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The modynamics o a Phase-D i en P oximi y Josephson Junc ion
© 2019 by he au ho s
Published e sion
Vischi, F ancesco; Ca ega, Ma eo; B aggio, Alessand o; Vi anen, Pauli;
Giazo o, F ancesco
Vischi, F., Ca ega, M., B aggio, A., Vi anen, P., & Giazo o, F. (2019). The modynamics o a
Phase-D i en P oximi y Josephson Junc ion. En opy, 21(10), A icle 1005.
h ps://doi.o g/10.3390/e21101005
2019
en opy
A icle
The modynamics o a Phase-D i en P oximi y
Josephson Junc ion
F ancesco Vischi 1,2,* , Ma eo Ca ega 1, Alessand o B aggio 1and Pauli Vi anen 3
and F ancesco Giazo o 1
1NEST, Is i u o Nanoscienze-CNR and Scuola No male Supe io e, Piazza S. Sil es o 12, I-56127 Pisa, I aly;
[email p o ec ed].i (M.C.); alessand [email p o ec ed].i (A.B.); [email p o ec ed] (F.G.)
2Dipa imen o di Fisica ”E. Fe mi”, Uni e si à di Pisa, La go B uno Pon eco o 3, I-56127 Pisa, I aly
3Depa men o Physics and Nanoscience Cen e , Uni e si y o Jy äskylä, P.O. Box 35 (YFL),
FI-40014 Jy äskylä, Finland; [email p o ec ed]
*Co espondence: [email p o ec ed]
Recei ed: 20 Sep embe 2019; Accep ed: 13 Oc obe 2019; Published: 15 Oc obe 2019


Abs ac :
We s udy he he modynamic p ope ies o a supe conduc o /no mal me al/supe conduc o
Josephson junc ion in he sho limi . Owing o he p oximi y e ec , such a junc ion cons i u es
a he modynamic sys em whe e phase di e ence, supe cu en , empe a u e and en opy a e
he modynamical a iables connec ed by equa ions o s a e. These allow concei ing quasi-s a ic
p ocesses ha we cha ac e ize in e ms o hea and wo k exchanged. Finally, we combine such
p ocesses o cons uc a Josephson-based O o and S i ling cycles. We s udy he ela ed pe o mance in
bo h engine and e ige a o ope a ing mode.
Keywo ds:
p oximi y e ec ; supe conduc i i y; Josephson junc ion; SNS junc ion; Josephson
he modynamics; Maxwell ela ion; quasi-pa icles en opy; quan um he modynamics; quan um
machines; quan um coole s
1. In oduc ion
The modynamic concep s ha e been ecen ly conside ed a he nanoscale, concei ing and ealizing
sys ems whe e quan um cohe en p ope ies a e mi o ed in he modynamic quan i ies a mesoscopic
le el [
1
–
14
] . Fu he mo e, one o he mos imp essi e examples o quan um ea u es e lec ed in
mac oscopic sys ems is ep esen ed by supe conduc i i y, whe e quan um cohe ence is mani es ed a
a mesoscopic scale. The e o e supe conduc ing sys ems a e in e es ing pla o m whe e in es iga ing
he in e play be ween he modynamic concep s and quan um cohe ences.
Supe conduc ing hyb id sys ems, i.e., cons i u ed o supe conduc ing pa s in elec ic con ac
wi h no mal (non-supe conduc ing) pa s, a e in p ac ice cohe en elec on sys ems wi h s iking
he modynamic equilib ium/ anspo p ope ies, esul ing in a wide a ie y o applica i e de ices:
low- empe a u e sensi i e he mome e s [
15
–
19
], sensi i e de ec o s [
20
–
32
], hea al es [
33
–
43
],
calo i onics (hea compu ing) [
11
,
37
,
44
–
48
], solid-s a e mic o- e ige a o s [
18
,
49
–
55
], solid-s a e
quan um machines [56–61], he moelec ic gene a o s [62–67].
In his pape , we e iew he equilib ium he modynamic p ope ies o a hyb id sys em based
on a Supe conduc o /No mal me al/Supe conduc o (SNS) Josephson Junc ion in he di usi e limi .
The beha io o such a sys em is uled by he p oximi y e ec , which consis s in a se o physical
phenomena owing o he p opaga ion o he supe conduc ing elec on co ela ions in he no mal
me al [68–70]
. In pa icula , guided by a ma e o he modynamic consis ency, we discuss a ela ion
be ween he elec onic and he mal p ope ies o he p oximized sys em. F om his ela ion, we de elop
a basic in es iga ion o he he modynamic p ope ies o such a sys em. These esul s a e hen exploi ed
En opy 2019,21, 1005; doi:10.3390/e21101005 www.mdpi.com/jou nal/en opy
En opy 2019,21, 1005 2 o 33
o in es iga e quasi-s a ic p ocesses and he modynamic cycles. We ocus wi hin a semi-classical egime
o Josephson coupling, i.e., we neglec non-commu a i i y be ween he phase and he numbe o pai s,
as usually done in he he modynamic limi .
We ema k ha , besides he sys em s udied in his pape , many equilib ium he modynamic
p ope ies ha e been in es iga ed in di e en condi ions, heo e ically and expe imen ally:
he modynamics o ings in e up ed by insula ing Josephson Junc ion [
71
–
75
], hea capaci y in SN
sys ems [
76
–
78
], ee ene gy in hyb id SN sys ems due o bounda y e ec s wi h app oaches di e en
o he quasi-classical heo y [79–81].
The pape is o ganized as ollows. Sec ion 2desc ibes he p oximized sys em unde s udy and
in oduces i s he modynamics, gi ing also an insigh in o he unde lying mic oscopical mechanism.
Sec ion 3s udies he he modynamic p ocesses. Hence, hese a e combined in Sec ion 4 o in es iga e
wo di e en he modynamic cycles. Finally, Sec ion 6summa izes and discusses he main indings. Fo
comple eness, Appendix Adiscusses he he modynamics o a Josephson junc ion close o he c i ical
empe a u e.
2. The modynamics o Hyb id Sys ems
2.1. Model
We conside a sys em as ske ched in Figu e 1, cons i u ed by a supe conduc ing ing
in e up ed by a Supe conduc o /No mal me al/Supe conduc o (SNS) p oximi y Josephson Junc ion.
The supe conduc o gap depending on empe a u e
T
is
∆(T)
and eaches
∆0
a
T=
0. The c i ical
empe a u e is
Tc
. The phase di e ence
ϕ
o he supe conduc ing o de pa ame e ac oss he junc ion is
uled by he magne ic lux h eading he ing, owing o he luxoid quan iza ion ela ion
ϕ=
2
πΦ/Φ0
,
whe e Φ0=h/2e≈2×10−15 Wb is he lux quan um.
Figu e 1.
(
a
) Ske ch o he SNS p oximized sys em. I consis s o supe conduc ing ing,
LS
long, pie ced
by a magne ic lux
Φ
. The ing is in e up ed by a no mal me al weak link. The elec on sys em o he
whole de ice is he mally and elec ically isola ed and a empe a u e
T
. The sys em is connec ed o
a he mal ese oi a empe a u e
T
h ough a hea al e
. (
b
) Magni ica ion o he SNS junc ion.
The no mal me al weak,
LN
long, is in clean elec ic con ac wi h he supe conduc ing leads.
Aj
,
Nj
a e
espec i ely he c oss-sec ion and he DoS a Fe mi ene gy o he
j=N
o
S
me al. The phase d op
ϕ
o he supe conduc ing o de pa ame e akes place ac oss he junc ion.
We assume ha he sys em is he mally and elec ically isola ed and a a homogeneous empe a u e
T
, neglec ing he mal g adien s. We conside only hea exchange wi h a ese oi a empe a u e
T
h ough he espec i e hea channel connec ed by a hea al e [33–42], as d awn in Figu e 1.
