Amplitudes of minima in dynamic conductance spectra of the SNS Andreev contact
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Amplitudes of minima in dynamic conductance spectra of the SNS Andreevcontact
Z. Popovi´c, S. Kuzmichev,
2, 3, a) and T. Kuzmicheva University of Belgrade, Faculty of Physics, Studentski trg 12, 11001 Belgrade,Serbia Lomonosov Moscow State University, Faculty of Physics, 119991 Moscow, Russia Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia (Dated: 6 July 2020)
Despite several theoretical approaches describing multiple Andreev reflections (MAR) effect insuperconductor-normal metal-superconductor (SNS) junction are elaborated, the problem of comprehensiveand adequate description of MAR is highly actual. In particular, a broadening parameter Γ is still unac-counted at all, whereas a ballistic condition (the mean free path for inelastic scattering l to the barrier width d ratio) is considered only in the framework of K¨ummel, Gunsenheimer, and Nikolsky (KGN), as well asGunsenheimer-Zaikin approaches, for an isotropic case and fully-transparent constriction. Nonetheless, aninfluence of l/d ratio to the dynamic conductance spectrum ( dI/dV ) features remains disregarded, thus beingone of the aims of the current work. Our numerical calculations in the framework of an extended KGN ap-proach develop the l/d variation to determine both the number of the Andreev features and their amplitudesin the dI/dV spectrum. We show, in the spectrum of a diffusive SNS junction ( l/d →
1) a suppression of theAndreev excess current, dramatic change in the current voltage I ( V )-curve slope at low bias, with only themain harmonic at eV = 2∆ bias voltage remains well-distinguished in the dI/dV -spectrum. Additionally, weattempt to make a first-ever comparison between experimental data for the high-transparency SNS junctions(more than 85 %) and theoretical predictions. As a result, we calculate the temperature dependences ofamplitudes and areas of Andreev features within the extended KGN approach, which qualitatively agreeswith our experimental data obtained using a “break-junction” technique.This article may be downloaded for personal use only.Any other use requires prior permission of the authorand AIP Publishing. This article appeared in Journal ofApplied Physics 128, 013901 (2020) and may be found athttps://aip.scitation.org/doi/10.1063/5.0010883 I. INTRODUCTION
There is a lack of qualitative comparison between theo-retical predictions and experimental practice for the mul-tiple Andreev reflections (MAR) effect manifestation inthe dynamic conductance spectrum of superconductor-normal metal-superconductor (SNS) junction. This isespecially evident in ballistic ( l > d ) high-transparent(85% < − B < l is electron mean free path forinelastic scattering, d is the width of thin normal N layeror the constriction, while B is the probability of the nor-mal reflection from the NS interface and (1 − B ) is thetransmission probability for normal current. The major-ity of MAR theories do not consider the finite ballis-tics of the contact along the current direction (hereafter z ), which could be introduced as l/d ratio, as well asthe broadening parameter Γ ≡ ~ /τ , where τ is the elas-tic scattering time of electron. On the other hand, theK¨ummel, Gunsenheimer, and Nicolsky theory (KGN), a) Electronic mail: [email protected] as well as a work by Gunsenheimer and Zaikin, whichnearly reproduces the Octavio, Thinkham, Blonder, andKlapwijk theory (OTBK) results, consider l/d ratio asan input parameter, instead of taking it infinite.The simple way to involve a time dependent scatter-ing is an approach, where one can define the probabil-ity of the electron to scatter during the MAR process as1 − exp( − t/τ ) that obviously gives zero for the movementtime t → t > τ . Multiply-ing both values t and τ by the Fermi velocity along thecurrent direction ( v F z ), one can change times to char-acteristic lengths and roughly estimate the probabilityto find the ballistic electron in normal metal undergo-ing n th Andreev reflection as P A ( n ) = exp( − n d/l ) d/l .This is because this electron has passed approximately n d distance since the moment of appearance at the NS-interface. At the same time, the probability to find inci-dent carriers along their mean free path is limited by thewidth of the normal metal, thus d/l (in a simplified caseall of them move normally to NS interface).It is well-known that the dI/dV dynamic conduc-tance spectrum of ballistic high-transparent SNS contactat any temperatures up to critical T c exhibits the seriesof minima at certain bias voltage eV n = 2∆ /n , where n is the integer number, and ∆ is the superconductinggap. These minima constitute so-called subharmonic gapstructure (SGS). In case of absence of the in-gap elec-tron states, a simplified KGN approach to the Andreevcurrent through fully transparent SNS contact could bedescribed as a sum of integrals of electronic density ofstates (DOS) over ( − ∆ − eV, − ∆) energy range, where eV is a contact bias voltage. According to the simpli-fied KGN approach, curving of current-voltage charac-teristics (CVC) at eV n , as well as the shape of the corre-sponding Andreev minima beneath dI/dV backgroundis defined by the single term of the sum for the An-dreev current. This term describes the very last act