Sardar, Manas

Sreeram, P A

IACS Institutional Repository

Indian J. Phys. 69A (1), 175 -  182 (1995)I J P  A— an international journalCharacteristics O f c Axis And ab  A xis Tunneling In S-Wave Superconductors.P. A. Sreeram 1 and M anas Sardar2Institute of physics, Bhubaneswar-751 005, IndiaAb,s1rart: T he peculiarities in tunneling characteristics have been studied in the light of the controversy between s-wave and d-wave character of High 7'c superconductivity. We show that anisotropic s-wave gap has the sam e low voltage power law conductance and two peak structure in the density of states as d-wave superconductors. The assym etric tunneling conductance and zero bias conductance for the c-axis tunneling is shown to occur because of finite band sp litting com ing from the interlayer hopping parameter. The charecteristics features which have been so far studied in this area have been attributed to d-wave super­conductors. Here we find that these features are not unique to the d-wave theory and can also be realised in the context of the s-wave theory as proposed recently by S.Chakraborty H al.K eyw ords: d-w ave s u p e rc o n d u c tiv ity ,tu n n e lin g  c o n d u c ta n c e , zero b ia s  c o n d u c ta n c ePACS N os. 74.20.Mn, 74.25Fy, 74.72.Bk, 74.80Fp1 e-mail: sreeram® iopb.ernet.in2c-mail:manas<&iopb.emet.in<£> 19951ACS176 P A Sreeram and Manas Sardar1 I n t r o d u c t io n .Anomolous single particle tunneling characteristics in NIS and SIS junctions of the high 7'c superconductors have remained a subject of great interest. The single particle tunneling conductance in both normal and superconducting states is a measure of the density of states in the normal and superconducting states, and so in principle one gets a lot of informations about the details of the superconducting gap parameter. This is a matter of great current interest, in view of the recent controversy about d^wave or anisotropic s-wave symmetry of the gap function. Whereas some experimental results [1],[2],[3] are in favour of a d-wave symmetry gap function, the recent experiments [4],[5],[6] are in favour of an anisotropic s-wave gap function.It has been argued th a t[7], the tunneling characteristics, specifically the quadratic rise of current with voltage a t low enough voltages and the two peak structure seen in the conductance voltage characteristics is an evidence for a d-wave superconductor In view of this, we investigated the single particle tunneling characteristics for the anisotropic s-wave superconductors , recently proposed by Chakraborty et. al.[8]Before we come to the specific problem we address, we highlight the main puzzling features seen in the tunneling spectroscopy of the high Tc superconductors[9].(1) At low temperature, V = 0 tunneling current is zero for tunneling along the ab axis, but nonzero along c axis tunneling. (2) The ab plane conductance becomes smooth at larger temperatures but the c axis conductance goes on incresing with temperature (3) At low temperature and large bias, ab plane tunneling conductance decreases or saturates whereas the c axis conductance goes on increasing roughly linearly with voltage. (4) The ab plane tunneling shows conventional gap like structures but the r axis tunneling shows a much broadened shoulder at the gap edge. (5) Both NIS and SIS junction shows assymetric I-V characteristics with respect to the sign of the bias voltage. (6) Both direction tunneling shows finite density of states for V  < A even at the lowest temperatures. For very low voltage current has a quadratic rise. (7) There is large broadening of gap in voltage , and a conductance overshoot for c axis tunneling. (8) For ab axis tunneling, zero bias conductance is zero. So there is no density of states at the Fermi energy, but for very small V  < A there is finite current, showing that there is no fully developed gap, or the gap is highly anisotropic with Afc being very small in a substantial region of the Brillouin zone. For the c axis tunneling the most common explanation for the ubiquitous zero bias conductance and the characteristic V shaped conductance versus voltage characteristics is explained , as due to either because of tunneling through localized states in the barrier or due to scattering by magnetic impurity inside the