function of the phase ϕ, implying that the entropy S is a function of the phase. In particular, due to the mini-gap dependence on phase, the entropy increases from ph = 0 to ϕ = π for a fixed temperature. Additionally, the entropy has to be an even function and 2π-periodic on ϕ. We investigate in detail the dependence of S on ϕ in the next
Figure 2.1. Entropy density and specific heat for N and S metals,in panels (a) and (b), respec-tively. N0,jis accordingly the DoS at Fermi level of the N or S metal.
subsection.
Current-entropy Maxwell relation and inconsistency
We have discussed how to obtain from ˆgRboth the CPR I(ϕ, T ) and the entropy
state function S(ϕ, T ) of a junction. Now, these two quantities are not indepen-dent: indeed, an argument of thermodynamic consistency requires that I and S are linked by a Maxwell relation. The thermodynamic consistency is a general constraint, and it is valid for any junction.
Let us consider the free energy F that describes the SNS junction. The free energy can be calculated within the quasi-classical theory of superconductivity [11, 12, 103, 104] (see also Pub. V). The CPR is related to the free energy with
~
2eI(ϕ, T ) =
∂F (ϕ, T )
∂ϕ . (2.11)
This relation underlines that the supercurrent is a thermodynamic variable and that the CPR is a thermodynamic equation of state connecting I, ϕ and T . In the same fashion, another thermodynamic equation of state is given by S(ϕ, T ), since
S(ϕ, T ) = −∂F (ϕ, T )
∂T . (2.12)
The thermodynamic consistency implies that the two cross derivatives of F are identical: ∂ϕ∂TF = ∂T∂ϕF. Hence, the following Maxwell relation is universally
valid −∂S(ϕ, T ) ∂ϕ = ~ 2e ∂I(ϕ, T ) ∂T . (2.13)
2.1. Thermodynamic properties of an SNS junction 27
Figure 2.2. Thermodynamic inconsistency within the RBC calculations. (a)Left and right-hand sides of equation (2.13) versus ϕ at fixed temperature T = 0.5Tc for ` = 1. (b) Left and
right-hand sides of equation (2.13) versus temperature at fixed ϕ = π/2 for ` = 1. (c) Relative discrepancy P defined in equation (2.14) as a function of the junction length ` = LN/ξ0.
the differential form dF would be not closed, implying that the potential F cannot be defined univocally. In physical terms, the total energy exchanged with the universe (heat+work) over a closed loop thermodynamic process is not zero, making impossible a description of the system.
However, a scaling argument shows that I and S calculated through the statistical integrals (2.1) and (1.35) within the RBC model do not satisfy the Maxwell relation (2.13) required by thermodynamic consistency. Indeed, the LHS of equation (2.13) scales like the volume of the junction ∂ϕS ∼ LNAN, while
the RHS of (2.13) scales like the resistance of the junction ∂TI ∼ AN/LN. The
different dependence on LNimplies then that the equality (2.13) does not hold.
This is even more evident in the limit LN → 0, where the supercurrent is
ex-pected to diverge while the entropy is exex-pected to go to zero.
The discrepancy between the two sides of the Maxwell relation can be tested also numerically. Figure 2.2(a,b) shows the result of a simulation consisting of calculating ˆgR numerically for ` = 1 and obtaining I and S through the
sta-tistical integrals shown above. The two curves, representing the two sides of the Maxwell relation as function of ϕ or T in the two panels, do not superpose, confirming the discrepancy. This is remarked in figure 2.2(c), displaying the
dependence on ` of the relative discrepancy P = max (ϕ,T ) ∂TI + 2e~∂ϕS ∂TI . (2.14)
It decreases with increasing junction length LN, and remains significant up to
LN several times the coherence length ξ0. The discrepancy becomes negligible
for long junctions (LN ξ0), where the proximity effect becomes weak.
2.2
Inverse proximity effect
The reason of the thermodynamic inconsistency of the numerical results relies on the form of the RBC in equation (1.34). In detail, let us consider a bound-ary condition expressed by an equation B[ˆgR(, x), ] = 0, where B is a
func-tional of ˆgR. In Pub. V it is demonstrated that if B is explicitly energy-dependent,
the entropy density expression (2.1) is not equivalent to −∂TF , being F is the
free energy density. Indeed, the RBC are explicitly energy-dependent, since B[ˆgR(), ] = ˆgR() − ˆgR
BCS().
