πstates in all-pnictide Josephson junctions

0.5 MB·

by  C. Nappi

π-States in all-pnictide Josephson Junctions
C. Nappi,1,∗ S. De Nicola,2,3 M. Adamo,1 and E. Sarnelli1,†
1CNR Istituto di Cibernetica ”E. Caianiello” Via Campi Flegrei 42 I-80078, Pozzuoli, Napoli, Italy
3
1 2CNR Istituto Nazionale di Ottica I-80078, Via Campi Flegrei 42,Pozzuoli, Napoli, Italy
0
2

n 3INFN Sezione di Napoli, Complesso Universitario di Monte S. Angelo, Via Cinthia, I-80126 Napoli, Italy
a
J

9
2
(Dated: January 30, 2013)

n] We study the Josephson effect in s±/I/s± junctions made by two bands reversed sign s-wave
o (s±)superconductivematerials. Wederiveanequation providingtheboundAndreevenergystates
parameterized by the band ratio α, a parameter accounting for the weight of the second band
c
- with respect to the first one at the interface. For selected values of the band ratio and tunnel
r barrier amplitude, we predict various features of the Josephson current, among which a possible
p
high temperature π state of thejunction (a doubly degenerate junction ground state) and a π 0
u →
crossover with decreasing temperature.
s
.
at PACSnumbers: 74.20.Rp,74.50.+r,74.70.Dd
m
- The importance of the Josephson effect as a tool to d-wave superconductor Josephson junction [24].
d
n probe the properties of new discovered superconductive We adopt the simplest model of a Josephson junc-
o materials can hardly be underestimated. This has been tion that shows the essential features of the Josephson
c true in the past in investigating the d-wave cuprate ma- effect in the presence of two gaps, namely we consider
[ terials and is nowadays the case for the multi-gap iron asuperconductor(S)-insulator(I)-superconductor(S)con-
3 based superconductors [1] whose complex behavior, be- tact. Theironbasedjunctionismodeledbyconsideringa
v yond the BCS theory, is presently a focus in condensed one-dimensional conductor, whose left (x<0) and right
9 matter theory [2, 3] and material science [4]. (x > 0) halves are both two band metals (two differ-
3 There is a rather solid indication now, supported by ent states at the Fermi level, one with the wave vector
0
experiments, that the pair potential symmetry in these p and the other with q). We assume that the motion of
4
. compoundsiss-wavewithsign-reversingorderparameter quasiparticles is described by the Bogoliubov de Gennes
2 (s±). Several theoretical models have already discussed (BdG)equation[25]andthattheorderparameterhasal-
1 the experimental consequences of an extended s-wave readybeen obtained self-consistentlyfromthe gapequa-
2
1 (s±- wave) order parameter symmetry on the Andreev tion. Therefore we choose a one dimensional model for
: conductance of an NS interface and on the Josephson the gap, such that the left and right two band supercon-
v effect in the iron-based superconductor junctions [5–19]. ductors have pair potentials given by
i
X InparticulartheJosephsoneffecthasbeenstudiedinhy-
briddevicesi.e. sIs junctions,assummarizedbySeidel
r ± ∆ (x)=∆ eiϕjθ( x)+∆ ei(ϕj+ϕ)θ(x), j =1,2 (1)
a in his review [20] and reference therein. j j − j
In this Letter we study an all-pnictide symmetric
s Is Josephsonjunctionandwediscussanumbernon- The normal region, where ∆ = 0, has an infinitesi-
± ± j
trivialphysicalconsequencesontheJosephsoneffectdue mal width and we also introduce a scattering potential
due to the presence of a second conduction band. We U(x)=U δ(x). Thepossibilityofnodesinthegapfunc-
0
show that, depending on the band ratio parameter α, tion is not considered. In the case of the two-gap model
whichaccountsfortheweightthesecondband,aπ-state withunequals-wavegaps,wewriteawavefunctionofthe
[21] candevelopfor a wide rangeofjunction transparen- same type introduced in the Blonder Tinkham Klapwijk
