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Microscopic barrier properties in electron-beam scribed YBCO Josephson junctions
Applied Superconductivity Vol. 5, Nos 7±12, pp. 277±284, 1998 0964-1807/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved
PII: S0964-1807(97)00057-4
MICROSCOPIC BARRIER PROPERTIES IN ELECTRONBEAM SCRIBED YBCO JOSEPHSON JUNCTIONS B. A. DAVIDSON*}, J. E. NORDMAN*, B. HINAUS$, M. RZCHOWSKI$, K. SIANGCHAEW% and M. LIBERA% *Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, U.S.A. $Department of Physics and Applied Superconductivity Center, University of Wisconsin, Madison, WI 53706, U.S.A. %Department of Materials Science and Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, U.S.A. AbstractÐWe present an analysis of the microscopic properties of the Josephson barrier in electronbeam junctions scribed at low temperatures in YBa2Cu3O7. These junctions behave as high-quality, uniform super-normal-superconductor (SNS) junctions which allows their characteristics to be compared in detail to well-established SNS theory. Combining this data with a Boltzmann transport model of the irradiated region, and treating the oxygen-sublattice defects as pair-breaking, allows quanti®cation of the interlayer's microscopic properties such as normal coherence length, quasiparticle mean-free path, and resistivity. We ®nd that the barrier exhibits properties of a dirty metal near the metal±insulator transition, with a Fermi surface area reduced an order of magnitude from that of the unirradiated ®lm. In addition, we analyze the limiting normal properties of irradiated YBCO, showing that the normal coherence length at the original transition temperature is constrained to less than twice the zerotemperature superconducting coherence length of the original ®lm. This analysis applies in general to weak-link structures in which the barrier is created from the electrode material through weakening by a pair-breaking mechanism. # 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION
It is well-established that electrons with energies between 60 and 0300 keV can selectively displace only the lightest atoms (i.e., oxygen) in the YBa2Cu3O7 ÿ d (YBCO) lattice, creating pointdefects on the oxygen sublattice in an otherwise undisturbed cation lattice [1, 2]. If irradiation of optimally-doped YBCO (O16.9) is performed at suciently low temperatures (E180 K) and suciently high ¯uences (e1021/cm2), the transition temperature Tc can be reduced to zero [1, 3, 4]. Electron irradiation with energies in the selective regime displaces oxygen on all lattice sites (chain, plane and apical) simultaneously, so that irradiation can both reduce the carrier concentration via chain- and plane-O vacancies [5, 6] and increase carrier scattering from in-plane defects [7, 8]. The latter eect has been shown to be an order of magnitude more eective at suppressing Tc than the former, and at high enough ¯uences both eects will be present. Suppression of Tc due to in-plane defects has been shown to be consistent with pair-breaking models [9] of nonmagnetic defects in an anistropic (d-wave) superconductor [8]. We have previously demonstrated that irradiation of YBCO6.9 at 140 K with a FEG electron source produces a completely normal (N) barrier (TcN=0) suciently con®ned to show SNS behavior for all temperatures below the electrode transition temperature of 90 K [4, 10]. At the lowest temperatures, excess current and hysteresis are observed in the current±voltage (IV) characteristics, suggestive of nonequilibrium eects [11]. Here we describe a transport analysis of our junctions interpreted in the context of a pair-breaking model applied to e-irradiated YBCO, allowing quanti®cation of the microscopic properties of these normal barriers. We show that the barrier exhibits properties of a dirty metal near its metal±insulator transition, a Fermi surface area reduced an order of magnitude from the unirradiated ®lm and a mean-free path reduced a factor of two to four. We determine the formal limits of these barrier properties, }Author to whom correspondence should be addressed. Now at Istituto di Cibernetica-CNR, via Toiano 6, 80072 Arco Felice, Italy. 277
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evaluating our present junctions in relation to these limits, and use these results to discuss the possible tailoring of the barrier properties to optimize IcRn or probe the d-wave nature of the YBCO electrodes. ANALYSES
A. Junction properties The transport analysis is based on the extraction of the relevant junction parameters from least-squares ®tting of the dynamic resistance Rd=dV/dI versus bias current I curves to the resistively-shunted junction model (RSJ) for temperatures T above 010 K (below which switching and hysteresis is observed). These parameters include the junction normal resistance Rn(T), the critical current Ic0(T) in the absence of thermal noise (the noise is characterized by g = 2EJ/ hIc/2e is the Josephson coupling energy), and any excess current Ixs(T). Fits for kBT where EJ= a typical junction are shown in Fig. 1. As reported elsewhere [11], a bias-independent excess current appears below 030 K and the temperature dependence of Ixs follows that of Vc=Ic0Rn, saturating at 00.7 Ic0. The typical temperature dependence of the normal resistance resulting from the ®ts is shown in Fig. 2. The small points represent the resistance at an ac bias of 1 mA, showing the electrode transition (90 K) and the temperature at which the Josephson coupling energy exceeds the thermal energy (065 K). The solid circles show the junction Rn taken from the RSJ ®ts. Above a temperature Tcrossover 1 30 K the resistance is characterized by metallic behavior which is ®t to a line, yielding a slope and a zero-temperature intercept. Below Tcrossover the junction resistance gradually increases.