junctions. In this paper we emphasize on the distinction between the a t plane and c axis single particle, tunneling channelsCharacteristics of c axis and ab axis tunneling etc 177for both supeconductor to normal and superconductor to superconductor (N1S and SIS) junctions. Specifically we shall consider a layer material like YBCO or Bi-2212 material. We model such superconductors by two planar BCS superconductors coupled by a single particle hopping term along the c axis. We consider also the case, when over and above the single particle hopping, there is a Joseph son coupling between the planes. We propose that the observed assymetry of the normal state in plane tunneling conductance, with respect to the sign of the bias voltage in MIC(metal- insulator-cuprate) junction is a consequence of the existence of nonbonding and bonding band with finite splitting between them, in the cuprates. In the split band picture, when the metal is held at positive bias with respect to the cuprate, then there are two channels of elastic tunneling into the nonbonding and the bonding bands. In the reverse bias situation, only one of the bands takes part in tunneling. So the conductance will be assymetric, for MIC and NIS junctions. On the other hand for SIS and CIO junctions the conductance voltage characteristics will be symmetric. Now for a CIC , NIS or for a SIS junction, when both sides of the junctions are cuprates, there is an important difference between tunneling along c and ab axis. In the ab axis tunneling geometry electrons tunnel only from antibonding to antibonding and bonding to bonding bands. Whereas for the c axis tunneling there is another additional channel for conduction, i.e from antibonding to bonding band. This tunneling will be present even in absence of a finite bias voltage either way. The chemical potentials for the two bands differ by 2t±, where <1 is the c axis hopping amplitude. So the tunneling along c axis will show a zero bias conductance, but the ab axis tunneling will have zero conductance at V =  0 and T  = 0.We find that the zero bias tunneling conductance observed along the c axis tunneling increases with temperature and do not show any sign of saturation at all. This is our main result.We shall also discuss, the reason why for T  > Tc the ab axis tunneling characteristics becomes smooth, while the c axis tunneling continues to be temperature dependent and rises with temperature. Lastly we predict that for MIC geometry tunneling (below Tc) the assymetry (or alternatively the background conductance) will be more for lesser value of the gap in the superconductor.1782 T u n n e l in g  C h a r e c t e r i s t i c s  in  s -w a v e  s u p e r c o n ­d u c to r sTo start we take the effective hamiltonian proposed by Chakraborty et. al.[8]k - P ) cl l clo + O ”*2) + V* c » ^ cl \ C- k l C- k ' l cl ' l + (1_>2)k t v-2 „2 / n+ * c*tC“ *i c- ■ 0 )kIn this model, there is no hopping term along the c axis from plane to pintle, even though the band theory estimates for the c axis hopping amplitude tj. is about -1 to - of the inplane hopping parameter. The reason is supposed to be that, due to strong correlation in the plane itself, the single particle band motion is absent in the c direction. The conduction along c axis is purely due to incoherent processes.' On the other hand coherent propagation of “singlet objects” ( pairs of elctrons)is possible That is the origin of the last term ( Josephson coupling of a very unusual kind) It should be emphasized that, it has not been proved within a realistic model for hig Tc superconductors.We prefer to keep the band term in the hamiltonian. The origin of subbands can be understood as follbws. We consider a two layer material like YBCO. The individual layers can be modelled by a 2-d tight binding band with dispersion,Pj4 Sreeram and Manas Sardarfit = -2f(cos(fcx) +  cos(ky)) +  4t,cos(kx)cos(ky)where t and t' are nearest and next to nearest neighbour hopping in the planes of some effective site. We take, t = 0.3eV and 4t± =  0.45eV. For two closely spaced planes, in interlayer matrix element t±(k) = t±(cos(kx) -  cos(ky))2 results in formation ofsubbands, E^ ^(k) = e(k) ±  fi(fc) where the and ip are antibonding and bondingband fermions defined as, $(fc), 4'(k) = The location of chemical