ciesandtemperatures. Thisπ-statecanpersistinthefull model[26]assolutionoftheBdGequationsandtreatthe
range of temperatures or can undergo a π 0 (inverse presence of the second band through the introduction of
→
0 π) crossover as the temperature decreases, depend- Blochwavefunctions[8]. Theboundstate(B)eigenfunc-
→
ingonthebandratioparameter. Thiscrossoverisanalo- tion with energy E <∆ (we assume ∆ <∆ ) can be
1 1 2
| |
goustothe0 π crossover[22,23],foundinmesoscopic written as
→
2
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
u (x) u u
ΨSL(x)= B =a 1 φ (x)+α 2 φ (x) +
v (x) B v e−iϕ1 −pe 0 v e−iϕ2 −qe
B 1 2
(cid:18) (cid:19) (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
v v
b 1 φ (x)+α 2 φ (x) x<0
B u e−iϕ1 ph 0 u e−iϕ2 qh
1 2
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
u (x) u u
ΨSR(x)= B =c 1 φ (x)+α 2 φ (x) +
v (x) B v e−i(ϕ1+ϕ) pe 0 v e−i(ϕ2+ϕ) qe
B 1 2
(cid:18) (cid:19) (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
v v
d 1 φ (x)+α 2 φ (x) x>0
B u e−i(ϕ1+ϕ) −ph 0 u e−i(ϕ2+ϕ) −qh
1 2
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
(2)
where ϕ is a global phase difference between the two condition requirements. In fact, the assumed wave func-
superconductiveregionsandϕ , ϕ arethephasesofthe tion Ψ is of the form Ψ = Ψ +α Ψ where, separately,
1 2 1 0 2
gaps ∆ , ∆ in both p and q bands respectively. In the Ψ andΨ solveBdGequationsforexcitationsofenergy
1 2 1 2
caseofs gapmodel, ϕ ϕ =π. Intheconsidereden- E for the gap ∆ and ∆ , respectively,. The boundary
± 2 1 1 2
−
ergy range the wave function has to decay exponentially conditions, Eqs. (5), are requested to be a constrain for
for x . Thecoefficientsa ,b ,c ,d aretheprob- the whole function Ψ.
B B B B
| |→∞
ability amplitudes transmission with branch crossing or The above matching procedure, with the require-
without branch crossing. Essential above is the intro- ment of non-triviality of the solution for the coefficients
duction of α , a mixing coefficient defining the ratio of a ,b ,c ,d provides for the s /I/s junction, the
0 B B B B ± ±
probabilityamplitudesforaquasiparticletobetransmit- spectral equation
tedtothefirst,(p),orsecond,(q),band[8];thefunctions
φ’s arethe Blochwavesinthetwo-bandsuperconductor; 1+2 1 E2 r2 E2α2 r2α4 2Z2+1 +
p and q are the Fermi vectors for the two bands corre- − − − −
sponding to the same energy E [8]: 2(cid:16)E2 Z2(p1+α4)+p(1 α2+α4) + (cid:17)(cid:0) (cid:1)
−
1(cid:2)+α2 2E2+2 1 E2 r2(cid:3) E2 r2α2
− − − − ×
φλ(x)= CG,λexp[i(λ+G)x] (3) chos(ϕ)=0(cid:16) p p (cid:17)i (6)
G
X
where we have introduced the gap ratio r =∆ /∆ , the
where λ = p ,q ,p ,q , and G represent any reciprocal 2 1
e e h h band ratio parameter α = α φ (0)/φ (0) and the bar-
lattice vector. u1, v1 and u2, v2 are the Bogoliubov co- rier strength Z = U /~v. Th0e qenergypis given in units
efficients for the first and second band, respectively 0
of ∆ (T), and the two gaps will be assumed to obey the
1
same BCS-like temperature law. Boundary conditions
1/2 1/2
1 Ω 1 Ω
u = 1+i 1 ,v = 1 i 1 on the wave function derivatives are usually discussed in
1 1
2 E 2 − E terms of Fermi velocities in the case of plane waves. For
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19)(cid:21)
1/2 1/2 Bloch waves we have to introduce ”interface velocities”
1 Ω 1 Ω
2 2
u = 1+i ,v = 1 i
2 2 E 2 2 − E i~φ′(0)
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19)(cid:21) v = λ (7)
λ
Ω = E2 ∆2,Ω = E2 ∆2 (4) −mφλ(0)
1 − 1 2 − 2
q q In deriving the spectral equation we have assumed, for
The globalwave function Ψ must satisfy the following
thesakeofsimplicity,equalbandinterfacevelocitiesv =
boundary conditions at the interfaces x=0 λ
v.