The junc ion, magni ied in Figu e 1b, consis s in he wo S leads in elec ic con ac wi h an N
weak link. The supe conduc o has a c i ical empe a u e
Tc
and BCS gap a ze o empe a u e
∆0
. The
N weak link and he S leads ha e espec i ely c oss-sec ions
AN
and
AS
, conduc i i ies
σN
and
σS
,
Densi y o S a es (DoS) pe spin a he Fe mi le el
NN
and
NS
. The leng h o he weak link is
LN
,
esul ing in a esis ance
RN=LN/ANσN
. The leng h o he supe conduc ing ing is
LS
. The whole
p oximized sys em ( ing+junc ion) olume is V.
En opy 2019,21, 1005 3 o 33
We make he ollowing assump ions abou he junc ion, in o de o make simple analy ical
p edic ions wi hin he Kulik-Omel’yanchuk (KO) ea men [
82
–
84
]. We conside di usi e cha ge
anspo wi h di usi i y
D
o bo h he S and N pa s. This equi es ha he weak link is longe
han he mean ee pa h
Lm p
:
LNLm p
. The KO ea men holds when he whole junc ion can be
ea ed in a quasi-1 dimensional app oxima ion, i.e., when
AS
,
ANξ2
. The di usi i y de ines he
cohe ence leng h
ξ=√¯hD/∆0
[
85
]. Mo eo e , we conside a sho cons ic ion weak link espec o
he supe conduc ing leads. Quan i a i ely, using he pa ame e s
l=LN/ξ
and
a=σSAS/σNAN
, we
conside a sho junc ion wi h l.1 and a cons ic ion wi h al 1.
The ypical alues o his kind o sys em a e he ollowing. The ing can be made o aluminium,
wi h
∆0≈
180 eV, co esponding o
Tc≈
1.2K [
86
–
88
]. The cohe ence leng h o hyb id Al-based
de ices is abou
ξ≈
150 nm [
89
–
91
]. In he ollowing, we se he Bol zmann cons an o
kB=
1,
implying ha he empe a u es ha e a physical dimension o ene gy while en opy and speci ic hea
a e dimensionless.
2.2. Hyb id Junc ion as The modynamic Sys em
Be o e in es iga ing he he modynamic beha io o ou sys em in de ail, we discuss abou he
he modynamic consis ency unde a gene al poin o iew ha is alid o any Josephson Junc ion (JJ).
In pa icula , we ocus on he ela ion be ween he cu en anspo and he junc ion en opy.
In a JJ, he Cu en Phase Rela ion (CPR) desc ibes he dissipa ionless supe cu en
I(ϕ
,
T)
lowing
ac oss i as unc ion o he phase di e ence
ϕ
and empe a u e
T
[
70
,
83
]. The p ecise o m o he CPR
depends on he geome y and on he ma e ials o he junc ion, and can be calcula ed om he ee
ene gy as
¯h
2eI(ϕ,T) = ∂F(ϕ,T)
∂ϕ (1)
whe e
F(ϕ
,
T)
has o be calcula ed wi hin quan um s a is ical me hods as a unc ion o he s a e
a iables
(ϕ
,
T)
. The CPR cons i u es an equa ion o s a e connec ing
I
,
ϕ
and
T
. Ano he equa ion o
s a e is gi en by he en opy S(ϕ,T)as a unc ion o phase di e ence and empe a u e
S(ϕ,T) = −∂F(ϕ,T)
∂T. (2)
The en opy and he CPR a e necessa ily linked by he modynamic consis ency. Indeed he wo
c oss de i a i es o
F
a e iden ical, i.e.,
∂ϕ∂TF=∂T∂ϕF
, owing o he Schwa z heo em. Hence, he
ollowing Maxwell ela ion is uni e sally alid
−∂S(ϕ,T)
∂ϕ =¯h
2e
∂I(ϕ,T)
∂T. (3)
Using his equa ion, he en opy o he JJ can be exp essed as
S(ϕ,T) = S0(T) + δS(ϕ,T)(4)
whe e S0(T)is he en opy a ϕ=0 and δS(ϕ,T)is he phase-dependen en opy a ia ion
δS(ϕ,T) = −∂
∂TE(ϕ,T)(5)
E(ϕ,T) = eR0
2πZϕ
0I(ϕ0,T)dϕ0(6)
whe e
E(ϕ
,
T)
is he Josephson ene gy s o ed in he junc ion a a gi en empe a u e
T
,
R0=h/
2
e2≈
12.9 kΩ
is he in e se o he quan um o conduc ance. We no e ha he p e ac o in Equa ion (6)
is usually exp essed as
Φ0/
2
π
. We chose he o m
eR0/
2
π
o allow an easie compa ison wi h he
junc ion esis ance RN.
En opy 2019,21, 1005 4 o 33
The en opy
S0(T)
a
ϕ=
0 canno be de e mined om he knowledge o he CPR. Indeed, any
unc ion
S0(T)
o he empe a u e is d opped by he phase de i a i e in Equa ion (3), hence sa is ying
he Maxwell equa ion. The physical solu ion o
S0(T)
can be ound wi hin a mic oscopic model ha
we show in he nex subsec ion.
2.3. P oximi y Induced Minigap
In his subsec ion, we gi e an insigh in o he mic oscopic mechanism which de e mines he
en opy in a hyb id junc ion. In pa icula , we show ha he en opy dependence on empe a u e and
phase is ela ed o he p esence o an induced phase-dependen minigap in he quasi-pa icle Densi y o
S a es (DoS). In a hyb id NS, co ela ed elec ons p opaga e om he supe conduc o in o he no mal
me al, s ongly modi ying he p ope ies o he la e wi h a se o phenomena called gene ically
unde he name o p oximi y e ec [
68
,
69
,
92
–
97
]. Among all possible consequences dic a ed by he
p oximi y e ec , he e we ocus on he induced mini-gap in he quasi-pa icle Densi y o S a es, i being
esponsible o he phase and empe a u e dependence o he en opy Sin an SNS junc ion.
Le us conside he N weak link in an SNS junc ion. When no p oximized, he weak link DoS is
homogeneous and app oxima ely cons an a i s Fe mi le el alue
NN
in he ene gy ange o in e es
o ew
∆0
a ound he Fe mi ene gy. Ins ead, when p oximized by he supe conduc ing leads, he DoS
is no mo e cons an nei he on ene gy no on posi ion, bu is gi en by
NNN(
,
ε
,
ϕ)
, whe e
N(
,
ε
,
ϕ)
is he no malized local DoS [
85
,
94
,
98
,
99
] ha is a unc ion o he posi ion
, ene gy
ε
and he phase
di e ence ϕ.
One way o calcula e he no malized local DoS is p o ided by he quasi-classical heo y o
supe conduc i i y [
85
,
94
–
96
,
98
–
100
]. Quali a i ely, a esul o his heo y is ha he no malized local
DoS is cha ac e ized by an induced gap in he N weak link, whose ampli ude
˜
∆
is smalle han he S
bulk gap
∆(T)
. Fo his eason,
˜
∆
is dubbed induced minigap. This induced minigap has he ollowing
p ope ies [
85
,
98
]: i s wid h
˜
∆
a
ϕ=
0 depends on he weak link leng h
LN
and eaches
˜
∆→∆(T)
when
LN
is well below he cohe ence leng h
ξ
. Mo eo e ,
˜
∆
depends on he phase
ϕ
h ough a unc ion
ha is e en and 2
π
pe iodic. The minigap is ully open a
ϕ=
0 and sh inks ill closu e a
ϕ=π
.
An analy ical solu ion o he local no malized DoS is a ailable o di usi e sho junc ions wi h igid
bounda y condi ions [
101
,
102
], yielding ha
˜
∆=∆(T)|cos(ϕ/
2
)|
. The p oximi y induced gap and i s
in e es ing p ope ies ha e been obse ed expe imen ally by unneling expe imen s [89,90,103].