ofthe in-gap Andreev reflection, which is very probable inthe high-transparent case (1 − B > − ( n + 1) eV ). In case of a classical energy disper-sion law E ( k ) and a BCS law of DOS on energy at zerotemperatures, infinitesimal probability B and broadeningparameter Γ, this term is proportional to I A ( V ) = (1) ev F L x L y P A ( n + 1) N (0) Z ∆ − ( n +1) eV − ∆ − eV | ε |√ ε − ∆ dε = VR N e − ( n +1) dl ( r eV − r ( n + 1) − ( n + 1) 2∆ eV ) , since the normal state resistivity R N of the SNS contact R N = ( e v F ( d/l ) L x L y N (0)) − , where L x L y is the con-tact area, d L x L y is a volume of the normal metal layer, N (0) is normal DOS and v F is the Fermi velocity.For the over-gap bias ( eV > n = 0, but some outgoing electronsstarting their motion through the constriction from theenergy range ( − ∆ − eV, ∆ − eV ), total 2∆ range, con-tinue to participate in a single Andreev reflection, thusproducing Cooper pairs and an additional (excess) cur-rent. The electrons starting from ( − ∆ , − ∆ − eV ) passtoward the conduction band, thus not undergoing An-dreev reflection. In this way Eq. (1) transforms to I exc ( eV ) = VR N ( r eV − r − eV ) e − dl , (2)which leading at high eV to the dependence I exc ( eV >> eR N e − dl (3)of the Andreev excess current I exc on 2∆ (total gap en-ergy range) and the exponent of inverse l/d ratio. Thisresult for the I exc deviates from the theories presentedin the Refs. 1 and 3 that use full expression for the An-dreev reflection probability on energy dependence, by thefactor 4 / ε > ∆.Since the part of the Andreev current that gives themain contribution to the curving of CVC at eV n volt-ages from Eq. (1) is proportional to exp( − n d/l ), oneshould expect a similar dependence of amplitudes of theAndreev minima in dI/dV dynamic conductance spec-trum of SNS contact. Whereas it is impossible to get ananalytical result for finite temperatures and appearanceof the in-gap states in the KGN theory framework, we used the numerical computation to estimate some gen-eral tendency of a finite l/d ratio, and the temperatureinfluence on the amplitudes of the Andreev minima andminima area. The latter can be compared with an exper-imental result of the “break-junction” technique, whichcould produce extremely transparent tunneling constric-tions (85% < − B < B ≈ − B decreasingcould be roughly taken into account in KGN frameworksfor small B values ( B < l/d ratio) by modifyingexponential multiplier by exp( n [ln(1 − B ) − d/l ]), whichtends to exp ( n [ − B − d/l ]) in B → T → | ε | → ∆ in the superconductingstate leads to a divergence of the amplitude of dI/dV Andreev minima. Unfortunately, none of MAR effecttheories do consider the influence of finite Γ. The theo-retical approach to this problem remains an open issue,since one need to consider the loss of carriers in unoc-cupated in-gap states (which do exist in this case), inaddition to the smearing of DOS distribution on energy.Because the loss of carriers strongly depends on energy,being maximal just below the gap edges, it makes the an-alytical description of this process nearly impossible forall known models of MAR effect.3) The “break-junction” technique achieves the bestresults when studying namely layered materials. Noteall layered superconductors are non classical ones. Theyhave frequency-dependent superconducting gap ∆( ω ),the in-gap occupied states at T → and needs further theoreticalinvestigation. II. MODEL AND METHODS
The theoretical model for calculation current voltagecharacteristic and dynamic conductance, used here isbased on formalism previously developed by K¨ummel,Gunsenheimer, and Nicolsky and further adopted byPopovi´c et al . Details of calculation can be foundin these references and we therefore will give a very briefinsight into the formalism used.The theory is applied to the physical situation of two
FIG. 1. a) Scheme of the considered SNS junction. b) Schemeof a break-junction formed on steps-and-terraces in a layeredsample. Typical terrace width is 20-500 nm. In such geom-etry, the current always flows through the constriction alongthe crystallographic c -axis, co-directional with z -axis. long superconducting electrodes (S) separated by a bal-listic normal metal layer (N) of thickness d (Fig. 1a).The SNS junction is connected to the voltage source bynormal conducting external current leads, so a constantelectric field F = − e z V /d ( V is applied bias voltage) ex-ist only in the N layer. We use the formalism of the timedependent Bogoliubov-de Gennes equations (BdGEs)for electron and hole wave function u ( r , t ) and v ( r , t ),respectively, which are i ~ ∂∂t u ( r , t ) = h m [ p + ec A ] − µ i u ( r , t ) ++ ∆( z )Θ( | z | − d/ v ( r , t ) ,i ~ ∂∂t v ( r , t ) = − h m [ p − ec A ] − µ i v ( r , t ) ++ ∆( z )Θ( | z | − d/ u ( r , t ) , (4)where Θ( z ) stands for the Heaviside step function, µ isthe chemical potential, while the temperature dependentvector potential is A = e z cV t/d Θ( d/ − | z | ). This equa-tions are combined with the relaxation time model forcharge transport. Through this model l enters into cal-culation. This implies l > d for the N layer.The KGN theory frameworks were extended by us ear-lier to introduce the anisotropy of the superconductinggap, as well as some effect of ferromagnet barrier on An-dreev transport. We approximate the spatial