ΨSL(0−)=ΨSR(0+) An analysis of the spectral equation shows that for
α = 0 there are, in general, four energy levels, E =
Ψ′SR(0+)−Ψ′SL(0−)= 2mℏ2U0ΨSR(0+) (5) ±baǫn61d(ϕc,aαs)ea(nαd=E20)=E±qǫ62(pϕr,oαv)id(esseethFeigw.e1ll).knIonwansrine1sgulelt-
for a conventional SIS junction [27, 28], namely E =
where primes denote derivative with respect to x. 1
Coupling between the bands is implicit in the boundary ǫ (ϕ,0) = ∆ (T) 1 Dsin2(ϕ/2) 1/2, where D =
1 1
± ± −
(cid:2) (cid:3)
3
1/(1+Z2) is the transmission probability through the For 0 < α < α , the two emerging levels E =
c 2
δ-function barrier, i.e. the junction transparency. ǫ (ϕ,α) and the levels E = ǫ (ϕ,α) have, in gen-
2 1 1
± ±
As α increases, the energy levels E = ǫ (ϕ,α) start eral, opposite dispersions, i.e. dE /dϕ> 0, dE /dϕ < 0
2 2 2 1
±
tobranchoffthelevelsE = ǫ (ϕ,α),whiletheselatter as can be seen in Fig. 1(a) and (b). The sign difference
1 1
±
gradually approach the zero energy state. As an exam- of the dispersions is the key point in determining the
ple, Fig. 1 (a) and (b) show the modifications of the Josephson current-phase relation, as it will be discussed
Andreev levels for increasing values of the band ratio, below.
for two values of the barrier parameter, Z = 0.7 and The Josephson current I carried by the discrete An-
d
Z = 2 and for the band gap ratio r = 2, respectively. dreev levels E through the contact can be found from
k
Theconditionfortheexistenceofazeroenergylevelcan thefreeenergy,accordingtothefollowingrelation[28,31]
be easilyobtainedfromthe spectralequation. Itisgiven
2e ∂E
by (1 rα2)2(1+2Z2+cos(ϕ))=0. Accordingly,a zero I (ϕ,α)= kf(E )=
− d ~ ∂ϕ k
energy state E = ǫ (ϕ,α ) = 0, is obtained for the
1 1 c k=1,2
± X
critical value of the band ratio α = 1/√r, for any Z
c 2e ∂ǫ ǫ
k k
value. The energy level E = ǫ (ϕ,α ) corresponding tanh (10)
2 ± 2 c − ~ ∂ϕ 2k T
to the critical value of the band ratio is given by [29] k=1,2(cid:18) B (cid:19)
X
where f(E ) is the Fermi distribution function and k
E = ǫ (ϕ,α )= k
2 2 c
± labels the Andreev level with energy E = ǫ (ϕ,α).
k k
2Dr2(2 D+Dcosϕ)sin2(ϕ/2) ±
− (8)
±s 1+r(2 2D+r)+2Drcosϕ
−
Z = 0.7
Z = 2
anditisshownasabluelineinFig. 1fortheindicated
values of D and r. For low transparencies (D 1),
≪
Eq. 8 reduces to E = ǫ (ϕ,α ) = ∆ (T)2r/(1 +
2 2 c 1
± ±
r)√Dsin(ϕ/2), a result closely resembling the midgap
statesofd-wavesuperconductorJosephsonjunction[30].
(cid:39) (cid:39)
For α > αc, there are no surface bound states with real 2 (cid:32)2 2 (cid:32)2
energy eigenvalues. (cid:39)1 (cid:39)1
(cid:72)
2
Z=2 FIG.2: current-phaserelationforas±superconductorS/I/S
(cid:72) junction with r =2, T/Tc =0.01 and increasing values of α.
Z=0.7 1
(a)Z =0.7, (blackline,α=0.01, redline,α=0.5, greenline,
α=0.7). (b) Z =2. (black line, α=0.01, red line, α=0.4,
(cid:16)(cid:72) green line, α=0.7)
1
Fig. 2(a) and (b) show the current-phase relations
(cid:16)(cid:72)
2 I (ϕ,α), at low temperature, corresponding to the dis-
d
crete spectrum of Andreev levels represented in Fig. 1
(a)and(b)respectivelyandcalculatedthroughEq. (10).