An impo an ea u e o his mic oscopic p oximi y DoS modi ica ion is ha i does no ake place
jus in he N weak link, bu also a ec s he S leads as well. The an i-p oximiza ion ope a ed by he
N weak link on he S leads is called in e se p oximi y e ec and plays he ole o a c ucial co ec ion
in sho junc ions, since i gi es an impo an con ibu ion o he o al en opy dependence on he
junc ion phase [84].
A nume ical example o he phase-dependence o he local no malized DoS
N
in a junc ion is
epo ed in Figu e 2, wi hin he quasi-classical me hods o Re e ence [
84
], calcula ed o a junc ion wi h
pa ame e s
l=
0.1 and
a=
10,
∆(T→
0
) = ∆0
. The colo plo s show he e olu ion o he no malized
local DoS
N
e sus ene gy
ε
and spa ial posi ion
x
o ou alues o
ϕ
om 0 o
π
. The blue a ea
co esponds o he gapped pa o he local DoS; he whi e a ea is he sa u a ion colo ha is associa ed
o he di e gence o he DoS a he gap edges. The posi ion is no malized o he cohe ence leng h
ξ
: as
shown in he i s panel, he cen al zone
x∈[
0, 1
]
coincides wi h he N weak link, while he la e al
zones a e he supe conduc ing leads. A
ϕ=
0, he DoS is homogeneous and is app oxima i ely gi en
by he BCS o m
NBCS(ε,T) = <|ε|
pε2−∆2(T). (7)
The spa ial homogenei y is due o he ac ha he calcula ion in ol es a sho junc ion, o he wise he
induced minigap would ha e been smalle han
∆0
[
84
,
101
,
102
]. Inc easing
ϕ
, he induced minigap
sh inks ill he comple e closu e a
ϕ=π
. I is possible o app ecia e also he in e se p oximi y e ec
in he S leads, ou side he s ipe delimi ed by he ed dashed lines in Figu e 2.

En opy 2019,21, 1005 5 o 33
Figu e 2.
Colo plo s o he quasi-pa icle local no malized Densi y o S a es (DoS)
N
in a
Supe conduc o /No mal me al/Supe conduc o (SNS) junc ion, e sus ene gy
ε
and posi ion
x
,
o
ϕ=
0,
π/
3, 2
π/
3,
π
. The dashed lines sepa a e he S egions (on he sides) o he N egion
(in he cen e ), as shown by he junc ion ske ch. The phase dependence o he DoS is mi o ed in a
phase-dependence o he junc ion en opy
S
. The nume ical calcula ion has been ob ained wi hin he
quasi-classical me hods o Re e ence [84] wi h a=10, l=0.1, ∆(T→0) = ∆0
The phase dependence o he quasi-pa icle DoS implies a phase dependence o he junc ion
en opy. The o al en opy is [84,104–107]
S(ϕ,T) = ZVol S(ϕ,T, )dV(8)
S(ϕ,T, ) = −4N Z∞
−∞N( ,ε,ϕ) (ε,T)log( (ε,T))dε(9)
whe e N is NSo NNwhe he is in he leads o he weak link.
A his poin , we ha e wo ways o calcula e he en opy a ia ion
δS(ϕ
,
T)
. One consis s in
calcula ing
δS(ϕ
,
T)
om
I(ϕ
,
T)
exploi ing he Maxwell ela ion h ough Equa ions (3)–(6). The o he
way is by means o Equa ions (8) and (9) gi en by he s a is ical a gumen abo e conce ning
he quasi-pa icle densi y o s a es. I is a non- i ial esul ha he wo app oaches gi e esul s
in
ag eemen [80,84,108]
. This is an equilib ium he modynamic ea u e due o he ac ha he
equilib ium supe cu en is ca ied by he And ee Bound S a es (ABS), whose spec al densi y is non
ze o below he supe conduc ing gap
|ε|<∆(T)
[
85
,
101
]. The quasi-pa icle DoS and he ABS spec al
densi y a e bo h unc ions o ϕ, ensu ing ha he wo app oaches a e equi alen .
We conclude his discussion by calcula ing
S0
. As discussed in Sec ion 2.2, his quan i y can no be
ob ained by he Maxwell ela ion (3), cons i u ing hence an unde e mined unc ion o he empe a u e
T
in Equa ion (4). Howe e ,
S0
can be de e mined wi h a s a is ical mechanics app oach. Gi en he
assump ions o Sec ion 2.1 o sho junc ion
l.
1 and
a
1, he local no malized DoS a
ϕ=
0 is
gi en by he BCS exp ession (7) [84]. Hence
S0(T) = −4VNSZ∞
−∞<|ε|
pε2−∆2(T) (ε,T)log( (ε,T))dε. (10)
In ob aining his exp ession om (9), we neglec ed ha
NS6=NN
in gene al. Howe e , since he
junc ion olume is negligible espec o he o al olume, we ha e app oxima ed he p e ac o wi h
NSVleads +NNVweak L ≈VNS.
Below, we discuss his esul wi hin he ull dependence o he o al en opy Son ϕand T.
En opy 2019,21, 1005 6 o 33
2.4. Kulik-Omel’yanchuk Theo y
The Kulik-Omel’yanchuk heo y, whose assump ions ha e been in oduced in Sec ion 2.1,
p o ides he CPR [82,83,101]
I(ϕ,T) = π∆(T)
eRN
cos ϕ
2Z∆(T)
|∆(T)cos(ϕ/2)|
1
pε2−∆2(T)cos2(ϕ/2) anh ε
2Tdε. (11)
This exp ession [
101
] is equi alen o he Ma suba a summa ion o m p esen ed in he i s pape
abou he KO CPR [
82
]. He e we adop he in eg al o m ha allows o ind simple closed exp essions
in he limi T∆0.
In he ze o- empe a u e limi T→0, he KO CPR educes o [82]
I(ϕ,T=0) = π∆0
eRN
cos ϕ
2a c anh sin ϕ
2. (12)
We use as scale o he supe cu en he c i ical cu en
Ic
a
T=
0, ob ained by maximizing (12).
Nume ical maximiza ion e u ns ha Icis
Ic=κπ∆0
2eRN
(13)
whe e κ≈1.33. The maximum is placed a phase ϕ≈1.97 ≈0.63π.
The KO CPR is shown in Figu e 3a, no malized o
Ic
. The
T=
0 cu e in Equa ion (12) is plo ed
in black do ed. As one can see he supe cu en dec eases e sus empe a u e, passing om a skewed
shape o a mo e sinusoidal shape [83].
Figu e 3.
Cha ac e is ics o he KO heo y, epo ed e sus phase
ϕ
o chosen empe a u es
T
in
legend. (
a
) Supe cu en
I(ϕ
,
T)
, in Equa ion (11). The do ed cu e a
T=
0 is gi en by Equa ion (12).
(
b
) Elec ic Ene gy
E(ϕ
,
T)
, in Equa ion (14). The do ed cu e a
T=
0 is gi en by Equa ion (15).
(c) En opy a ia ion δS(ϕ,T), in Equa ion (5).
Acco ding o he p esc ip ion gi en in (6), he associa ed Josephson ene gy o he KO CPR is
E(ϕ,T) = R0
RNZ∆(T)
|∆(T)cos(ϕ/2)|log "∆(T)|sin(ϕ/2)|+pε2−∆2(T)cos2(ϕ/2)
p∆2(T)−ε2# anh ε
2Tdε(14)
whe e R0=h/2e2≈12.9 kΩis he in e se o he conduc ance quan um.
The cha ac e is ics o
E(ϕ
,
T)
a e plo ed in Figu e 3b. Being he in eg al o he supe cu en ,
he Josephson ene gy inc eases e sus empe a u e. A T=0, E educes o
E(ϕ,T=0) = eR0∆0
2eRNhlog 1−sin2ϕ
2+2 a c anh sin ϕ
2sin ϕ
2i. (15)
En opy 2019,21, 1005 7 o 33
The maximum Josephson ene gy is E0=E(ϕ=π,T=0), gi en by
E0=log 4
2
R0
RN
∆0=log 4
κπ eR0Ic(16)
ha is abou E0≈0.33eR0Ic.