varia-tion of the pair potential by step function ∆( θ )Θ( | z | − d/ θ ) = ∆ max (1+0 . A (cos(4 πθ ) − k xy momen-tum space (corresponds to the ab -plane of the real space),such that tan( θ ) = k y /k x . The coefficient A reflects thegap anisotropy in percentages, while ∆ max is the maxi-mum amplitude. Here we use some minor amount of theanisotropy ( A = 2%) to get rid of the dI/dV divergenceat the Andreev minima positions eV n = 2∆ /n . For apure s-wave case of superconducting electrodes the coeffi-cient A is equal zero, and ∆ max is the bulk superconduct-ing gap ∆. The temperature dependence of ∆ is givenapproximately by ∆( T ) = ∆(0) tanh(1 . p T c /T − u ± k and v ± k which move in the electric fielddue to the applied voltage V . This solutions are used tocalculate average current density which in the relaxation-time model is given by h j i = − e m ˆ X k n f ( E k ) h h u + ∗ k P u + k i + h u −∗ k P u − k i i + (5)+(1 − f ( E k )) h h v + k P v + ∗ k i + h v − k P v −∗ k i io . Here f is the Fermi distribution function and P =[ − i ~ ∇ + e A /c ] is the gauge-invariant momentum oper-ator. The averaged momentum densities h u ±∗ k P u ± k i and h v ± k P v ±∗ k i are calculated in the same way as in Refs. 5and 10–13 and they are proportional to correspondingmultiple Andreev reflection probability amplitudes h u ±∗ k P u ± k i ≈ ∞ X n =0 (cid:12)(cid:12)(cid:12) A ± n ( E ± eV (cid:12)(cid:12)(cid:12) , (6) h v ± k P v ±∗ k i ≈ ∞ X n =0 (cid:12)(cid:12)(cid:12) A ± n +1 ( E ± eV (cid:12)(cid:12)(cid:12) , (7)where A ± n ( E ) = Q nν =1 γ ( E ± νeV ∓ eV ), A ± n +1 ( E ) = Q n +1 ν =1 γ ( E ± νeV ∓ eV ). So, A ± n A ± n +1 are the proba-bility amplitude that a quasiparticle at energy E startsto move as an electron against (+) or opposite to (-)the field will reappear in the normal metal barrier as anelectron after 2 n Andreev reflections and as a hole af-ter 2 n + 1 Andreev reflections, respectively. Therefore, | γ ( E ) | is the probability that an AR occurs at energy E in the phase boundary of a semi-infinite superconductingelectrode and γ = ( E − i (∆ − E ) / ) / ∆ for E < ∆(whereas for
E > ∆ it may be approximate by zero as inRef. 5).Note that, the solution of time dependent BdGEs inthe limit of vanishing voltage must turn into the sta-tionary quasiparticle wave function of an superconduc-tor/normal metal junction. In the theory used here voltage dependent solutions evolve from the bound states(while in Ref. 6 the solutions evolving from the scatter-ing states play the dominant role) and quasiparticle starttheir motion in the electric field from energy | E | < ∆ af-ter each Andreev reflection. Since, the SNS junction isconnected by normal conducting external current leadsto the voltage source almost all quasiparticle excitationshave energies | E | < ∆ and they are completely Andreevreflected at the external interfaces between the leads andthe junction.After very extensive calculations presented in detail inRefs. 5 and 10 it can be obtained that the total cur-rent density h j i (which has only z component) is the sumof Ohmic current density h j N i and current density dueto Andreev reflection h j AR i . In the following we calcu-late the total current I (and corresponding conductance dI/dV ) which is connected to the total current density h j i z via cross section area L x L y I = I N + I AR = L x L y ( h j N i z + h j AR i z ) . (8)The total current is normalized by the temperature de-pendent current I = 2∆( T ) / ( eR N ).Experimentally, in order to make SNS junctions forAndreev spectroscopy studies, we used a break-junctiontechnique. Its specialities, some details and discussionscould be found elsewhere. The sample prepared as a thinrectangular plate with dimensions about 3 × . × . was attached to a springy sample holder by four-contactpads made of In-Ga paste at room temperature. Af-ter cooling down to T = 4 . c is a constriction. The resulting constriction turns farfrom potential and current leads, which prevents junc-tion overheating and provides true four-point probe. Inmagnesium diborides, the used technique providesconstrictions with various transparency.Under fine tuning the curvature of the sample holder,the two cryogenic clefts slide apart touching onto var-ious terraces. Such tuning enables to adjust the con-striction area in order to realize a desired tunneling SNSregime. During the experiment, the cryogenic surfacesremain tightly connected when sliding that prevents im-purity penetration into the crack and protects the purityof cryogenic clefts.In a layered sample the crack naturally splits the ab -planes, with a formation of steps and terraces. The heightof the step is usually about (10 − c unit cell pa-rameters, whereas the typical terrace size appears about20 −