We note that the consistency is broken by explicit dependence of B, and not by implicit dependence on contained in ˆgR itself. An example of boundary
condition that does not depend explicitly on is provided in an NSN junction, where B[ˆgR(), ] = ˆgR() − ˆgR
N, where ˆgNR is the bulk Green function of a
nor-mal metal. Another example of energy-independent boundary equation is an insulating barrier, given by n · ∇ˆgR= 0.
As a consequence, in order to have consistent thermodynamic results for an SNS junction, a different model with energy-independent boundaries must be elaborated.
Continuity boundary conditions
We introduce here the Continuity Boundary Conditions (CBC) model for an SNS junction, where the SN interfaces are described by energy-independent bound-aries. The solution to this model reveals that an important contribution to the total entropy S comes from the inverse proximity effect, consisting of the in-fluence of the N weak link on the S leads. As a consequence, the state of the leads close to the SN interface does not recover abruptly the BCS form ˆgR
BCS, as
imposed in the RBC model.
The CBC model is depicted in figure 2.3. In this model, the junction is di-vided in 5 regions, forming an SS’NS’S scheme, where N is the weak link, S’ is the inverse proximized region, and S is the bulk. In this model, the anti-proximized region has a fictitious boundary with the bulk at a distance LS from
semi-2.2. Inverse proximity effect 29
Figure 2.3. Continuity boundary conditions model.The model consists of three regions where ˆ
gRis calculated, which are the weak link N and the two inverse-proximized superconducting
leads S’. In the scheme, the Green functions ˆgR
S0,1, ˆgRN, ˆgSR0,2in the three regions are matched at
the SN interfaces with the continuity boundary conditions in equations (2.15) and (2.16). The scheme is complemented by rigid boundaries at a distance ˜LSfrom the SN interface.
infinite leads approaches zero when ˜LS → ∞. In the simulations, we assume
˜
LS = 5ξ0.
The solution to the model consists of calculating ˆgR using the Usadel
equa-tions in the three regions S’NS’. The compatibility of the results with the Maxwell relation holds independently of the form of the pair potential. For simplicity, some calculation are performed non-self-consistently with constant ∆ = ∆eiφ, where φ = 0 or ϕ according to considered lead. In the following plots, we specify if the pair potential is calculated self-consistently or not.
In the N weak link and in the inverse proximized leads S’. At the SN inter-faces, x = 0 and x = LN), the CBC consist in two matching conditions. The first
imposes the continuity of the Green function ˆ
gR(, x)|x→0− = ˆgR(, x)|x→0+ , (2.15)
and similarly in x = LN. The second imposes the continuity of the matrix current
ˆ gR∇ˆgR[6, 9, 105]: σSASˆgR(, x)∂xgˆR(, x) x→0− = σNANˆgR(, x)∂xgˆR(, x) x→0+ , (2.16)
and similarly in x = LN. The boundary condition at the interface S’S define the
superconducting phase ˆ gR(, x)|x→− ˜L+ S = ˆg R BCS(, φ = 0) (2.17) ˆ gR(, x)|x→( ˜L S+LN)− = ˆg R BCS(, φ = ϕ) . (2.18)
Figure 2.4 reports a comparison between the numerical results within both the RBC and CBC models, with ` = 1 and a = 1. It is worth to underline that a is
Figure 2.4. DoS calculated with RBC or CBC.In the calculations, it is ` = 1, a = 1, and ∆ is non-self-consistent. The phase difference ϕ and the used model are written in each panel. The dashed vertical lines delimit the N weak link region.
a parameter that appears only in the CBC model. Two features can be noticed: first, the induced mini-gap in the weak link is reduced in the CBC model, due to the fact that the inverse proximization on the leads increases the “effective length” of the junction. Second, the DoS in the leads is no more BCS in the CBC case and it is phase-dependent.