Fig. 1. (a) Dynamic resistance Rd=dV/dI versus bias current I at various temperatures in the shortjunction regime (w/lJE4) for Y829L link ]]3 (w = 3.3 mm, ®lm thickness d = 50 nm). Open circles are the data, solid lines are the ®t to thermally-rounded RSJ model; (b) same for wide-junction regime (w/ lJ>4), including excess current.
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279
Fig. 2. Junction normal resistance Rn as a function of temperature for Y829L link ]]3. Solid circles are from ®ts to RSJ model, small dots at 1 mA ac bias. Crossover temperature Tcrossover and associated value of kFl is indicated, as well as linear ®t to Rn above Tcrossover.
Figure 3 show the critical voltage Ic0Rn of the junction versus temperature of the same junction as Figs 1 and 2. Included is a ®t to the de Gennes formula IcRn A D2i (L/xn(T))eÿL/xn(T) using both the clean (xn/l << 1, xn A T ÿ1) and dirty (xn/l >> 1, xn A T ÿ1/2) limit formulae for the normal coherence length xn in the interlayer (L is the electrode separation, and Di the order parameter at the SN interface). Both limits ®t the data well and yield a ratio L/xn(Tc)1 9±9.6 for this junction, reasonable since the junction is between these two limits in this temperature range (as discussed below). A BCS temperature dependence for Di is assumed in these ®ts. When rigid boundary conditions hold at the SN interface, no further temperature dependence of Di(T) is expected [12, 13]. This is consistent with the small value of gap suppression parameter p g= Nn rs =Ns rn for these junctions, which determines whether rigid (g << 1) or soft (g >> 1) boundary conditions apply (here Nn,s is the single-spin DOS in the barrier and electrodes). The barrier analysis given below shows rs << rn and Nn E Ns, implying g << 1. Values of L/xn(Tc) and Di(0)/kBTc for all junctions are given in Table 1. B. Barrier properties We utilize a Boltzmann approach in analyzing the metallic behavior of the interlayer, which is more general than the free-electron (Drude/Sommerfeld) model that assumes a spherical or cylindrical Fermi surface. As discussed below, in HTS near the metal±insulator transition the character of the Fermi surface can change drastically. For a more detailed description of the
Fig. 3. Critical voltage Ic0Rn as a function of temperature for the same link as Figs 1 and 2. Solid circles are from ®ts to RSJ model, lines are ®ts to de Gennes formula in both clean and dirty limits (with rigid boundary conditions) which give values for L/xn(Tc). Inset shows same on logarithmic scale.