500 nm. These features, as well as the geometryof the break-junction setup are presented in Fig. 1(b).While the constriction area “c” of planar ScS-contact in ab crystallographic plane can be of any shape, its areavariation under the fine tuning defines just normal re-sistance R N . The current passes along c -direction, andnamely the junction width d defines ballistic l/d ratio(note, l is inelastic scattering length along c -direction, or z -axis in Fig. 1). Typically this is the case for polycrys-talline sample of layered compound as well .Since the constriction appears as a part of a terrace(see Fig. 1(b)), the typical in-plane dimensions of thebreak-junction L x , L y supposed to be much more thanthe Cooper pair size ξ . Since the macroscopic scale(which could be estimated on accounting relatively lowresistance for some contacts obtained), there is no needin accounting of the dimensional quantization effect. De-spite the impossibility of direct lengths measuring or vi-sualization of the break-junction obtained, it can be as-sumed that along the c -direction, a couple of distorted(“broken”) layers of crystal structure act as the con-striction separating the intact superconducting banks.Then, the effective constriction width d is compared with(2 − c unit cell parameters.Here we present the experimental data for the ballistic high-transparency barriers (95%–98%), which are elec-trically equivalent to a thin layer of normal metal (SNS)with the thickness d about the superconducting coher-ence length ξ . In the majority of the break junctions inFe-based superconductors we studied, the resulting con-striction formally act as normal metal, with the I ( V ) and dI ( V ) /dV typical for the clean classical SNS junction. We consider the presence of the excess current and clearlyvisible SGS, as well as their disappearing over T c , as thebenchmark of the MAR regime developing and the ballis-tic character of our break-junctions. As mentioned above,using mechanical readjustment, it is possible to producecontacts of various geometry and area. As a commonpractice, considered are only the break-junctions whichSGS position do not depend on the contact in-plane ge-ometry (as an example, see Figs. 9, 12, 16 in review ),and which CVC is symmetric above and below T c .In our studies, the dynamic conductance spectrawere measured directly by a standard modulationtechnique. We used a current source with an admix-ture of ac frequency about 1 kHz from the external oscil-lator. The results obtained with this kind of setup areinsensitive to the presence of parallel ohmic conductionpaths; if any path is present, the dynamic conductancecurve shifts along the vertical axis only, while the biasstay unchanged. As a result, the “break-junction” tech-nique is a precise and high-resolution local probe of thesuperconducting order parameter, its temperature de-pendence and a fine structure. III. RESULTS AND DISCUSSION
The numerical results of CVC calculation with the l/d ratio variation ( T → which accounts for a realistic angle dependent three-dimensional geometry of the contact, are presented inFig. 2(a). Note, the bias voltage is normalized to su-perconducting gap value ∆. One can say, this l/d ratiodemonstrates the ballistic “quality” of a microcontact.Since KGN approach is valid only for fully transparentconstrictions ( B = 0), contacts with the largest l/d ratioare in deep ballistic limit. In the latter case, KGN theorypredicts the extreme rise of the Andreev current at low bi-ases (so-called “foot” region), as it could be seen for threeupper CVCs (violet, cyan and blue) in Fig. 2(a). Therange of l/d values, which produces the situation, whenthe current at the edge of the foot structure (“the edgecurrent”) overcomes a total current of the SNS-contactat ∆ / < eV < n ln (1 − B )] = (1 − B ) n correction to be made for theabsolute ballistics l/d → ∞ case (see the 1 st point of thelist of difficulties presented above), this limit roughly cor-responds to the normal reflection probability B ≈ − B ≈
88% trans-parency, even absolutely ballistic, will not demonstrateany drastic rise of current in foot region, which, how-ever, agrees with Averin-Bardas theory predictions forthe case of high transparency (more than 80%) constric-tion. This fact is in good agreement with our experimen-tal findings. In general, Fig. 2(a) shows a tendency ofdramatic flattering of the CVC slope in the foot regionwith l/d or (1 − B ) decrease. As for high bias voltages eV ≫ − d/l ], thus corresponding to Eq. (3).All I ( V ) characteristics from Fig. 2(a), as well as theirderivatives dI/dV from Fig. 2(b), show well visible sub-harmonic gap structure at eV n = 2∆ /n , where n is theinteger number. It was shown, the SGS position isdetermined by this formula at any temperatures up to T c . As one can see from the numerical calculations ofdynamic conductance spectra presented in Fig. 2(b), itis possible to make the same conclusion for the l/d ratiovariation. This variation changes a dynamic conductancebackground (especially for biases eV < ∆), as well as theamplitudes and areas of the Andreev minima, but doesnot change their locations. (a) l/d
100 15 10 5 2 1 I( V ) (b) l/d
100 15 10 5 2 1 d I( V ) / d V eV / FIG. 2. The numerical result for (a) CVCs I ( V ) and (b) cor-responding dynamic conductance spectra dI ( V ) /dV , for the fully transparent SNS contact ( T →
0) with the l/d ratiovariation, calculated in the framework of KGN theory. Thesubharmonic gap structure (SGS) is well visible in all CVCsand conductances. The location of the SGS corresponds to eV n = 2∆ /n , where n is integer number, ∆ is the super-conducting gap amplitude. Normal state ohmic dependence( T > T c ) is shown by dashed line in panel (a). The divergence of BCS density of states near the gapedge leads to the divergence of minima amplitudes, re-sulting in KGN frameworks. To overcome such the theo-retical divergence, we used as small as 2% anisotropy forthe superconducting s -wave order parameter distortion.The visible amplitude of the n = 1 minima (“2∆” fea-ture) in dynamic conductance spectra (Fig. 2(b)) dom-inates over other minima only for black and red curveswith l/d ≤