The latter characteristic implies that an important phase-dependent contri-bution to the junction total entropy comes from the leads. Let us indeed consider the entropy contributions from the weak link SWLand from both the leads SLe
SWL(ϕ, T ) = Z weak link S(ϕ, T, x)dx (2.19) SLe(ϕ, T ) = Z leads S(ϕ, T, x)dx . (2.20)
Figure 2.5 reports ∂ϕSWL and ∂ϕSLe calculated numerically for ` = 1, a = 1
and non-self-consistent ∆, hence through the numerical integration of the DoS in figure 2.4 in the CBC. We note that ∂ϕSLe is greater than ∂ϕSWL, showing
that for the specific simulation parameters the contribution from the leads is bigger than the contribution from the weak link. Finally, the curve of the total entropy ϕ-derivative ∂ϕS = ∂ϕSWL + ∂ϕSLe is superposed on the curve of the
2.2. Inverse proximity effect 31
Figure 2.5. Numerical test of thermodynamic consistency with inverse proximity effect.The panels show −eRN∂TI and 2e2RN∂ϕS, respectively the right hand side and left hand side of
equation (2.13). The entropy contributions from the weak link and the leads, S = SLe+ SWL,
are shown separately. (a) The two sides of the Maxwell relation versus ϕ for fixed temperature T = 0.5Tc. (b) The two sides of the Maxwell relation versus T for fixed phase ϕ = π/2. The
simulation parameters are a = 1, ` = 1, and non-self-consistent ∆.
are thermodynamically consistent.
The amount of entropy that is distributed among the weak link and the leads depends on ` and a. In particular, for short junctions, the supercurrent increases as ∼ 1/LN while the volume decreases as ∼ LN, suggesting that in the short
junction limit LN → 0 the phase-dependent part of entropy is “pushed into the
leads”.
Heat capacity
As an application, we consider now the difference between the RBC and the CBC results for an observable thermodynamic quantity, the heat capacity. Fi-gure 2.6 reports the heat capacity variation δC(ϕ, T ) = C(ϕ, T ) − CBCS(T )
cal-culated numerically for a junction with ` = 1, a = 500 within the RBC and CBC models. At ϕ = 0 is almost zero, since C(ϕ = 0, T ) ≈ CBCS(T ) due to the fact
that the chosen parameters `, a make the junction well proximized. Then, in both the models δC increases due to the closure of the mini-gap. However, the calcu-lated heat capacity variation in the RBC is three orders of magnitude lower com-pared to the CBC result, where the contribution of the superconducting leads is taken into account. Moreover, it is shown in Pub. V that the numerical result within the CBC is compatible with the heat capacity calculated from the KO CPR through the Maxwell relation, as expected since the chosen a, l are compatible with the assumptions of the KO theory. More details about the calculation of the heat capacity from the KO CPR will be studied in the next chapter.
Figure 2.6. Heat capacity variation on phase.The heat capacity variation δC(ϕ, T ) = C(ϕ, T ) − CBCS(T )calculated for ` = 0.1, a = 500, self-consistent ∆ within the RBC model (dashed lines)
and the CBC model (solid lines). The CBC yields a heat capacity variation greater three orders of magnitude compared to the RBC result, due to the fact that it takes into account the entropy variation in the leads.
2.3
Analytical results for the inverse proximity
Let us consider the inverse proximity effect in more detail. In order to quantify it, we define the proximity-induced DoS variation δN
δN (, x) = N (, x) − NBCS() , (2.21)
and the proximity-induced entropy variation in the leads δSLe(T, x) = S(T, x) − SBCS(T ) = = −4N0,SAS Z ∞ −∞ d Z leads dx δN (, x, ϕ)f (, T ) ln f (, T ) . (2.22) We limit the following considerations to the entropy in the leads, since the ana-lytical solutions can be found in the short junction limit, which makes the contri-bution from the weak link negligible. We note that the quasi-1D approximation transforms the volumetric integral over the leads in an integral over x multi-plied the cross-section AS.
Figure 2.7 shows δN as a function of energy and position. As expected, δN is localized around the junction; far from the junction, the DoS recovers the BCS form and δN goes to zero. Interestingly, the plot of δN remarks the variation of the available states due to the modulation of the mini-gap.