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Table 1. Summary of calculations from 2-dimensional Boltzmann analysis for all links (Sections II and III.A). Rows 1± 4: L/xn(Tc) and Di(0)/kBTc from ®ts of Ic0Rn(T) to de Gennes theory; Row 5: te from linear ®t of Rn(T) in metallic regime; Rows 6 and 7: xn(Tc)/ l(Tc) and L/l(Tc) assuming ltr=l0tr=0.3; Rows 8 through 10: ratios of irradiated to unirradiated quantities, using n0F=1.4 107 cm/s and @r0/@T = 0.83 mO-cm/K; Rows 11±14: using r0=75 mO-cm Link parameter L xnc Tc L xnd Tc
825-2
825-3
825-4
829-2
829-3
829-4
846-1
846-3
9.0
9.7
12.1
11.0
9.0
10.8
10.2
11.3
9.9 0.36 0.66
10.3 0.56 0.77
11.6 0.23 0.47
11.4 0.37 0.71
9.6 0.34 0.60
12.0 0.43 0.86
10.8 0.45 0.69
12.0 0.55 0.85
r 0 te=r`T c xn Tc l Tc L l Tc
1.05 0.41
0.29 0.29
1.79 0.51
0.97 0.40
1.06 0.41
0.62 0.35
0.40 0.31
0.29 0.29
3.9
2.9
6.1
4.5
3.8
3.7
3.2
3.4
SF S0F l Tc l0 Tc r Tc r0 Tc
0.11
0.05
0.09
0.09
0.07
0.06
0.13
0.11
Di(0)/kBTc (clean) Di(0)/kBTc (dirty)
r(Tc) (mO/cm) L(nm) xn(Tc) (nm) l(Tc) (nm) Tcrossover (K)
SF lc0 4p2
0.24
0.39
0.18
0.25
0.24
0.31
0.36
0.39
38
53
62
46
57
51
22
23
2.9 5.9 0.63 1.5 25 1.6
4.0 7.0 0.70 2.4 40 1.3
4.6 6.8 0.57 1.1 19 0.9
3.4 7.1 0.64 1.6 27 1.4
4.3 5.8 0.63 1.5 34 1.0
3.8 7.0 0.67 1.9 34 1.3
1.6 7.3 0.69 2.2 25 3.8
1.7 8.2 0.70 2.4 27 3.8
Boltzmann treatment, see [14]. Thus, instead of using carrier concentration and eective mass, the resistivity is expressed (in two dimensions) as: r
16p2 RQ 16p2 RQ ÿ1 timp gT; SF F SFl
1
with SF is the Fermi surface area, RQ=h/4e2 1 6.45 kO is the resistance quantum, and the mean-free path l and the Fermi velocity nF are averaged over the Fermi surface [15]. The total scattering rate is the sum of the impurity scattering rate (denoted tÿ1 imp) and the T-linear term (denoted g T). There is still debate about the origin of the T-linear behavior, which persists in YBCO up to the ortho-tetragonal phase transition (1600 K) and which is seen in our irradiated barriers down to Tcrossover. We note that when such linearity is attributed to an electron±phonon scattering mechanism, g = (2pkB/h)ltr where ltr is a dimensionless `'transport'' coupling coecient that is estimated ltr 1 0.3 in optimally-doped YBCO [16]. It has been shown that ltr does not change under electron irradiation in the selective regime and Tc suppression under such irradiation can be analyzed within the pair-breaking model with a constant ltr [8]. It is convenient to de®ne for each barrier a dimensionless parameter te as the ratio of an ``eective temperature'' to Tc of the electrode: teff
Teff r 0 1 0 ; Tc r Tc gtimp Tc
2
where r' = @r/@T. Given ltr, te can be determined from the Rn(T) data for each barrier; typically te 1 0.5±1.5 in our high-¯uence (11022eÿ/cm2) barriers, as summarized in Table 1. Using (1) and (2), simple forms can be derived for the ratios of many of the important microscopic barrier properties. For example, using the standard de®nition for the clean-limit coherence length, the ratio of coherence length to mean-free path is xn/l = (1 + te)ltr A tÿ1 at Tc. ÿ2 ÿ2 The interpolation formula xÿ2 n =xnc +xnd then gives xn Tc 1 teff ltr p : l Tc 1 2 1 teff ltr
3
Typical values of xn(Tc)/ l(Tc) = 0.3±0.5 are observed in our samples, revealing that the barriers are near the clean limit at Tc but crossover to the dirty limit at lower temperatures. Using this ratio multiplied by L/xn(Tc) obtained in Section II.A allows a calculation of L/l(Tc)1 3±6,
Microscopic barrier properties in Josephson junctions
281
showing that the physical length is typically 3±6 mean-free paths long at Tc. These are summarized in Table 1. Using (1), one can calculate the ratio of the Fermi surface area SF in the irradiated material to that of the unirradiated material S0F in terms of L/l(Tc), the junction RnA product (at Tc), and the known properties of unirradiated YBCO such as resistivity slope @r0/@T 1 0.83 mO/cm (here the superscript ``naught'' indicates unirradiated material). For these barriers, this ratio SF/ S0F10.05±0.13; that is, the Fermi surface area has been reduced an order of magnitude from the unirradiated ®lm. This appears reasonable for YBCO that has been pushed to near the metal± insulator transition, as discussed below. Using the ratio of Fermi surface areas, the ratio of the mean-free paths before and after irradiation can be calculated (typically l/l010.2±0.4 at Tc), ®nally yielding the ratio of resistivities r/r0=S0Fl0/SFl120±55 at Tc for these barriers. Given these ratios, values for the barrier material's mean-free path l(Tc)1 1±2.5 nm, resistivity r(Tc)1 1.5±4 mO-cm, normal coherence length xn(Tc)1 0.6±0.7 nm, and physical length L 1 6±8 nm are estimated using known values for unirradiated YBCO r0(Tc)1 75 mO-cm and l0(Tc)16 nm [17]. DISCUSSION