2, and the number of a detectable minima islimited there, even we consider the case of T → →
0. The flattering of the foot region is well visiblein this case. The usage of finite bias step in Figs. 2 (a)and (b) leads to an averaging of beating behaviour of thedynamic conductance at small bias voltage eV ≪ ∆ (seeAppendix A for details and some technical issues).Figure 3 presents the results for the numerical esti-mation of amplitude of the dI/dV Andreev minima offully transparent SNS contact (having contact diameterrange d = (0 . − ξ , which ensure that no one An-dreev in-gap level is visible) at T → l/d ratiovariation. Note that, l and d are normalized by coher-ence length ξ , and n are natural numbers, which definea location of the SGS minima eV n = 2∆ /n . Verticalbars depict the numerical uncertainty of the result. For-mally, the KGN theory does not consider any inelasticprocesses for the l/d >
1, presented here, and all elec-trons and holes pass normal metal ballistic. It is clearfrom Fig. 3, the considered l/d range is separated by twoparts at l/d ≈
3, where all Andreev minima comprisingthe subharmonic gap structure have nearly the same vis- ~exp(-2d/l) n=2n=3 m i n i m a a m p lit ud e , ( d I / d V ) / R N l / d ~exp(-3d/l)~exp(-d/l ) n=1 FIG. 3. Visible amplitudes of SNS Andreev dynamic con-ductance minima for n = 1 , , l/d ratio(black squares, red triangles, green circles, correspondingly)at T → d = (0 . − ξ . Points of thesame hue, but different shade demonstrate the range of re-producibility for the finite dV -step of numerical calculations.Lines demonstrate exp( − n d/l ) dependencies (is shown withthe corresponding color). T/T C c = 1 a m p lit ud e / a m p lit ud e ( ) Andreev minima n=1 n=2 n=3 c = 0.9 1.2 a r ea / a r ea ( ) ( ) / T / T C FIG. 4. Temperature dependence of Andreev minima am-plitude of n = 1 , , T →
0. Comparison with the tanh[∆ / (2 k B T )]dependence is shown by bold gray arc curve. The inset showstemperature dependence of the area of n = 1 , , T ), as well as to their area (0) / ∆(0)value. Comparison with tanh[ c ∆ / (2 k B T )] function with thefree parameter c = 0 . − . d = (0 . − ξ and l/d = 3 / −
100 vari-ation. ible amplitude. For l/d < n = 1 dominates over all otherminima of SGS. The opposite situation could be seenfor the high-quality ballistic junction with l/d >
3. Inreal constriction having finite transparency, this limit-ing value could be higher and approaches, for example,( l/d ) lim = 4 − − B ≈
90% normal transmission(according to the exp( n [ln(1 − B ) − d/l ]) estimation fromthe 1 st point of the list of difficulties presented above).It is possible to make an analytical approximation forthe numerical result presented in Fig. 3 for T →
0. If wedenote an amplitude of the n th Andreev minima for infi-nite l/d ratio (absolute ballistic constriction) as A n ( ∞ ),then we can expect the A n ( l/d ) = A n ( ∞ ) exp( − n d/l )dependence (plotted as lines in Fig. 3) according to theexpression for the current Eq. (1). Thus, at B → T → r A ( l/d ) ≡ A n +1 /A n = r A ( ∞ ) exp( − d/l ).The temperature dependence of Andreev minima am-plitude A n ( T ) in KGN theory frameworks was checkedfor the variety of normal metal thicknesses (the same d = (0 . − ξ ) and ballistic quality of the constric- h a l f- w i d t h / h a l f- w i d t h ( ) ( ) / Experimental dataMg(Al)B samples Kr10b (n=2) MB6 (n=2) MB12 (n=1) MBA1a (n=1)Sm,Th-1111 NZ21b (n=1) T / T C FIG. 5. Temperature dependence of relative half-width ofAndreev minima normalized to their magnitudes at T → T ). Experimental results for Mg(Al)B and Sm-1111 superconductors. tion (3 / < l/d < A n ( T ) /A n (0) results for the first ( n = 1), second ( n = 2)and third ( n = 3) minima are qualitatively the same (seethin curves in Fig. 4, where their colors represent theSGS number “ n ”). The scattering range of our numer-ical estimation is presented in Fig. 4 by bold gray arctending to tanh[∆ / (2 k B T )], which temperature depen-dence is close to the BCS superconducting ∆( T ). Notethat in this work all the amplitudes are initially normal-ized to 2∆( T ) value, since the current is normalized to I = 2∆( T ) / ( eR N ). The amplitude residual is shown inFig. 8 of Appendix B.The inset of Fig. 4 demonstrates the temperaturetrends of Andreev minima areas normalized to their T = 0 value for the same numerical estimations in therange of normal metal width d up to 3 ξ , limiting the vi-sual appearance of 1 st Andreev bound state, as separateCVC feature. As an envelope curve for dI ( V ) /dV spec-tra, we used a piecewise function constructed as a seriesof straight line segments, which connect all points withmaximal dynamic conductivity at eV → /n from theleft. Being the easiest solution, such the piecewise func-tion produces a background uncertainty. The area is es-timated between the dI ( V ) /dV spectra and its envelopecurve. For details, see the second paragraph of AppendixB.All the numerical results lay within bold yellow re-gion (see the inset of Fig. 4), which is constructed bythe tanh[ c ∆ / (2 k B T )] function with the variation of afree parameter c = 0 . − .