F om
E
i is possible o calcula e
δS
. Figu e 3c epo s he en opy a ia ion
δS(ϕ
,
T)
calcula ed
nume ically wi h
δS(ϕ
,
T) = −∂TE(ϕ
,
T)
, o chosen empe a u es in legend. I can be no iced ha
δS
dec eases wi h he empe a u e, consis en ly wi h he hi d law o he modynamics.
A low empe a u es, whe e
∂T∆(T)→
0, a closed o m o
δS
can be ob ained by he empe a u e
de i a i e o Equa ion (14), yielding [26,84,101]
δS(ϕ,T) = R0
2RNZ∆0
∆0|cos ϕ
2|log 

∆0|sin(ϕ/2)|+qε2−∆2
0cos2(ϕ/2)
q∆2
0−ε2

ε
T2sech2ε
2Tdε. (17)
The beha io o he en opy can be quali a i ely g asped wi h he minigap mechanism. Le us
conside a ixed empe a u e
T
. Hence, he dis ibu ion unc ion
log
in Equa ion (9) has a ce ain
bandwid h o he o de
T
. A low empe a u e
T∆0
and
ϕ=
0, he DoS gap has wid h
∆0
and
he dis ibu ion bandwid h is smalle han he gap. Hence, he lack o a ailable s a es exponen ially
supp esses he en opy. When
ϕ
mo es om
ϕ=
0 o
ϕ=π
, he minigap sh inks gi ing new
a ailable s a es o he dis ibu ion
log
, inc easing he en opy. A
T∆0
and sho junc ion, i is
app oxima ely
˜
∆=∆0|cos(ϕ/
2
)|
[
102
], he ma ching phase be ween he minigap and he dis ibu ion
bandwid h is 2
a ccos(T/∆0)
, a which he en opy inc eases. This is pa icula ly e iden in he cu e
T=0.1Tcin Figu e 3c, whe e δSis negligible excep close o ϕ→π.
2.5. To al En opy
Gi en he mic oscopic and he KO CPR conside a ions o he las subsec ions, we can s udy he
o al en opy, ha is
S(ϕ,T) = S0(T) + δS(ϕ,T)(18)
whe e
S0
is gi en by he BCS en opy in Equa ion (10) and
δS=−∂TE
whe e
E
is gi en by exp ession (14).
We no e ha he i s e m scales as
∆0NSV
, while he second as
eR0Ic/∆0
. Fo his eason, i is
con enien o in oduce a pa ame e αo he sys em ha se s he a io be ween hese wo quan i ies:
α=eR0Ic
NS∆2
0V. (19)
α
cha ac e izes he ela i e in luence o he phase-dependen e m
δS
o e he emaining e m
S0
.
The quan i y
α
can be expe imen ally de e mined by hea capaci y measu emen s, as explained in
Sec ion 3.2. Mo eo e ,
α
con ols he empe a u e o a i s -o de ansi ion o he no mal s a e when
ϕ6=0, discussed in de ail in Appendix A.
Figu e 4 epo s he o al en opy o
α=
0.6. Di e en alues
ϕ
in he legend a e plo ed, showing
he inc ease o
S
om
ϕ=
0 o
ϕ=π
. The ou cu es co espond o he DoS s a es in he ames o
Figu e 2. As expec ed, he closu e o he minigap om
ϕ=
0 o
ϕ=π
implies an inc ease o en opy.
The scale o his inc ease is se by α.
En opy 2019,21, 1005 8 o 33
Figu e 4.
To al en opy
S
o he sys em o
α=
0.6. (
a
)
S
e sus empe a u e
T
o chosen phases
ϕ
in
legend. The case
ϕ=
0 co espond o he BCS en opy
S0(T)
in Equa ion (10). (
b
) Magni ica ion o panel
(
a
) a ound
T=
0.2
Tc
, highligh ing he passage om a exponen ial supp essed beha io a
ϕ=
0 o a
linea beha io a ϕ=π. The dashed cu e is he analy ical low- empe a u e in exp ession (23), (24).
In he ollowing, he calcula ions a e ob ained wi h
α=
0.6. This alue e idences he en opy
a ia ion and he ela ed esul s while keeping a p oximized olume negligible espec o he o al
olume, as shown below in his subsec ion, and keeping he unwan ed i s -o de ansi ion abo e he
empe a u e 0.7
Tc
, as discussed in Appendix A. Conside ing ha
NS≈7×1046 m−3J−1
[
109
],
α=
0.6
co esponds o a a io I/V≈20 mAµm−3.
The beha io o he en opy can be s udied in mo e de ail a low empe a u e
T∆0
, whe e
closed exp essions can be ob ained. A
ϕ=
0, he DoS has he BCS o m in he whole olume o he
de ice, e u ning he exponen ially supp essed beha io o en opy desc ibed by he ed cu e in
Figu e 4. Hence, a low empe a u es
T∆0
and
ϕ=
0 he en opy can be app oxima ed by he
exp ession [110,111]
S0(T)≈√2π ∆0
Te−∆0/TVN0∆0. (20)
A
ϕ=π
, he minigap is closed and a p oximized spa ial egion a ound he weak link has a
me allic-like DoS. The en opy densi y (9) is hen exponen ially supp essed in he leads and wi h a
linea -in- empe a u e dependence in he p oximized egion. This is con i med by an analy ical exp ession
o δS ha can be ob ained a low empe a u es a ϕ=π. Subs i u ing ϕ=πin (17) we ob ain
δS(ϕ=π,T) = −R0
4RNZ∆0
0log ∆0−ε
∆0+εε
T2sech ε
2Tdε. (21)
De eloping he loga i hm a ound ε=0 as log(1−2ε/(∆0+ε)) ≈ −2ε/∆0and subs i u ing ε/T=z,
δS(ϕ=π,T) = R0
2RN
T
∆0Z∆0/T
0z2sech2z
2dz. (22)
Fo T→0, we ob ain
δS(ϕ=π,T→0) = π2
3
R0
RN
T
∆0
=2π
3
eR0Ic
κ∆0
T
∆0
. (23)
The linea beha io o
δS(ϕ=π
,
T→
0
)
allows o neglec he exponen ially supp essed
S0
con ibu ion
o he o al en opy S, allowing he ollowing app oxima ion
S(ϕ=π,T)≈δS(ϕ=π,T). (24)
Figu e 4b epo s he low- empe a u e beha io o he en opy o 0
≤T≤
0.4
Tc
. The
ϕ=
0 and
ϕ=π
cu es show he exponen ially supp essed and linea beha io espec i ely. The pu ple dashed
cu e epo he analy ical exp ession (23), e ealing a good ag eemen a T<0.2Tc.
En opy 2019,21, 1005 15 o 33
Since he isen opic CPR is cons ained be ween he iso he mal CPRs a
Ti
and
T
, i.e.,
I(ϕ
,
Ti)<
IS(ϕ
,
Ti)<I(ϕ
,
T )
as shown in Figu e 6d, he isen opic wo k is equally cons ained be ween he
iso he mal wo ks a Tiand T .
4. The modynamic Cycles
The combina ion o di e en he modynamic p ocesses, s udied in he p e ious sec ion, allows
cons uc ing he modynamic cycles. In his sec ion, we p esen wo possible examples o he modynamic
cycles ha can be buil based on he a ious p ocesses discussed abo e. In pa icula , we ocus on wo
cycles ha we call Josephson-O o cycle and Josephson- S i ling cycle, hanks o hei analogy wi h
classic he modynamic coun e pa . We i s explain hei implemen a ion and hen we discuss hei
pe o mances.