A. Fermi surface area and the metal±insulator transition Optimal doping in HTS materials (de®ned as highest Tc) is found almost universally to occur at hole concentrations of 0.16 per Cu±O2 plane per unit cell away from half-®lling [18]. At this doping level the Fermi surface is large, roughly cylindrical and centered around (p, p) points in the Brillouin zone. Though the details of the Fermi surface evolution with doping are still under scrutiny [19±22], photoemission evidence suggests that as the insulting phase is approached, pockets form around (p/2, p/2) by which point the Fermi surface area is drastically reduced. This behavior is consistent with the above calculated reduction of Fermi surface area by an order of magnitude in the barrier. The usual criteria kFl 0 1 at the metal±insulator (MI) transition can be expressed in terms of SF as SFlc0/4p2 0 1, where c0 is the c-axis lattice constant. Using the values generated from the Boltzmann analysis gives SFlc0/4p2 11±4 for these eight barriers (01.1 for the junction of Figs 1 and 2), as shown in Table 1. This oers further evidence that the barriers are approaching such an MI transition at the crossover temperature. The metallic behavior of these barriers, crossing over into semiconducting behavior at a temperature 0 Tc/2 has been observed in underdoped HTS materials [23] with similar kFl. Large resistivities and near-unity kFl values of these barriers scribed at low temperatures suggest higher defect densities than comparable room-temperature irradiation [24]. B. Limits of normal properties of irradiated YBCO In conventional proximity-coupled junctions, IcRn is primarily determined through the exponential dependence on the reduced length L/xn, as well as the suppression of the gap at the SN interface: IcRn A D2i(L/xn)eÿL/xn. The discussion above shows that conventional SNS theory is sucient to describe the behavior of these e-beam junctions, without modi®cation for the possible d-wave symmetry of the order parameter in the electrodes. This is likely a result of the long barrier (L/xn(Tc) >> 1), as eects of d-wave symmetry are theoretically predicted to appear only at shorter reduced lengths [25, 26]. Consequently, to probe this eect the reduced length should be made as small as possible. In addition, applications of Josephson junctions almost universally bene®t from a high IcRn product to improve performance, speed or sensitivity. Since the physical extent of the irradiated region L 1 5 nm estimated above by the Boltzmann analysis is consistent with Monte Carlo simulations of spreading of a 200 keV electron beam in a 50 nm ®lm of YBCO on a LAO substrate, L is quite possibly already at the limits of the fabrication technology. Thus one is brought immediately to the issue of increasing the normal coherence length in the interlayer as a means of increasing IcRn. This raises a more general question: what is the limit of normal coherence length in a metal which is produced solely by local weakening of a single, homogenous superconductor? Electron irradiation weakens superconductivity and modi®es the normal state properties of YBCO through increased impurity scattering from in-plane defects
282
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(which act as pair-breaking) and a reduction of the Fermi surface area. (This is in contrast to the case where the normal layer is deposited separately from the superconducting electrodes, in which arti®cial interfaces occur and the normal layer can have radically dierent normal properties from those of the superconductor.) We show below for the case in which weakening occurs via a pair-breaking mechanism, xn at Tc is limited to less than 11.5 the superconducting coherence length xs(0) of the original superconductor. Lower limits for xn and l are determined by the unit cell size, below which a description in terms of metallic conduction is not appropriate and these quantities are no longer meaningful. (There may be more esoteric considerations involved which prevent an insulating transition, however, in the general case). The upper limit of the quasiparticle mean-free path in a weakened superconductor is naturally given by that of the original superconductor, as the introduction of pair-breaking defects can only serve to increase the impurity scattering. In order to place an upper limit on the normal coherence length, we ®rst analyze separately the clean (xn << l) and dirty (xn >> l) limits and then combine them using a formula valid for arbitrary l. In the clean limit, the ratio of the normal coherence length at Tc to the coherence length of the original superconductor at T = 0 is given by xnc Tc hF pD 0 F D 0 3 F 0 1 ; 2pkB Tc h0F xs 0 F 2kB TC 2 0F