2. The results for n = 1minima (black thin lines) slightly differ to those for n = 2 ,
3, but can be roughly approximated by the sametanh[ c ∆ / (2 k B T )] function.Here we use the experimental data obtained for break- Experimental dataMg(Al)B samples Kr10b (n=2) MB6 (n=2) MB12 (n=1) MBA1a (n=1)Sm,Th-1111 NZ21b (n=1)Theoretical results BCS (T) KGN numerical tanh[ /(2 k B T)] m i n i m a a m p lit ud e / a m p lit ud e ( ) T / T C FIG. 6. Temperature dependence of Andreev minima am-plitudes normalized to those at T →
0. Bold gray curvedemonstrates the range of reproducibility for d = (0 . − ξ according the KGN theory (data is taken from Fig. 4; nobroadening parameter Γ is accounted). Experimental resultsfor Mg(Al)B and Sm-1111 superconductors are shown bypoints. Weak-coupling BCS theory result for ∆( T ) is pre-sented by dash-dotted curve for comparison. junctions formed in MgB , MgB + MgO, Mg − x Al x B superconducting samples, as well as in Sm − x Th x OFeAsiron-based superconductor (so called 1111). For thesematerials, the in-plane ξ is as low as several unit cell pa-rameters, therefore one could roughly estimate the con-striction width d ≈ ξ . Being two-gap superconductors,magnesium diborides as well as the 1111 system are ex-pected to have isotopic s-wave order parameters, whichis important to get rid of the uncertainty that producesany gap anisotropy. SNS Andreev spectra showing twodistinct SGS (caused by the ∆ L and ∆ S gaps) for sev-eral break-junctions were recorded by us earlier with avariation of T . These spectra were selected to plottemperature dependence of half-widths, amplitudes, andareas for n = 1 , L / ∆ S ratio (about 3-6), the two SGSare non-overlapped, thus facilitating the observation ofundistorted Andreev minima. With it, SGS for the small∆ S gap is usually located in the area of the drastic rise ofthe dynamic conductance at low biases (“foot”), whichmakes the amplitude and half-width estimation ambigu-ous. In order to exclude that factor, we use the large gapSGS to made such estimate.The temperature dependence of the half-width of theAndreev minima (various n numbers) normalized toboth, its zero value and ∆ L ( T ) / ∆ L (0) are shown on Fig.5. Results were obtained in several superconducting sam-ples. All the data plotted here was estimated directlyfrom the dips in our experimental spectra and was notaveraged. Noteworthily, the dependence showed by olive m i n i m a a r ea / a r ea ( ) ( ) / T / T C Experimental dataMg(Al)B samples Kr10b (n=2) MB6 (n=2) MB12 (n=1) MBA1a (n=1)Sm,Th-1111 NZ21b (n=1) KGN theory tanh[c /(2k B T)] c = 0.9 - 1.2
FIG. 7. Temperature dependence of Andreev minima areasnormalized to their magnitudes at T →
0, as well as ∆( T ).The range of the numerical results in the framework of KGNtheory are presented by bold yellow lines. Experimental re-sults for Mg ( Al ) B and Sm-1111 superconductors are shownby points. up triangles and blue down triangles similarly evolve withtemperature, up to ∼ T c /
2, as one can see in Fig. 5 aswell as in Fig. 6 showing temperature dependence of An-dreev minima amplitude. This demonstrate the temper-ature trend of their relative minima amplitudes, despitethese data correspond to the behavior of Andreev minimafor the superconductors of different families (MgB andFe-based 1111). The slowest growing dependence (openblack and cyan symbols) in Fig. 5 seem correspondingto the most homogeneous contact points, with, therefore,the minimum broadening Γ. The latter obviously makessuch data an ideal candidate to compare with MAR the-ories, those considering zero Γ.The data range shown in Figs. 6 and 7 by solid pointsrepresents the experimental uncertainty. For the mostqualitative SNS-contacts on magnesium diborides (blackand cyan points), the data keep close to the variety ofnumerically obtained data using the extended KGN the-ory (bold arches). The slightly averaged experimentaldependence of amplitudes A n ( T ) /A n (0) are strain afterthe theoretical predictions (bold gray line in Fig. 6). Thedeviation from this trend is apparently caused more bythe changes in the scattering rate than the subharmonicorder n . This means the larger Γ, the lower the temper-ature dependence of the relative minima amplitude. Itseems, the experimental results tends to tanh[∆ / (2 k B T )]function, rather