To his aim, we conside he hyb id sys em a ached o wo di e en ese oi s, iden i ied as
Le Rese oi (L) and igh Rese oi (R), in he ske ch o Figu e 7. The wo ese oi s a e a ixed
empe a u e
Tj
and can elease hea
Qj
o he sys em h ough a hea channel con olled by a hea al e
j
, whe e he subsc ip
j
can be
L
o
R
, espec i ely. We conside
Qj
posi i e when he hea lows om
he ese oi o he sys em, in ag eemen wi h he sign con en ion de ined in Sec ion 3. The ea e ,
we will s udy cycle cha ac e is ics as a unc ion o he empe a u es
(TL
,
TR)
. In pa icula , we will
show ha he e a e egions o
(TL
,
TR)
whe e he cycles can ope a e as engine o e ige a o . The
ese oi oles depend on he ope a ing mode: when a cycle ope a es as engine, he wo ese oi s play
he ole o he Ho Rese oi (HR) and Cold Rese oi s (CR), a empe a u es
Th >Tc
espec i ely.
In a cycle, he sys em abso bs an amoun
Qh
om he HR and eleases
|Qc |<Qh
hea o he CR.
In p ac ical sys ems, he cold ese oi can be cons i u ed by he ambien , i.e., he la ge subs a e
he malized o he c yos a , while he ho ese oi can be a hea ed subsys em, like a la ge me allic pad
hea ed by Joule e ec .
Figu e 7.
Ske ch o he sys em connec ed o wo ese oi s, iden i ied as Le Rese oi (L) and Righ
Rese oi (R), h ough wo hea al es
L
,
R
espec i ely. The modynamic cycles can be implemen ed
a ying con igu a ions be ween di e en empe a u es
TL
and
TR
, achie ing also opposi e ope a ional
modes such as engine o e ige a o con igu a ions (see ex ).
Con e sely, when he cycle is conside ed as a e ige a o , he wo ese oi s play he ole o
he Cooled Subsys em (CS) and Hea Sink (HS), a empe a u es
Tcs <Ths
espec i ely. In a cycle,
he sys em abso bs an amoun o
Qcs
om he CS and eleases
|Qhs|>Qcs
o he HS. In p ac ice,
he CS is an isola ed subsys em om which he hea is ex ac ed, whe e he hea capaci y is assumed
o be la ge enough o conside he CS as a ese oi wi hin one cycle. The CS can be cons i u ed o a
me allic pad ha can be used as cooled subs a e o nanode ices. In p ac ical sys ems, he hea sink is
ypically cons i u ed by he ambien , i.e., he subs a e he malized o he c yos a in ou de ice.
The cycle pe o mances a e cha ac e ized by inspec ing se e al igu es o me i . In he case o he
engine we in es iga e he wo k eleased pe cycle Wand i s e iciency, de ined as
η=W
Qh
. (47)

En opy 2019,21, 1005 16 o 33
This quan i y is physically limi ed by he Ca no e iciency
ηC=1−Tc
Th
. (48)
In he ollowing subsec ions, we show
W
and
η
e sus bo h he empe a u es
Tc
,
Th
. We discuss in
de ail he dependence o
W
and
η
as a unc ion o
Th
o ixed
Tc
, since in eal sys ems i is mos likely
possible o une he HR empe a u e while he CR empe a u e Tc is ixed by he ambien .
In he e ige a o mode, he igu es o me i we conside a e he ex ac ed hea
Qcs
om he CS
pe cycle, and he Coe icien o Pe o mance (COP), de ined as
COP =Qcs
|W|. (49)
Like he e iciency, he COP is limi ed physically by he Ca no COP limi
COPC=Tcs
Ths −Tcs . (50)
In he ollowing subsec ions, we show
Qcs
and he COP e sus bo h empe a u es
Tcs
,
Ths
. We discuss
in de ail he dependence o
Qcs
and he COP as a unc ion o
Tcs
o ixed
Ths
, since in eal sys ems
he HS empe a u e
Ths
is gi en by he ambien and can no be uned, while
Tcs
dec eases om he
ambien empe a u e in he e ige a ion p ocess.
No ice ha he wo k
W
and he hea ex ac ed
Qcs
a e quan i ies de ined pe cycle. Hence,
a cycling equency
ν
, he engine e u ns a Powe
˙
W=Wν
and he e ige a o e u ns a Cooling
Powe CP =Qcsν.
4.1. Josephson-O o Cycle
He e we s udy he Josephson-O o cycle, by s a ing wi h he engine mode o sake o simplici y.
The Josephson-O o engine is desc ibed by he scheme in Figu e 8, whe e he panels a and b show
espec i ely he p ocesses in he
(T
,
S)
and
(ϕ
,
I)
planes. The cycle is cons i u ed by wo isen opic
p ocesses, i.e.,
1→2
and
3→4
, and wo isophasic p ocesses, i.e.,
2→3
and
4→1
, see Figu e 8. We
choose by con en ion ha he s a e
1
and
3
a e he malized o he R and L ese oi , espec i ely. In
his way, he R and L ese oi s play espec i ely he ole o he HR and CR.
The cycle is gi en by he succession o he ollowing p ocesses:
•Isen opic 1 →2
. All he mal al es a e closed o make he sys em he mally isola ed. The sys em
is d i en om he s a e
(ϕ1=
0,
T1=TR)
o
(ϕ2=π
,
T2)
, whe e
T2=T (ϕ=π
,
T1)
. In his
p ocess he uni e se spends a wo k
|W12|
(
W12 <
0 acco ding o he con en ion de ined in Sec ion
3). |W12|is ep esen ed by he g een a ea in Figu e 8b. No hea is exchanged, Q12 =0.
•Isophasic 2 →3
. By opening he he mal al e
L
, he sys em goes om he s a e
(ϕ=π
,
T2)
o
(ϕ=π
,
T3=TL)
. The sys em eleases hea
|Q23|
o he le ese oi (magen a a ea in Figu e 8a).
No wo k is pe o med, W23 =0.
•Isen opic 3 →4
. All he mal al es a e again closed o make he sys em he mally isola ed. The
sys em is d i en om he s a e
(ϕ3=π
,
T3=TL)
o
(ϕ4=
0,
T4)
. By cons uc ion, i
T2>TL
hen
i is
T4<TR
. In his p ocess he sys em e u ns a wo k
W34
(
W34 >
0 acco ding o ou con en ion),
ep esen ed by he sum o he g een and blue a eas in Figu e 8b. No hea is exchanged, Q34 =0.
•Isophasic 4 →1
. By opening he he mal al e
R
, he sys em goes om he s a e
(ϕ=
0,
T4)
o
(ϕ=
0,
T1=TR)
. The sys em abso bs hea
Q41
om he ese oi a
TR
(magen a+pink a ea in
Figu e 8a). No wo k is pe o med, W41 =0.
The o al wo k eleased pe cycle is
W=W12 +W34 (51)
En opy 2019,21, 1005 17 o 33
The hea Qh abso bed om he HR (co esponden o R) is
Qh =QR=Q41 . (52)
Figu e 8.
O o cycle scheme. The example conside s an engine om a ho ese oi a 0.6
Tc
, cold
ese oi a 0.2
Tc
and
α=
0.6. (
a
) Scheme in he
(T
,
S)
plane. The colo ed a eas help o he discussion
in he ex o he hea exchanges. (
b
) Scheme in
(ϕ
,
I)
plane. O he ou p ocesses o he O o cycle,
only he wo isen opic a e isible, since he wo isophasics a e collapsed a he poin s
(ϕ=
0,
I=
0
)
and
(ϕ=π
,
I=
0
)
. The colo ed a eas help o he discussion in he ex o he wo k exchanges. Fo
comple eness, he do ed cu es ep esen pa ial iso he mal CPRs a he labelled empe a u e in he plo .
F om he wo schemes p esen ed in Figu e 8i can be no iced ha he cycle ope a es as an engine
i
T2>T3
. This condi ion equi es ha
TL<T (ϕ=π
,
TR)
, i.e., a empe a u e gap be ween he wo
ese oi s is equi ed. When
TL
app oaches
T (ϕ=π
,
TR)
he cycle ends o he degene a e case
epo ed in Figu e 9a, whe e he wo adiaba ic cu es end o supe impose. Also in he
(ϕ
,
I)
plane he
wo adiaba ic cu es end o supe impose, meaning ha he ne wo k is
W=
0 a
TL=T (ϕ=π
,
TR)
.