potentialswill be detrmined by the doping.To illustrate the difference between tunneling along ab axis and along c axis wc write down the tunneling hamiltonian without any explicit dependence of the tunneling amplitude on momenta or energy. For tunneling along ab axis, the hamiltonoan will be+  0  —* vo*P kpwhere a  denotes layer index (1 and 2). The c axis tunneling hamiltonian on the other hand will be^ 2 TkpcP°ck»*p kp,+ iCharacteristics afc axis and ab axis tunneling etc 179It is clear, that for c axis tunneling there is an extra channel for conduction , i.e from the nonbonding to bonding band which is absent for the ab axis tunneling. At this point, we compare our model with that of Levin and Quader[10], who also consider a split band picture. We insist that there is a major difference between our viewpoints as regards the role of the split bands. Levin et. al.[10] assume that the bonding band(V> band) is almost submerged below the Fermi surface. For the underdoped case, the ip hole band is completely filled and frozen much below the Fermi surface, and do not take part in tunneling to the metal on the other side of the junction. Consequently there will not be any conductivity asgymetry for underdoped case. For larger doping case, both the bands will be partially filled and take part in tunneling. Moreover one needs additional assumptions, that the ip band is actually a band of nondegenerate band of fermions since their number is so small. One needs to have, in an adhoc fashion, different dispersion for <p and ip fermions(linear and quadratic in momenta) to reproduce some normal state properties. This picture is approximately right when <1 is large, giving rise to large band splitting. We assume , on the other hand that the band splitting is small (small t 1 ). So, even at small doping concep tions , both bands will be partially filled.Within the interlayer tunneling mechanism of superconductivity, even though the intralayer BCS coupling gives a small Tc =  5K  on its own, a very small t± is enough to raise the Tc to large values =  90A' through the Josephson coupling term. For CIC junctions when both the electrodes are high Tc materails (break junctions), a look at the tunneling hamiltonians for the ab and c axis tunneling shows that, for ab tunneling, the elctrons tunnel from 0 to ^ and from ip to ip bands only. For the c axis tunneling, cross tunneling also takes place. If T^  And are the tuneling matrix elementsbetween the respective subbands of both electrodes, we getG M V )  =  \T „ \2N i[(l + a l)  -  a U ( ~ ~ )] andG.0 0  =  | 7 ^ |2jV*[(l +  « i +  2Q2) _  Q 2 /( Z * £ ^ )  _  (a2 +  a 2 ) / ( z 2 i £ + I ) ]where a \  =  N ^ T ^ / N ^ T ^  and a 2 =  N ^ T ^ / N ^ T ^ .  The main features of this expression are: (1) The conductance voltage characteristics is symmetric with respect to bias for both ab and c axis tunneling. (2) For c axis tunneling, there is a zero bias current coming from cross tunneling, which is operative even at zero bias because of finite band splitting. For ab axis tunneling there is no zero bias current. (3) There is a z^ro bias conductance for. both ab and c axis tunneling At T  =  0 and V =  0,180 P A Sreeram and Manas SardarG ( V )  =  T ^ A ^ (1  +  a j ) .  T h e  m ean  field h a m ilto n ia n  in th e  su p e rc o n d u c tin g  phase is, +  tj.)olaQka +  ~k k+ ( ^ + y ) ^ Z [ ( + &<t>k\,i>-kl) + 4 -"P ]  (2)kT h e  h a m ilton ian  looks like a su m  o f  tw o B C S reduced h a m ilto n in s  for th e  bon d ing  an j an tib on d in g  electron sy s tem s . T h e  g en era lised  gap  eq u a tio n  w ill be I1 1 tanh(ta 0 E *  / 2) 1 * t a n h(0 E\2 E l + £? / 2) \2 E *(3)where,E t ’v = v ^ k ± t ± r- +W e have solved the gap  e q u a tio n  n u m erica lly  for different tem p era tu res . In N1S junc tions, the tu n n e lin g  current is g i \e n  by,In i s  =  =  £  |T |2[ . ^ ( eV' +  E k -  *p) [ / ( E t ) -  / « , ) ]kp+ u|6(ev - f : fc- ^ ) [ i - / ( f : t ) - / « r ) l ]  (i)For the SIS ju n ctio n  the corresp o n d in g  exp ressio n  is,l s , s  =  ^ m 2[ ( l - / ( £ t ) - / ( E p))(t;,2Up26(el / - f ; p - £ t )kp-  u l ' l H c V  +  EP +  £ , ) )  +  ( f ( k )  -  f ( E p ))(uj:iip6(eV +  E k -  E P)-  v l i j 6 ( e V  +  E p -  E k))}Characteristics of c axis and ab axis