than standard BCS theory result ∆( T )(dash-dot curve). All experimental data plotted in Fig.6 demonstrate the decrease of the Andreev minima am-plitudes starting from the lowest temperatures, in accor-dance with OTBK and KGN theories. The results for the relative area of the Andreev minimaestimation from our experimental data for the dynamicconductance of SNS contacts in Mg(Al)B superconduc-tor show some dispersion due to the distinct level of scat-tering (see points in Fig. 7). The resulting dependenceof relative area on T for green rhombs ( n = 2 minima)shows the signs of some kind of a “tail” that tends to T c starting from T /T c = 2 /
3. Taking into account thatthe corresponding dependences in Figs. 5 and 6 for thisSNS contact (green rhombs) deviates significantly fromthe set of all others, one can conclude, it has the shortestsmearing time, thus the strongest Γ, which can result inthe loosing of states and the suppression of this Andreevdip well before the T c . In case when Γ value is com-parable to ∆, the number of empty states appears insidethe superconducting gap region, and electrons involved inMAR process may be lost in the in-gap range of energies,thus decreasing the Andreev part of the current, as wellas the amplitude and area of the corresponding Andreevminima. However the variety of these experimental datapoints correspond to the predictions of the KGN theorywell (the range limited by bold yellow curves) and can beroughly approximated by the tanh[ c ∆ / (2 k B T )] functionhaving the only free parameter c = 0 . − . IV. CONCLUSIONS
In conclusion, we made qualitative comparison be-tween theoretical predictions of the extended KGNtheory in isotropic case and experimental data bythe “break-junction” technique for areas and amplitudes A n of the Andreev minima in the dynamic conductancespectra of ballistic high-transparent ( B < T )and, thus, the pretty small Γ.Our experimental A n ( T ) dependences do not showany region of the minima amplitude increase with tem-perature, and tend to the theoretical one A ( T ) ∼ ∆( T ) tanh[∆ / (2 k B T )]. The temperature dependences ofthe minima area correspond to the KGN theoretical pre-dictions well, and can be qualitatively described withtanh-like function with the single free parameter.We estimated, how the minima amplitudes in the dy-namic conductance of SNS contact depend on the meanfree path to the constriction width ratio A n ( l/d ) at T → − n d/l ) analytical approx-imation fits the numerical result. We have shown thatthe amplitude ratio of any adjacent Andreev minima is r A ( l/d ) ≡ A n +1 /A n = r A ( ∞ ) exp ( − d/l ) at T → ACKNOWLEDGMENTS
We are grateful to S. I. Krasnosvobodtsev, as well as L.G. Sevastyanova, K. P. Burdina, V. K. Gentchel, B. M.Bulychev, and N. D. Zhigadlo for the samples providedat our disposal. The work of S.A.K. ware supported byRFBR project 18-02-01075a. T.E.K. acknowledges thestate assignment of the Ministry of Science and HigherEducation of the Russian Federation (topic “Physics ofhigh-temperature superconductors and novel quantummaterials”, No. 0023-2019-0005). The work of Z. P. wassupported by Serbian Ministry of Education, Science andTechnological Development, Project No. 171033.
DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest.
V. APPENDIX A
To overcome the issue with vanishing broadening pa-rameter Γ (the 2 nd point of the list of difficulties pre-sented above) leading to some divergences in the theoryfrom one hand, and finite Γ that smears features of ex-perimental dynamic conductance spectra, from the otherhand, we use the finite dV -step for the partial DOS onenergy, as well as for CVC calculations. The results ofcalculation obtained for various normal metal width d ,distinct l/d ratio, have shown the counterintuitive en-hancement of the numerical reproducibility with an in-crease in the dV -step of calculation up to 0 . dI/dV minima amplitudes and the re-producibility of its shapes. The other numerical way toovercome the minima divergences is to extend the KGNmodel and involve some superconducting gap anisotropyin the k -space. We used 2% anisotropy and standardfour-fold superconducting s-wave order parameter distri-bution.The absence of well-defined minima series in a footregion of dynamic conductance spectra for l/d = 10 to15 (blue and cyan curves in Fig 2(b), respectively) comesfrom the numerical limiting of the bias step ( dV = 0 . /n − / ( n + 1) = 2 /n ( n + 1) ≈ ( eV ) , since n is an integer partof 2∆ /eV . To overcome this technical issue one need toincrease the density of calculated points as 1 / ( eV ) . VI. APPENDIX B