On he con a y, i
TL>T (ϕ=π
,
TR)
, he cycle is in e ed as in Figu e 9b. In his case, he cycle
wo ks as a e ige a o and he wo k is W<0, i.e., made by he uni e se on he sys em.
Hence, he cu e in he plane (TL,TR)whe e W=0 can be de ined as he cha ac e is ic cu e o
he O o cycle. I sepa a es he egions whe e he cycle is in he engine o e ige a o mode and i is
gi en by he equa ion
TL=T (ϕ=π,TR). (53)
Close o he cha ac e is ic cu e, in he case shown in Figu e 9a, i is e iden ha
QL
,
QR
end o
ze o bu hei a io ends o
QR/QL→TR/TL
. This p ope y is exploi ed below o calcula e he limi s
o ηand COP close o he cha ac e is ic cu e.
En opy 2019,21, 1005 18 o 33
Figu e 9.
Pa icula cases o he O o cycle on
TL
,
TR
. (
a
) App oaching he degene a e case o
T (TR) =
TL. (b) O o cycle as e ige a o o T (TR)<TL.
Le us conside he e ige a o mode in he case
TL>T (TR)
, ep esen ed in Figu e 9b. In his
case, he cycle is clockwise and ope a es as a e ige a o . The wo ese oi s play a di e en ole:
he R ese oi ep esen s he Hea Sink, while he L one ep esen s he Cooled Subsys em. The case
TL>T (TR)coincides wi h he ollowing cycle
•Isen opic 1 →2
. All he mal al es a e closed o make he sys em he mally isola ed. The
sys em is d i en om he s a e a he ambien empe a u e
(ϕ1=
0,
T1=TR)
o
(ϕ2=π
,
T2)
,
whe e
T2=T (ϕ=π
,
T1)
. In his p ocess, he uni e se spends a wo k
|W12|
(
W12 <
0 o o
Sec ion 3). No hea is exchanged, Q12 =0.
•Isophasic 2 →3
. By opening he he mal al e
L
, he sys em goes om he s a e
(ϕ=π
,
T2)
o
(ϕ=π
,
T3=TL)
, emo ing he hea
Q23
om he CS (magen a a ea in Figu e 9b). No wo k is
pe o med, W23 =0.
•Isen opic 3 →4
. All he mal al es a e closed. The sys em is d i en om he s a e
(ϕ3=π
,
T3=
TL)
o
(ϕ4=
0,
T4)
. Now,
T4>TR
. In his p ocess, he sys em e u ns a wo k
W34
. No hea is
exchanged, Q34 =0.
•Isophasic 4 →1
. By opening he he mal al e
R
, he sys em goes om he s a e
(ϕ=
0,
T4)
o
(ϕ=
0,
T1=TR)
. The sys em eleases hea
Q41
o he ese oi a
TR
, since
T4>TR
, which
co espond o he magen a+pink a ea in Figu e 9b. The empe a u e
T4
plays an analogous ole o
he ho hea exchange ha is p esen in he e ige a o s. No wo k is pe o med, W41 =0.
In he e ige a o mode, he wo k eleased is s ill gi en by
W=W12 +W34
. The hea
Qcs
abso bed by he CS is
Qcs =QL=Q23 . (54)
Figu e 10 is a summa y o he wo k eleased
W
and he hea abso bed
Qh
and
Qcs
. Panels a,b
a e colo plo s o hese quan i ies e sus
(TL
,
TR)
. The dashed ed cu e ep esen s he cha ac e is ic
cu e de ined in Equa ion (53), co esponding o
W=
0. Abo e i , o
TL<T (ϕ=π
,
TR)
, he cycle
ope a es as engine, while below i (
TL>T (ϕ=π
,
TR)
) he cycle wo ks as e ige a o . The o ange
do -dashed cu e epo s he he mal equilib ium
TL=TR
. We can no ice ha below his cu e, i.e.,
o
TR<TL
, he e is a egion whe e wo k is spen by he uni e se o pump hea om he L ese oi
( he ho e one) o he R ese oi ( he colde one). Hence, wo k is spen o pe o m a p ocess ha can
be pe o med spon aneously. We de ine his egion as a cold pump, ollowing he de ini ion gi en in
Re e ences [112,113].
Figu e 10c epo s he eleased wo k e sus he HR empe a u e
TR
o di e en alues o he
CR empe a u e
TL
as epo ed in he legend. The cu es each he alue ze o co esponding o he
cha ac e is ic cu e plo ed in Figu e 10a. We obse e ha he gene al end o he wo k is o inc ease
wi h he empe a u e di e ence
TR−TL
be ween he wo ese oi s. The o de o magni ude o he
wo k pe cycle is ∼0.1eR0Ic.
En opy 2019,21, 1005 19 o 33
Figu e 10d epo s he abso bed hea
Qcs =QL
e sus he CS empe a u e
TL
o di e en alues
o he HS empe a u e
TR
. The black cu e epo s he case o he hea abso bed
Qcs
a
TL=TR
. The
cu es wi h ixed
TR
a e limi ed on he igh a
TL=TR
, o no include he Cold Pump case, see colo
plo s in Figu e 10. The cu es wi h ixed
TR
goes o ze o in co espondence o he cha ac e is ic cu e,
de ining he minimum achie able empe a u e o he e ige a o . The e ige a o can no physically
cool below he minimum achie able empe a u e, since he abso bed hea eaches Qcs =0.
Figu e 10.
(
a
) Wo k eleased in a Josephson-O o cycle as a unc ion o
(TL
,
TR)
. The dashed ed cu e,
gi en by Equa ion (53), epo s
W=
0 and sepa a es he egion whe e he cycle ope a es as engine
o e ige a o . (
b
) Hea abso bed in a Josephson-O o cycle. As an engine, he hea
QR
om he Ho
ese oi is ep esen ed by he R ese oi . As a e ige a o , he hea
QL
om he CS is ep esen ed by
he L ese oi . The dash-do ed line ep esen s he he mal equilib ium
TL=TR
, below which he
sys em is a cold pump. (
c
) Cu s o he wo k in panel (
a
) e sus he Ho Rese oi empe a u e
TR
o
ixed empe a u es
TL
o he Cold Rese oi . (
d
) Cu s o he abso bed hea e sus he CS empe a u e
TL
o ixed empe a u es
TR
o he Hea Sink. The black solid cu e epo s he abso bed hea a
TL=TR
. The iole dash-do ed cu e epo s he analy ical esul o Equa ion (55). The cu es ha e
been ob ained wi h α=0.6.
The black cu e epo ing
Qcs
a
TL=TR
is impo an since i epo s he hea abso bed pe cycle
when he e ige a o s a s o ope a e a he he mal equilib ium. Hence, o a cycling equency
ν
, he co esponding cooling powe
˙
Qcs =Qcsν
o
TL=TR
gi es he maximum hea ing powe
leakage ha he e ige a o can sus ain. I he hea leakage is abo e he cooling powe a he he mal
equilib ium, no ne e ige a ion can be accomplished. The hea abso bed pe cycle has he same o de
o he wo k pe cycle, ∼0.1eR0Ic.
En opy 2019,21, 1005 20 o 33
I is possible o ind an analy ic exp ession o
Qcs
alid o
TR
,
TL∆0
. Conside ing he scheme
in Figu e 9b, i can be no iced ha a low empe a u e he hea abso bed by he CS is uled by he
pu ple a ea de ined by he linea exp ession o en opy in Equa ions (23) and (24). App oxima ing he
T2
empe a u e o 0, due o he s ong isen opic cooling a low empe a u es, he
Qcs
is gi en a he
leading o de by he CS empe a u e
Qcs ≈π
3κTL
∆02
eR0Ic. (55)
This exp ession is plo ed in Figu e 10d as a iole dash-do ed cu e. The ag eemen wi h he
nume ical esul s is good a TL<0.2Tc, co esponding o he ag eemen ange in Figu e 4b.