The current in this figure is normalized with respect to
FIG. 1: Andreev levels for a s± superconductor S/I/S junc- I = e∆ (0)/~, which is the zero temperature maxi-
tion with r = 2 and increasing values of α. (a): Z = 0.7. 0 1
mum Josephson current through a one dimensional s-
Black line, α = 0.01, red line, α = 0.5, green line, α = 0.7.
(b): Z =2. Blackline,α=0.01,redline,α=0.4,greenline, wavejunction in the clean limit. The three curves corre-
α=0.7. Theblacklinecurvescorrespondtoanearlyperfect spond to values of the band ratio approaching the crit-
(α = 0) ”conventional” s-wave junction. The blue lines are ical value αc = 0.7071. The current-phase relation rep-
theAndreev level for α=αc =0.7071. resented by the green lines (α = 0.7) in both Figs 2
(a) and (b) shows a clear π phase-shift (the maximum
In the nearly insulating limit (Z ), the system Josephson current for 0 < ϕ < π has a negative value).
→ ∞
decouples and we obtain information on the two sepa- For the considered α value this π state persists in the
rateelectrodes. Moreprecisely,inthislimit, thespectral whole temperature range (see Fig. 3 (a) and (b)). The
equation (6) describes surface bound states of energy mechanism of formation of this state is the following.
As the band ratio increases an upper +ǫ (ϕ,α) and a
2
1 r2α4 lower ǫ2(ϕ,α) extra Andreev bands gradually emerge.
E = − (9) −
B ± 1 α4 These two bands have different character compared to
r − the low energy bands ǫ (ϕ,α): they transport super-
1
already discussed in ref. [8] for a junction N/I/s±. currents in the opposit±e directions. As the temperature
4
decreases,onlylowenergylevelarepopulatedwhilethose
at higher energy are empty. In the competition between
Z=0.001
these opposite carrying current energy levels ǫ (ϕ,α)
2
±
and ǫ (ϕ,α), which coexist for any value of the band
± 1 (cid:68)
ratio,itis the temperaturethatdeterminesthe direction
ofthetotalcurrentandthepossibleexistenceofaπ 0 0.001
→
crossoverin the considered two gap superconductor one-
0.5
dimensional junction.
Thesituationissimilar,althoughnotidentical,tothat 0.65
(cid:39)
found in a two-dimensional d-wave π junction. In this 2 (cid:32) 2 0.7
case [24], two kinds of bands, conventional and midgap, (cid:39)
alternate, without coexisting and compete each other in 1
(a)
determining the supercurrent,depending on the angle of
incidence of the Andreev quasiclassical trajectories with
the interface. Thereforeitisthe two-dimensionalityhere
that plays the key role.
In Fig. (3), (a) and (b), we report, for increasing val-
ues of α , the dependencies of the normalized Josephson
criticalcurrentI /I (I =max I (ϕ))fromthereduced
c 0 c ϕ d
temperatureT/T ,derivedbythediscreteAndreevspec- 0.015 (cid:68)
c
tra for a nearly clean junction (Z = 0.001) and a low II0c0.010(cid:115) 0.001
tinradTnichsapetaenrseegnthactayitvjuethnsecigtjinuonnocf(tZiIocn=(bfr3leu)e,erelenisneperesgc,yt,Fivii.egel..y.3thae)qaunadntbit)y) criticalcurrent00..000005 (cid:39) 00..65
tFh(aϕt)t=heΦg0r/o2uπnd0ϕsdtaϕtIe(ϕo)f,thhaesjaunmcitnioimnuims πatsϕhif=teπd.suAchs 0r.e0du0c.e2dt0e.m4pe0r.a6tu0re.8T1T.c0 (cid:39)2 (cid:32) 2 0.7
R 1
discussed above, the π 0 crossover occurring with de-
creasing temperature fo→r α=0.65 in figure 3 (a) and for (cid:115)Z=3 (b)
α = 0.6 (green line) in figure 3 (b)(see inset) is due to
the competition between the contribution to the current
from the high energyband E andthat low energyband
2
E .