Theoretical dependence of Andreev minima amplitudeversus temperature normalized to their values at T → A n ( T ) /A n (0), in Fig. 4 find quantitative coincidence.This is presented in Fig. 8 where their amplitude residual l/d > 10 -1.0-0.50.00.5 n=1 n=2 n=3 minima l/d < 10d < < d < 3 ; l/d < 10 r e s i du a l o f a m p lit ud e , % T / T C FIG. 8. Temperature dependence of amplitude residual inpercents for all curves from Fig. 4 (shows better then 1%correspondence). in comparison with tanh[∆ / (2 k B T )] function is shown.As an example see solid and dashed lines that representresidual to one and the same function in this figure. Asone can see at Fig. 8 the residual between them for allcurves from Fig. 4 does not exceed ≈ n = 1Andreev minima, similarly for relative areas of minimaon temperature (from the inset of Fig. 4). The best oneis obtained for n = 3 (error < . T c produces the largest errors, since we needto compare the values tending to zero. One can concludefrom Fig. 8 that both l/d and d/ξ variation has no in-fluence on the correspondence quality between amplitude A n ( T ) dependence and tanh[∆ / (2 k B T )] from Fig. 4.For the estimation of an area of the first n = 1 min-ima (presented by thin black lines in the inset of Fig. 4)we used the upper limit 3∆, in order to make its inte-gration range the same as for n = 2 minima, instead ofinfinite. An accurate estimation of this area is a difficulttask both for theory and for experiment, since the dI/dV background uncertainty. M. Octavio, M. Tinkham, G. E. Blonder, and T. M. Klapwijk,Phys. Rev. B , 6739 (1983). G. B. Arnold, J. Low Temp. Phys. , 1 (1987). D. Averin and A. Bardas, Phys. Rev. Lett. , 1831 (1995). J. C. Cuevas, A. Mart´ın-Rodero, and A. Levy Yeyaty, Phys. Rev.B , 7366 (1996); A. Poenicke, J. C. Cuevas, and M. Fogelstrøm,ibid. , 220510(R) (2002). R. K¨ u mmel, U. Gunsenheimer, and R. Nicolsky, Phys. Rev. B U. Gunsenheimer and A.D. Zaikin, Phys. Rev. B , 6317 (1994). D. M. Gokhfeld, Supercond. Sci. Technol. , 62-66 (2007). J. Moreland and J. W. Ekin, Appl. Phys. Lett. , 175 (1985);J. Moreland and J. W. Ekin, J. Appl. Phys. , 3888 (1985). S. A. Kuzmichev and T.E. Kuzmicheva, Low Temp. Phys. , 1284 (2016)]. Z. Popovi´c, L. Dobrosavljevi´c - Gruji´c, and R. Zikic, Phys. Rev.B Z. Popovi´c, L. Dobrosavljevi´c-Gruji´c, and R. Zikic, J. Phys. Soc.Jpn. , 114714 (2013). Z. Popovi´c, R. Zikic, and L. Dobrosavljevi´c-Gruji´c, Prog. Theor.Exp. Phys. , 103I01 (2015). Z. Popovi´c, P. Miranovi´c, and R. Zikic, Phys. Status Solidi b , 1700554 (2018). R. K¨ u mmel and W. Senftinger, Z. Phys. B , 275 (1985). A. Jacobs, R. K¨ u mmel, and H. Plehn, Superlattices and Mi-crostructures , 669 (1999). S. A. Kuzmichev, T. E. Shanygina, S. N. Tchesnokov, S. I. Kras-nosvobodtsev, Solid State Commun. , 119 (2012). S. A. Kuzmichev, T. E. Kuzmicheva, and S. N. Tchesnokov,JETP Letters , 295 (2014) [Pisma ZheTF , 339 (2014)]. Ya.G. Ponomarev, S.A. Kuzmichev, N.M. Kadomtseva, M.G.Mikheev, M.V. Sudakova, S.N. Chesnokov, E.G. Maximov, S.I.Krasnosvobodtsev, L.G. Sevastyanova, K.P. Burdina, and B.M.Bulychev, JETP Lett. , 484 (2004) [Pisma ZheTF 79, 597(2004)]. Ya.G. Ponomarev, S.A. Kuzmichev, M.G. Mikheev, M.V. Su-dakova, S.N. Tchesnokov, N.Z. Timergaleev, A.V. Yarigin, E.G.Maksimov, S.I. Krasnosvobodtsev, A.V. Varlashkin, M.A. Hein,G. M¨uller, H. Piel, L.G. Sevastyanova, O.V. Kravchenko, K.P.Burdina, B.M. Bulychev, Solid State Comm. , 85 (2004). T.E. Kuzmicheva, S.A. Kuzmichev, M.G. Mikheev, Ya.G. Pono-marev, S.N. Tchesnokov, Yu.F. Eltsev, V.M. Pudalov, K.S. Per-vakov, A.V. Sadakov, A.S. Usoltsev, E.P. Khlybov, and L.F. Ku-likova, Europhys. Lett. , 67006 (2013). T.E. Kuzmicheva, S.A. Kuzmichev, K.S. Pervakov, V.M. Pu-dalov, N.D. Zhigadlo, Phys. Rev. B , 094507 (2017). T.E. Kuzmicheva, S.A. Kuzmichev, N.D. Zhigadlo, Phys. Rev. B , 144504 (2019). Ya.G. Ponomarev, S.A. Kuzmichev, M.G. Mikheev, M.V. Su-dakova, S.N. Tchesnokov, O.S. Volkova, A.N. Vasiliev, T. H¨anke,C. Hess, G. Behr, R. Klingeler, and B. B¨uchner, Phys. Rev. B79