4
with nF the Fermi velocity in the normal material, n0F that of the original superconductor, and assuming 2D(0)/kBTc 1 6 in optimally-doped YBCO [27]. This is the proper form for xs since the ÿ1 ÿ1 ÿ1 Pippard coherence length is given by xÿ1 s =x0 +l 1 x0 and l diverges at low temperatures [28]. Thus in the clean limit, xnc can be at most about twice xs, or lower if the Fermi velocity is reduced during the modi®cation. In the dirty limit, the ratio appears 1=2 xnd Tc hFl pD 0 F 3p D 0 1=2 : xs 0 2 2 h=t 4pkB Tc 0F h0F
5
We can identify h/t 0 a, the pair-breaking energy according to the Abrikosov±Gorkov theory [29±32] here evaluated at Tc. Application of the pair-breaking theory to irradiated HTS materials has been studied in detail [33, 34, 8]. For the case in which the material is converted completely normal (TcN=0), 2a/D(0) 1 1, and the last term in parenthesis in Equation (5) is p 3p=2 1 2.2. If the material retains a ®nite TcN (lower irradiation dose), 0<2a/D(0)<1, and the p term in parenthesis is 3p=2. Thus F xnd Tc F 3p D 0 1=2 0 2:2 0 : 6 xs 0 2 2a F F As discussed previously, in the general case the interpolation formula for the normal coherence length valid for arbitrary l can be used to express the ratio xn/xs(0). Then xn/xs(0) is a maximum when 2a/D(0) 4 0: " 2 #ÿ1=2 F F xn Tc 4 2a 0:21 0 1:5 0 : 7 xs 0 9 D F F Thus the upper limit on xn/xs(0) occurs when nF/n0F=1 and 2a/D(0) is small; in this case xn(Tc) can be a factor of 1.5 larger than xs(0). This might pertain, in the case of selective oxygen disordering of YBCO, at very light ¯uences and within a small temperature range near Tc. In general, when Tc is signi®cantly reduced (i.e., high ¯uences) xn(Tc) will be smaller than xs(0), consistent with the numerical results obtained in the analysis here. Equation (7) applies in the general case where the normal material is produced by modi®cation of the superconductor, and the modi®cation mechanism can only weaken the superconductivity by reducing the Fermi surface area and/or increasing pair-breaking.
Microscopic barrier properties in Josephson junctions
283
CONCLUSION
The results deduced from the Boltzmann analysis paint a consistent picture of the properties of the irradiated barrier material, and show that these high-¯uence barriers are near the metal± insulator transition with signi®cantly reduced Fermi surface area and increased resistivity. Combining this data with the discussion of limiting normal properties of irradiated YBCO allows a discussion of tailoring the microscopic properties of the normal barrier. For example, as the ¯uence is decreased from the high levels studied here, the mean-free path can be varied between l(Tc) 11.5 nm to that of the initial ®lm l0(Tc)1 6 nm. Similarly, the normal coherence length xn can be varied between 11.5 xs(0) 1 2.5 nm at light ¯uences to 10.6 nm as seen here. Increasing xn may decrease L/xn and yield higher IcRn products, and decreasing L/l may promote ballistic instead of diusive transport. This may also open the possibility of observing the eects of the d-wave order parameter of the electrodes on the SNS junction properties. It seems likely that suppression of the gap at the SN interface Di is ®xed primarily by the graded interface produced by the Gaussian beam pro®le, since the width of the graded region should be much less than xs in order to maximize de Gennes' extrapolation length b and hence maximize Di. Unless there exists some saturation or threshold eect for e-beam induced modi®cation, or unless recovery of damage by annealing has a dose dependence, it seems unlikely that the interface can be sharpened signi®cantly. Thus a reduced Di due to the graded nature of the interface may be an inherent limitation, and it seems reasonable that attempts to increase IcRn will likely be achieved through control of xn, l and L, and not Di. These improvements may also be more pronounced in HTS materials with intrinsically longer coherence lengths (BSCCO and TBCCO), since xn(Tc) is fundamentally limited by xs(0) in the original ®lm. In the opposite direction, higher ¯uences may push the barrier beyond the MI transition and allow the possibility of coherent tunneling across an insulating barrier. Observation of such will likely require more stringent control of the length L than is seen in the junctions analyzed here.