Analytical solutions can be obtained in the limiting cases of short junction ` 1at phase differences ϕ = 0 and ϕ = π. A solution to the Usadel equation in a semi-infinite superconducting wire with uniform ∆ = ±|∆| is given in the
2.3. Analytical results for the inverse proximity 33
Figure 2.7. DoS variation in a proximized junction. The color plots show δN = N (, x) − NBCS()in the N weak link and in the inverse-proximized S leads. In the simulation, ` = 1, a =
1, non-self-consistent ∆. The x-axis indicates the distance from the NS interfaces. The two vertical dashed white lines delimit the weak link. (a) Phase ϕ = 0 (mini-gap open). (b) Phase ϕ = π(mini-gap closed). θparametrization by θ(x) = θBCS− 4 artanh e− √ 2(x−LN)/ξtanhθBCS− θ(LN) 4 , (2.23)
and ξ = (1−2/|∆|2)−1/4ξ0and θBCS = artanh(|∆|/ + i0+)is the θ representation
of the bulk solution ˆgBCSR . The spatially integrated change in the leads DoS can be evaluated based on this solution:
Z leads dx δN (x, ) =√2Re ξcosh θBCS cosh θBCS− θ(LN) 2 − 1+ −ξsinh θBCSsinh θBCS− θ(LN) 2 (2.24) For LN ξE, the Usadel equation in the N region can be approximated as
∂x2θ(x) = 0. Matching to the boundary condition σNAN∂xθN = σSAS∂xθBCS at
the two SN interfaces results to θ(LN) = ( θBCS for ϕ = 0, √ 2(ξ/ξ0)a` sinh θBCS−θ(LN) 2 for ϕ = π , (2.25)
from which θ(LN) can be solved. For the entropy at ϕ = 0, this gives a trivial
solution δSLe = 0. On the other hand, at ϕ = π, we have for temperatures
T |∆|, δSLe(ϕ = π) ' 4π2 3 T N0,SASξ0 × ( 1 , for a` 1, π 2a`, for a` 1. (2.26)
Figure 2.8. Entropy variation δSLein the superconducting leads. (a)Temperature dependence
at ϕ = 0 for a = 1 and chosen ` in legend. (b) Same as panel (a) at ϕ = π. Result (2.27) for ` = 0is also shown (dashed). (c) Dependence of δSLe(ϕ = π)on ` and a, at T /Tc= 0.1. Dashed
curves report the limit results of equation (2.26).
The full temperature dependence for ` → 0 reads δSLe(ϕ = π) = − 16 √ 2N0,SAS× Z ∞ −∞ d Re (coshθBCS 2 − cosh θBCS)ξ f (, T ) ln f (, T ) . (2.27) For cross-over regions, the boundary condition matching would need to be solved numerically.
The behavior in the rigid boundary condition limit (a → ∞) can be un-derstood based on the above result. For the entropy, the short-junction rigid-boundary limit a → ∞, ` → 0 is not unique, but results depend on the product a`. Generally, the entropy is proportional to ~/(e2R
tot), where Rtot is the
resis-tance of the ξ0-length superconductor segment in series with the normal wire,
as can be expected a priori [5].
Figure 2.8(a,b) shows the geometry dependence of the inverse proximity effect contribution δSLe to the entropy, for ϕ = 0 and ϕ = π. Generally,
δSLe(ϕ = 0)decreases with decreasing junction length and approaches the limit
of δSLe(ϕ = 0) → 0for l → 0. The temperature dependence of δSLe(0)is largely
affected by the presence of a mini-gap in the spectrum, S(0) ∼ e− ˜∆/T. For
2.4. Discussion 35 increases with decreasing length, in accordance with the increase of the super-current with decreasing junction resistance. For very short junctions, ` . a−1,
δSLe saturates as indicated in equation (2.26). The behavior of δSLe(ϕ = π) at
phase difference ϕ = π as a function of the product al is shown in figure 2.8(c). It is interesting to note that the results are essentially converged to the short-junction limit ` 1 already at ` = 1.
2.4
Discussion
In this chapter, we have shown numerically and with a scaling argument that energy-dependent boundary conditions in the calculation of the Usadel equa-tions can yield a current-phase relation and a quasi-particle entropy which do not satisfy the thermodynamic consistency expressed by a Maxwell relation be-tween these two quantities. On the contrary, we have demonstrated that a dif-ferent model able to take into account the inverse proximity effect in the super-conducting leads returns thermodynamically consistent results. In particular, we have found out numerically and analytically the relevant role of inverse proximity in short junctions. As an application, we calculated the heat capac-ity numerically, both including and neglecting the contribution from the inverse proximized leads: as a result of the calculation, the heat capacity modulation on phase is three orders of magnitude bigger when including the contributions due to the inverse proximity effect.
A proper quantitative calculation of entropy and thermodynamic quantities taking into account the inverse proximity effect is thus of importance both for fundamental and application purposes. The results obtained can be used in designing superconducting devices concerning caloritronics, bolometric photo-detection, and are in general relevant also for other devices based on thermody-namic working principles.
Contributions
This chapter is based on Publication V, realized by the collaboration of F.V., author of this thesis, with P. Virtanen (first author), E. Strambini, M. Carrega, F. Giazotto. P.V. conceived the article; F.V. and P.V. carried out the numerical calculations; F.V. and E.S. worked on the figures; F.G. supervised the work. All authors participated to discussion and writing.