F om he cha ac e is ics o
W
,
Qh
,
Qcs
in Figu e 10 i is possible o calcula e nume ically he
engine e iciency and he e ige a o COP. Figu e 11 epo s he e iciency and he COP o he s udied
O o engine. Figu e 11a shows a colo plo o
η
, COP e sus
TL
,
TR
. The wo quan i ies a e con ined
espec i ely in he engine and e ige a o egions o
(TL
,
TR)
. The g ay a ea co esponds o he
Cold Pump case. Figu e 11b epo s cu s o he e iciency
η
e sus he Ho Rese oi empe a u e
TR
o chosen ambien empe a u es
TL
. The cu es end on he le in co espondence o he O o
cha ac e is ic cu e, whe e he e iciency sa u a es a he Ca no limi . Figu e 11c epo s cu s o he
COP e sus he CS empe a u e
TL
o chosen HS empe a u es
TR
, showing he e olu ion o he COP
when he CS is cooled down owa d he minimum achie able empe a u e, ha delimi s he COP
cu es on he le . The COP cu es a e limi ed on he igh by he he mal equilib ium s a e
TL=TR
,
whe e he COP eaches he heo e ical Ca no limi .
Figu e 11.
E iciency and COP o he O o machine. (
a
) Colo plo o
η
and
COP
e sus
(TL
,
TR)
,
wi h di e en colo pale es. The g ay egion ep esen s he s a e whe e he cooled subsys em
empe a u e is abo e he hea sink empe a u e. (
b
) Cu s o O o cycle e iciency
η
e sus
TR
o
chosen
TL
in legend. The do -dashed line epo s he Ca no limi o e iciency. The cu es end a he
O o cha ac e is ic cu e, Equa ion (53), whe e he e iciency eaches he Ca no limi . (
c
) Cu s o O o
cycle COP e sus
TL
o chosen
TR
in legend. The do -dashed line epo he Ca no limi o COP. The
cu es a e limi ed on he igh by he he mal equilib ium s a e
TL=TR
; on he igh , he cu es a e
limi ed by he O o cycle cha ac e is ic cu e. On his cu e, he COP eaches he COP Ca no limi .
An in e es ing p ope y o he Josephson-O o cycle is ha close o he cha ac e is ic cu e, bo h
η
and he COP each he Ca no limi , e en hough he wo k eleased o he hea abso bed goes o ze o.
This poin can be explained by e e ing o he degene a e case o Figu e 9a. Close o he cha ac e is ic

En opy 2019,21, 1005 21 o 33
cu e, he quan i ies
QL
,
QR
end o ze o bu hei a io ends o
|QL/QR| → TL/TR
. Exploi ing he
ene gy conse a ion QL+QR−W=0, i is
η(TL→T (ϕ=π,TR)) = 1+QL
QR→1−TL
TR
(56)
ha is he Ca no limi . Wi h simila conside a ions, we ob ain he analogous limi o he COP:
COP(TL→T (ϕ=π,TR)) = QL
QR+QL→TL
TL−TR
. (57)
4.2. Josephson-S i ling Cycle
In his sec ion, we analyze ano he possible he modynamic cycle, i.e., a Josephson-S i ling cycle,
ha has in p ac ice se e al p ac ical applica ions, in pa icula as e ige a o [
114
]. The S i ling cycle
is a di e en combina ion o he s udied p ocesses, being buil wi h wo isocho ics and wo isophasics.
In an ideal gas sys em, i consis s o wo isocho ic hea addi ion/ ejec ion p ocesses and wo iso he mal
(comp ession + expansion). Real S i ling engines a e e en ually equipped by egene a o s ha inc ease
he e iciency [115,116]; he e we s udy he simple case wi hou he egene a o s.
Fi s o all, le us conside he engine case and hen mo e o he e ige a o one.
The Josephson-S i ling engine is desc ibed by he scheme in Figu e 12, whe e panels a and b show
espec i ely he p ocesses in he
ST
diag am and
Iϕ
diag am. The cycle is cons i u ed by wo iso he mal
p ocesses (
1→2
and
3→4
in Figu e 12) and wo isophasic p ocesses (
2→3
and
4→1
in Figu e 12).
The s a es
1
,
2
and
3
,
4
a e espec i ely he malized o he igh and le ese oi s. When ope a ing as
S i ling engine, he le and igh ese oi s play he oles o ambien and hea sou ce espec i ely.
Figu e 12.
Josephson-S i ling cycle scheme. The plo ed example conce ns an engine be ween a ho
ese oi
TR=
0.6
Tc
and a cold ese oi
TL=
0.3
Tc
and
α=
0.6. (
a
) Scheme in he
(T
,
S)
plane.
The colo ed a eas help o he discussion in he ex abou he exchanged hea s. (
b
) Scheme in
(ϕ
,
I)
plane. O he ou p ocesses o he Josephson-S i ling cycle, only he wo iso he mals a e isible, since
he wo isophasics a e collapsed a he poin s
(ϕ=
0,
I=
0
)
and
(ϕ=
0,
I=
0
)
. The colo ed a eas help
o he discussion in he ex abou he exchanged wo ks.
In summa y, he Josephson-S i ling engine is gi en by he succession o he ollowing p ocesses:
•Iso he mal 1 →2
. The he mal al es
R
is open and
L
is closed, so ha he sys em is in he mal
con ac wi h he igh ese oi . The sys em is d i en om he s a e
(ϕ1=
0,
T1=TR)
o
(ϕ2=
π
,
T2=TR)
. He e a wo k is spen
|W12|
ep esen ed by he g een a ea in Figu e 12b. The hea
Q12
is abso bed om he ese oi , ep esen ed by he g een + da k pu ple a ea in Figu e 12a.
•Isophasic 2 →3
. By closing
R
and opening
L
, he sys em goes om he s a e
(ϕ=π
,
T2)
o
(ϕ=π
,
T3=TL)
. The sys em eleases hea
Q23
o he le ese oi , ep esen ed by he ligh
pu ple + da k pu ple a ea. No wo k is pe o med, W23 =0.
En opy 2019,21, 1005 22 o 33
•Iso he mal 3 →4
. The al es a e kep in he same s a e:
R
open and
L
closed. The sys em is
d i en om he s a e
(ϕ3=π
,
T3=TL)
o
(ϕ4=
0,
T4=TL)
. In his p ocess he sys em e u ns
a wo k
W34
ep esen ed by he sum o he g een and blue a eas in Figu e 12b. The hea
|Q34|
is
eleased o he le ese oi , ep esen ed by he blue a ea in Figu e 12a.
•Isophasic 4 →1
. By closing
L
and opening
R
, he sys em goes om he s a e
(ϕ=
0,
T4)
o
(ϕ=
0,
T1=TR)
. The sys em abso bs he hea
Q41
om he ese oi a
TR
, gi en by he sum o
he a eas in blue, ed and ligh pu ple in Figu e 12a. No wo k is pe o med, W41 =0.
The o al wo k pe cycle is gi en by
W=W12 +W34
. The hea abso bed om he Ho R
( ep esen ed by he R ese oi ) is
Qh =QR=Q12 +Q41 . (58)
In o de o wo k as an engine, i mus be
TR>TL
, as shown in Figu e 12a. I
TL>TR
, he cycle is
e e sed as displayed in Figu e 13. Panels (a) and (b) show he case o
TR=
0.6
Tc
and
TL=
0.35
Tc
and
TL=
0.25
Tc
espec i ely. In his case, he machine can wo k as a e ige a o wi h he CS ep esen ed
by he R ese oi and HS ep esen ed by he L one (di e en ly om he case o he Josephson-O o
cycle).
Figu e 13.