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24. We note that the athermal recombination model of defect accumulation under electron irradiation recently proposed to account for the saturation of Tc suppression to ®nite values under room-temperature irradiation [8] if interpreted as originally formulated (Ref. [26] of [8]) predicts a smaller defect concentration at lower irradiation temperatures. 25. H. X. Tang, Z. D. Wang and Jian-Xin Zhu, Phys. Rev. B 54, 12509 (1996). 26. YuS Barash, H. Burkhardt and D. Rainer, Phys. Rev. Lett. 77, 4070 (1996). 27. A. A. Abrikosov, Phys. Rev. B 55, R6149 (1997). 28. M. Tinkham, Introduction to Superconductivity, ®rst edition, Krieger, Florida, (1975). 29. A. A. Abrikosov and L. P. Gorkov, Sov. Phys. JETP 12, 1243 (1961). 20. P. de Gennes, Superconductivity of Metals and Alloys, Chapter 8, Addison-Wesley, U.S.A., (1989). 31. A. A. Abrikosov, Physica C 214, 107 (1993). 32. H. Kim, G. Preosti and P. Muzikar, Phys. Rev. B 49, 3544 (1994). 33. R. J. Radtke, K. Levin, H-B. Schuttler and M. R. Norman, Phys. Rev. B 48, 653 (1993). 34. E. M. Jackson, B. D. Weaver, G. P. Summers, P. Shapiro and E. A. Burke, Phys. Rev. Lett. 74, 3033 (1995).
PII: S0964-1807(97)00057-4
MICROSCOPIC BARRIER PROPERTIES IN ELECTRONBEAM SCRIBED YBCO JOSEPHSON JUNCTIONS B. A. DAVIDSON*}, J. E. NORDMAN*, B. HINAUS$, M. RZCHOWSKI$, K. SIANGCHAEW% and M. LIBERA% *Department of Electrical and Computer Engineering, University of Wisconsin, Madison, WI 53706, U.S.A. $Department of Physics and Applied Superconductivity Center, University of Wisconsin, Madison, WI 53706, U.S.A. %Department of Materials Science and Engineering, Stevens Institute of Technology, Hoboken, NJ 07030, U.S.A. AbstractÐWe present an analysis of the microscopic properties of the Josephson barrier in electronbeam junctions scribed at low temperatures in YBa2Cu3O7. These junctions behave as high-quality, uniform super-normal-superconductor (SNS) junctions which allows their characteristics to be compared in detail to well-established SNS theory. Combining this data with a Boltzmann transport model of the irradiated region, and treating the oxygen-sublattice defects as pair-breaking, allows quanti®cation of the interlayer's microscopic properties such as normal coherence length, quasiparticle mean-free path, and resistivity. We ®nd that the barrier exhibits properties of a dirty metal near the metal±insulator transition, with a Fermi surface area reduced an order of magnitude from that of the unirradiated ®lm. In addition, we analyze the limiting normal properties of irradiated YBCO, showing that the normal coherence length at the original transition temperature is constrained to less than twice the zerotemperature superconducting coherence length of the original ®lm. This analysis applies in general to weak-link structures in which the barrier is created from the electrode material through weakening by a pair-breaking mechanism. # 1998 Elsevier Science Ltd. All rights reserved
INTRODUCTION
It is well-established that electrons with energies between 60 and 0300 keV can selectively displace only the lightest atoms (i.e., oxygen) in the YBa2Cu3O7 ÿ d (YBCO) lattice, creating pointdefects on the oxygen sublattice in an otherwise undisturbed cation lattice [1, 2]. If irradiation of optimally-doped YBCO (O16.9) is performed at suciently low temperatures (E180 K) and suciently high ¯uences (e1021/cm2), the transition temperature Tc can be reduced to zero [1, 3, 4]. Electron irradiation with energies in the selective regime displaces oxygen on all lattice sites (chain, plane and apical) simultaneously, so that irradiation can both reduce the carrier concentration via chain- and plane-O vacancies [5, 6] and increase carrier scattering from in-plane defects [7, 8]. The latter eect has been shown to be an order of magnitude more eective at suppressing Tc than the former, and at high enough ¯uences both eects will be present. Suppression of Tc due to in-plane defects has been shown to be consistent with pair-breaking models [9] of nonmagnetic defects in an anistropic (d-wave) superconductor [8]. We have previously demonstrated that irradiation of YBCO6.9 at 140 K with a FEG electron source produces a completely normal (N) barrier (TcN=0) suciently con®ned to show SNS behavior for all temperatures below the electrode transition temperature of 90 K [4, 10]. At the lowest temperatures, excess current and hysteresis are observed in the current±voltage (IV) characteristics, suggestive of nonequilibrium eects [11]. Here we describe a transport analysis of our junctions interpreted in the context of a pair-breaking model applied to e-irradiated YBCO, allowing quanti®cation of the microscopic properties of these normal barriers. We show that the barrier exhibits properties of a dirty metal near its metal±insulator transition, a Fermi surface area reduced an order of magnitude from the unirradiated ®lm and a mean-free path reduced a factor of two to four. We determine the formal limits of these barrier properties, }Author to whom correspondence should be addressed. Now at Istituto di Cibernetica-CNR, via Toiano 6, 80072 Arco Felice, Italy. 277