1
So far,byusing Eq. (10), wehaveincludedthe contri-
butiontotheJosephsoncurrentfromthediscreteenergy FIG. 3: Critical current as a function of temperature for a
spectrum only. However the continuous-spectrum states s± superconductor S/I/S junction with r = 2, T/Tc = 0.01
make their owncontribution to the current which has to (a)Z = 0.001; black line, α = 0.001, red line α = 0.5, green
be accounted for properly. Following the approach de- line,α = 0.65, blue line, α = 0.7. (b) Z = 3; black line,
α = 0.001, red line, α = 0.5, green line, α = 0.6, blue line,
veloped by Furusaki Tsukada [32] the total Josephson
current, including contributions from both the Andreev α = 0.7. The negative sign of Ic indicates that the junction
minimum is at ϕ = π (π-junction). The inset in panel (b)
bound states and the continous spectrum is given by
shows, on a larger scale, thetransition π 0 for α=0.6
→
e∆ k T 1
1 B
I = [a(ϕ,iω ,α) a( ϕ,iω ,α)] (11)
ℏ Ω n − − n
1
Xωn
Z=0.001 Z=3
Here we have defined Ω = ω2 +∆2 and introduced
1 n 1 (cid:39)
2 (cid:32)2
the Matsubara frequencies ωn = πkBT(2n+1), with n (cid:39)
p 1
rangingfrom to+ . a(ϕ,iω ,α)isascatteringam-
n
−∞ ∞
plitude coefficient for the process in which an electron-
like quasiparticle traveling from the left of the junction (cid:39)2 (cid:32)2
is reflected back as a hole-like quasiparticle in the pres- (cid:39)1
(a) (b)
ence of two bands. This coefficient is derived by solving
the BdG equations under the assumption E > ∆ and
1
dropping the requirement of exponentially decay of the
solutions for x . The details of this procedure will
| |→∞
be given elsewhere [33]. The results of the calculated FIG. 4: Critical current as a function of temperature. The
totalcriticalcurrentI asafunctionofthe reducedtem- contribution of the continuous energy spectrum states has
c
perature, are shown in Fig. 4, (a) and (b), for different been added. (a)Z = 0.001;(b)Z = 3; in both figures, (a)
valuesofthe bandratioαandalreadyconsideredinFig. and (b), the black, red, green, brown, blue lines correspond
to thecurveswith α=0.001,0.5,0.6,0.65,0.7, respectively.
3.
5
The general features of the curves I vs T, shown theAndreevlevels(withoutconsiderationforthe contin-
c
in Fig. (4) qualitatively reproduce those of Fig. (3). uous energy spectrum) shows the formation of a note-
Most notably they confirm the π 0 junction crossover. worthy π-state in the junction coupling. For selected
−
Howeversomenoticeabledifferencesmaybepointedout. values of the band ratio, a π 0 crossover may occur
→
For instance, the π 0 step-like crossover,calculated for as the temperature decreases. In this case the π-state
−
α=0.65andshowninFig. (4)(a),issmoothedoutwhen is observed at high temperature whereas the 0-state is
consideringthecontinuousspectrumcontribution,ascan observed below the crossover temperature. In this re-
be seen by comparing with the corresponding curve in gionthe junctionrecoversthe behaviorofa conventional
Fig. (3), (b). ′′0′′ junction. These results are confirmed by means of
In conclusion the model investigated in this paper for a more exhaustive evaluation of the Josephson current
a multiband superconductor symmetric s±Is± junction, englobing the contribution of the continuous spectrum
predicts a number of non trivial details related to the energy states.
Josephson effect in these systems. These results are of
relevant interest for the case of all-pnictide Josephson
micro-junctions. The spectralequation provides the An-
Acknowledgments
dreevboundlevelsasafunctionofthebandratioparam-
eterα. Themaineffectofthepresenceofthesecondband
isthebuildingupoftwoextraAndreevlevelswhichdrive We thank A. A. Golubov for illuminating discus-
Cooper pairs in a direction opposite to that observed in sions on this topic. The financial contribution of EU
the presence of a single band. NMP.2011.2.2-6 IRONSEA project nr. 283141 is grate-
The phase-current relations predicted on the basis of fully acknowledged.
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