tunneling etc 181The normalised conuctance versus voltage for the ab and c axis tunneling are plotted in Fig.l and Fig.2 respectively. The parameter values used in all the calculations are: Vbcs = 0.125 eV, <1 =  0.1 eV, d =  0.02 eV, chemical potential (/i) = 0.45 eV. This gives a Tc a  84° Kelvin.The notable features are:19cS§OoRVoltage (in eV)Fig.3 : Parellel (a-b axis) Current Charectensocs0 10 20 30 40 50 60 70Temperature (in Deg. K)Fig.4 : Zero Bias Conductance (ZBC) Characteristics(1) At T  = 0 there is a sharp voltage threshold for conductivity for the ab axis tunneling, whereas there is a finite zero bias conductance for the c axis tunneling. (2) The sharp voltage threshold for ab axis tunneling gets washed out at a very small temperature (4/v). (3) The plots for conductance at T  =  4,10 clearly shows the characteristic two peak structures seen in experiments. For d-wave superconductors also one gets similar two peak structures.In Fig.3 we plotted the current versus temperature for ab axis tunneling for 2 and 5 degree Kelvin. We emphasize that, even at very low temperatures ( 5 K) the current rises quadratically with voltage at very low voltages. Fig.4 shows the temperature dependence of the normalised zero bias conductance for both ab and c axis tunneling.3 C o n c lu s io nOne extraordinary feature of the interlayer tunneling gap function1 is that the gap along T -  M  direction is large and almost temperture independent upto about 90% of the Tc. On the other hand gap in any other direction falls very fast compared to the usual BCS gap*. This agrees with the recent photoemission experiment. This is true when Tj  > Vicj. For weaker Tj  or with larger in plane , the averaged gap falls faster with temperatures and slowly approach the usual BCS temperature dependence. All these182 P A Sreeram and Manas Sardarpeculiarities are only because of the one ■ k summation in the interlayer josephson coupling term, as emphasized by Anderson. Two things follow automatically from above discussion. One is that, in the interlayer mechanism, the gap magnitude in most part of the BZ is very low (1-3 meV) and also very fragile as far as thermal fluctuation is concerned. The gap in these regions falls faster than in the usual BCS gap. This would mean that we shall not get any sharp gap features at all at any finite temperatures in tunneling experiments. This is what is observed in our numerical calculations at finite temperatures. In Fig. 3. we show the I - V  characteristics at T  =  2 and T  =  5 degrees for tunneling along the ab plane. We See clearly that already at T  =  5 degrees there is finite current at very low voltages. In other words, indeed it will be very difficult to distinguish between, the situation where the gap function has gap nodes on the Fermi surface like in d-wave superconductors, and the interlayer case. For tunneling along, the c axis there will be a finite current for arbitrarily small voltages. So m ceramic materials, where we measure some average current along both directions, wc shall always get an I-V characteristics looking just like a superconductors with gap nodes on the Fermi surface even at T  = 0. For single crystal measurements , and for ab plane tunneling there will be a sharp voltage threshold, but no sharp threshold for small but finite temperatures. It is worth emphasizing that, we do not really know how impurities and inhomogeneities suppress the interlayer tunneling gap.R e fe re n c e s[1] P. C. Hammel et al, Phys. Rev. Lett. 63, 1992(1989)[2] S. L. Cooper et al, Phys. Rev. B39, 5920(1988)[3] D. R. Harshman et al, Phys. Rev. B39, 851(1989)[4] P. Chaudhury and S. Y. Lin, Phys. Rev. Lett. 72, 1084(1994)[5] Jian Ma et al, Cond-Mat 9494096, Submitted to PRL[6] M. Holezer et al, Phys. Rev. Lett. 67, 152,161(1991)[7] C. Zhow and H. J. Schulz, Phys. Rev. B45, 7397(1992)[8] S. Chakraborty et al, Science 261, 337(1993)[9] J. R. Kirtley, Int. Jour. Mod. Phys, B 4, 201(1990) and references therein.[10] G A. Levin and K. F. Quader, Phys. Rev. B 48, 16184(1993)

Characteristics of c Axis and ab Axis Tunneling in S-Wave Superconductors

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Characteristics of c Axis and ab Axis Tunneling in S-Wave Superconductors

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