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evaluating our present junctions in relation to these limits, and use these results to discuss the possible tailoring of the barrier properties to optimize IcRn or probe the d-wave nature of the YBCO electrodes. ANALYSES
A. Junction properties The transport analysis is based on the extraction of the relevant junction parameters from least-squares ®tting of the dynamic resistance Rd=dV/dI versus bias current I curves to the resistively-shunted junction model (RSJ) for temperatures T above 010 K (below which switching and hysteresis is observed). These parameters include the junction normal resistance Rn(T), the critical current Ic0(T) in the absence of thermal noise (the noise is characterized by g = 2EJ/ hIc/2e is the Josephson coupling energy), and any excess current Ixs(T). Fits for kBT where EJ= a typical junction are shown in Fig. 1. As reported elsewhere [11], a bias-independent excess current appears below 030 K and the temperature dependence of Ixs follows that of Vc=Ic0Rn, saturating at 00.7 Ic0. The typical temperature dependence of the normal resistance resulting from the ®ts is shown in Fig. 2. The small points represent the resistance at an ac bias of 1 mA, showing the electrode transition (90 K) and the temperature at which the Josephson coupling energy exceeds the thermal energy (065 K). The solid circles show the junction Rn taken from the RSJ ®ts. Above a temperature Tcrossover 1 30 K the resistance is characterized by metallic behavior which is ®t to a line, yielding a slope and a zero-temperature intercept. Below Tcrossover the junction resistance gradually increases.
Fig. 1. (a) Dynamic resistance Rd=dV/dI versus bias current I at various temperatures in the shortjunction regime (w/lJE4) for Y829L link ]]3 (w = 3.3 mm, ®lm thickness d = 50 nm). Open circles are the data, solid lines are the ®t to thermally-rounded RSJ model; (b) same for wide-junction regime (w/ lJ>4), including excess current.
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Fig. 2. Junction normal resistance Rn as a function of temperature for Y829L link ]]3. Solid circles are from ®ts to RSJ model, small dots at 1 mA ac bias. Crossover temperature Tcrossover and associated value of kFl is indicated, as well as linear ®t to Rn above Tcrossover.
Figure 3 show the critical voltage Ic0Rn of the junction versus temperature of the same junction as Figs 1 and 2. Included is a ®t to the de Gennes formula IcRn A D2i (L/xn(T))eÿL/xn(T) using both the clean (xn/l << 1, xn A T ÿ1) and dirty (xn/l >> 1, xn A T ÿ1/2) limit formulae for the normal coherence length xn in the interlayer (L is the electrode separation, and Di the order parameter at the SN interface). Both limits ®t the data well and yield a ratio L/xn(Tc)1 9±9.6 for this junction, reasonable since the junction is between these two limits in this temperature range (as discussed below). A BCS temperature dependence for Di is assumed in these ®ts. When rigid boundary conditions hold at the SN interface, no further temperature dependence of Di(T) is expected [12, 13]. This is consistent with the small value of gap suppression parameter p g= Nn rs =Ns rn for these junctions, which determines whether rigid (g << 1) or soft (g >> 1) boundary conditions apply (here Nn,s is the single-spin DOS in the barrier and electrodes). The barrier analysis given below shows rs << rn and Nn E Ns, implying g << 1. Values of L/xn(Tc) and Di(0)/kBTc for all junctions are given in Table 1. B. Barrier properties We utilize a Boltzmann approach in analyzing the metallic behavior of the interlayer, which is more general than the free-electron (Drude/Sommerfeld) model that assumes a spherical or cylindrical Fermi surface. As discussed below, in HTS near the metal±insulator transition the character of the Fermi surface can change drastically. For a more detailed description of the
Fig. 3. Critical voltage Ic0Rn as a function of temperature for the same link as Figs 1 and 2. Solid circles are from ®ts to RSJ model, lines are ®ts to de Gennes formula in both clean and dirty limits (with rigid boundary conditions) which give values for L/xn(Tc). Inset shows same on logarithmic scale.