Pa icula examples o he Josephson-S i ling cycle o
TR<TL
. (
a
) S i ling in e se cycle
wo king as e ige a o . The hea abso bed om he R ese oi in he p ocess
1→2
, ep esen ed by
he a ea de ined by he ela ed g een a ow, is bigge han he hea eleased o R ese oi in he p ocess
4→1
, ep esen ed by he a ea de ined by he ela ed ed a ow. (
b
) S i ling in e se cycle wo king as
Joule pump, exploi ing wo k o elease hea o bo h ese oi s.
E en hough he cycles in bo h panels a e clockwise, only he cycle in panel (a) wo ks as
e ige a o . Indeed, he e a e u he condi ions ha de ine he
(TL
,
TR)
egion whe e he cycle
can wo k as a e ige a o . Le us conside he hea exchanged wi h he cold igh ese oi , gi en by
p ocesses
4→1
and
1→2
. F om Figu e 13 i can be no iced ha in
4→1
he hea is eleased om
he sys em o he R ese oi , while in
1→2
he hea is abso bed by he sys em. Cooling hen can
ake place i
Qcs =QR>
0, i.e., i
|Q12|>|Q41|
. This is ue when
TL
is closely below
TR
,
TL/TR
;
hen, when he CS is cooled down,
|Q12|
dec eases, since
δS(ϕ=π
,
T)
in an iso he mal hea exchange
(31) dec eases, while
|Q41|
inc eases wi h he inc ease o he empe a u e di e ence
Ths
4
−Tcs
in
he isophasic p ocess. As a consequence, i exis s a minimum achie able empe a u e
TMAT
ha is
cha ac e ized by a null cooling powe QR=0, i.e.,
TMAT =TR .c. Q12(TL,TR) + Q41(TL,TR) = 0 . (59)
No e ha
TMAT
is a unc ion o he HS empe a u e
TL
. I
TR<TMAT
, he o al hea
QR
exchanged
wi h he R ese oi is nega i e, and he CS is hea ed. This case co esponds o he
(T
,
S)
diag am in
Figu e 13b, whe e he ed a ea ep esen ing he eleased hea o he igh ese oi includes he g een
En opy 2019,21, 1005 23 o 33
a ea o he abso bed hea om he igh ese oi . As be o e, we call he cu e
(TL
,
TR=TMAT(TL))
he
cha ac e is ic cu e o he Josephson-S i ling cycle. We obse e o comple eness ha when
TR<TMAT
and
QR<
0, also he le ese oi can abso b o elease hea
QL=Q23 +Q34
, depending on
(TL
,
TR)
.
I
QL>
0 he cycle abso bs wo k o ans e hea om he ho o he cold ese oi , cons i u ing a Cold
Pump simila o he si ua ion desc ibed in he Josephson-O o cycle. On he o he hand, i
QL<
0, he
machine eleases hea o bo h he ese oi s, con e ing comple ely he wo k in hea . Following he
de ini ion o Re e ences [112,113], we call his ope a ing mode as Joule pump.
In he e ige a o case, he o al wo k is
W=W12 +W34
and he hea ex ac ed is
Qcs =QR=
Q12 +Q41, like he engine case.
The eleased wo k
W
and he hea abso bed
Qcs
,
Qh
a e summa ized in Figu e 14. In he
colo plo s in panels a,b, he cu es
W=
0,
QR=
0,
QL=
0 sepa a e he egions o he engine,
he e ige a o , he Joule pump and he cold pump. The cu e
W=
0 co esponds o
TL=TR
.
The e ige a o egion is be ween he cu e TL=TRand he cha ac e is ic cu e TMAT(TL).
Figu e 14.
(
a
) Wo k eleased in a S i ling cycle as a unc ion o
(TL
,
TR)
. The dashed cu e
W=
0
co espond o he he mal equilib ium cu e
TL=TR
and sepa a es he egion whe e he cycle ope a es
as engine o e ige a o . Mo eo e , he cu es
QR=
0 and
QL=
0 u he dis inguish egions whe e
he cycle is a Joule Pump (JP) o a Cold Pump. (
b
) Hea abso bed in a S i ling cycle. In bo h engine and
e ige a o modes, he hea
QR
is abso bed om he R ese oi ha plays he ole o Ho Rese oi o
CS in he espec i e egions. (
c
) Cu s o he wo k in panel (
a
) e sus he Ho Rese oi empe a u e
TR
o ixed empe a u es
TL
o he Cold Rese oi . The black dashed line epo s exp ession (60). (
d
) Cu s
o he abso bed hea
QR
e sus he CS empe a u e
TR
o ixed empe a u es
TL
o he Hea Sink. The
black solid cu e epo s he abso bed hea a TL=TR. The cu es ha e been ob ained wi h α=0.6.
Figu e 14c epo s cu s he eleased wo k pe cycle e sus he HR empe a u e
TR
o ixed CR
empe a u es
TL
. The cu es each ze o a
TL=TR
. The gene al end is ha he wo k inc eases wi h
inc easing he empe a u e di e ence TR−TLbe ween he wo ese oi s.
En opy 2019,21, 1005 24 o 33
An analy ical exp ession o
W
can be calcula ed. Le us conside a S i ling cycle wi h
TL
TR∆0
. The eleased wo k can be app oxima ed by
W=E(ϕ=π
,
TR)−E(ϕ=π
,
TL≈
0
)
. Using
app oxima ion (33) o E, we ob ain
W≈π
3κeR0IcTR
∆02
. (60)
This exp ession is plo ed in Figu e 14c and is in good ag eemen wi h he nume ical esul s.
Figu e 14d epo s he hea abso bed pe cycle
Qcs =QR
e sus he CS empe a u e
TR
o ixed
HS empe a u es
TL
. The cu es go o ze o on hei le in co espondence o he cha ac e is ic cu e.
The cu es a e limi ed on he igh by he black cu e o
Qcs
a
TL=TR
. The o de o magni ude o
he abso bed hea pe cycle is ∼0.1eR0Ic.
F om he
W
,
Qh
,
Qcs
cha ac e is ics i is possible o calcula e he
η
and he COP, as epo ed in
Figu e 15. Figu e 15a shows a colo plo o
η
and
COP
e sus
(TL
,
TR)
. The wo quan i ies a e plo ed
o e he engine and e ige a o egions espec i ely. The g ay a ea is whe e he cycle wo ks as cold
pump o Joule pump.
Figu e 15.
E iciency and COP o he S i ling machine. (
a
) Colo plo o
η
and
COP
e sus
(TL
,
TR)
,
wi h di e en colo pale es. The g ay egion ep esen s whe e he cycle is a Joule pump o Cold pump.
(
b
) Cu s o S i ling cycle e iciency
η
e sus
TR
o chosen
TL
in legend. The do -dashed line epo s
he Ca no limi o e iciency. The cu es end a
TR=TL
. (
c
) Cu s o S i ling cycle COP e sus
TR
o
chosen
TL
in legend. The do -dashed line epo s he Ca no limi o he COP. The cu es go o in ini y
on he igh a he he mal equilib ium s a e
TL=TR
; on he le , he cu es a e limi ed by he S i ling
cha ac e is ic cu e.
Figu e 15b epo s cu s o he e iciency
η
e sus he ho ese oi empe a u e
TR
o chosen
ambien cold ese oi empe a u es
TL
. The cu es end on he le a he s a e
TR=TL
. Fo
TL→TR
,
bo h he cycle e iciency and Ca no limi end o ze o. Indeed, he wo k W=Q=QR+QL ends o
ze o, since QR→ −QLas can be no iced om Figu e 13a.
Figu e 15c epo s cu s o he COP e sus he CS empe a u e
TR
o chosen HS empe a u es
TL
. The cu es end on he le a
TR=TMAT
. Fo
TL→TR
, bo h he cycle COP and i s Ca no
limi end o in ini y. Wi h he same geome ical a gumen used o he e iciency, i is
W→
0 and
QR=TRδS(ϕ=π,T), implying ha he COP =QR/|W| → ∞.
En opy 2019,21, 1005 31 o 33
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