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Table 1. Summary of calculations from 2-dimensional Boltzmann analysis for all links (Sections II and III.A). Rows 1± 4: L/xn(Tc) and Di(0)/kBTc from ®ts of Ic0Rn(T) to de Gennes theory; Row 5: te from linear ®t of Rn(T) in metallic regime; Rows 6 and 7: xn(Tc)/ l(Tc) and L/l(Tc) assuming ltr=l0tr=0.3; Rows 8 through 10: ratios of irradiated to unirradiated quantities, using n0F=1.4 107 cm/s and @r0/@T = 0.83 mO-cm/K; Rows 11±14: using r0=75 mO-cm Link parameter L xnc Tc L xnd Tc
825-2
825-3
825-4
829-2
829-3
829-4
846-1
846-3
9.0
9.7
12.1
11.0
9.0
10.8
10.2
11.3
9.9 0.36 0.66
10.3 0.56 0.77
11.6 0.23 0.47
11.4 0.37 0.71
9.6 0.34 0.60
12.0 0.43 0.86
10.8 0.45 0.69
12.0 0.55 0.85
r 0 te=r`T c xn Tc l Tc L l Tc
1.05 0.41
0.29 0.29
1.79 0.51
0.97 0.40
1.06 0.41
0.62 0.35
0.40 0.31
0.29 0.29
3.9
2.9
6.1
4.5
3.8
3.7
3.2
3.4
SF S0F l Tc l0 Tc r Tc r0 Tc
0.11
0.05
0.09
0.09
0.07
0.06
0.13
0.11
Di(0)/kBTc (clean) Di(0)/kBTc (dirty)
r(Tc) (mO/cm) L(nm) xn(Tc) (nm) l(Tc) (nm) Tcrossover (K)
SF lc0 4p2
0.24
0.39
0.18
0.25
0.24
0.31
0.36
0.39
38
53
62
46
57
51
22
23
2.9 5.9 0.63 1.5 25 1.6
4.0 7.0 0.70 2.4 40 1.3
4.6 6.8 0.57 1.1 19 0.9
3.4 7.1 0.64 1.6 27 1.4
4.3 5.8 0.63 1.5 34 1.0
3.8 7.0 0.67 1.9 34 1.3
1.6 7.3 0.69 2.2 25 3.8
1.7 8.2 0.70 2.4 27 3.8
Boltzmann treatment, see [14]. Thus, instead of using carrier concentration and eective mass, the resistivity is expressed (in two dimensions) as: r
16p2 RQ 16p2 RQ ÿ1 timp gT; SF F SFl
1
with SF is the Fermi surface area, RQ=h/4e2 1 6.45 kO is the resistance quantum, and the mean-free path l and the Fermi velocity nF are averaged over the Fermi surface [15]. The total scattering rate is the sum of the impurity scattering rate (denoted tÿ1 imp) and the T-linear term (denoted g T). There is still debate about the origin of the T-linear behavior, which persists in YBCO up to the ortho-tetragonal phase transition (1600 K) and which is seen in our irradiated barriers down to Tcrossover. We note that when such linearity is attributed to an electron±phonon scattering mechanism, g = (2pkB/h)ltr where ltr is a dimensionless `'transport'' coupling coecient that is estimated ltr 1 0.3 in optimally-doped YBCO [16]. It has been shown that ltr does not change under electron irradiation in the selective regime and Tc suppression under such irradiation can be analyzed within the pair-breaking model with a constant ltr [8]. It is convenient to de®ne for each barrier a dimensionless parameter te as the ratio of an ``eective temperature'' to Tc of the electrode: teff
Teff r 0 1 0 ; Tc r Tc gtimp Tc
2
where r' = @r/@T. Given ltr, te can be determined from the Rn(T) data for each barrier; typically te 1 0.5±1.5 in our high-¯uence (11022eÿ/cm2) barriers, as summarized in Table 1. Using (1) and (2), simple forms can be derived for the ratios of many of the important microscopic barrier properties. For example, using the standard de®nition for the clean-limit coherence length, the ratio of coherence length to mean-free path is xn/l = (1 + te)ltr A tÿ1 at Tc. ÿ2 ÿ2 The interpolation formula xÿ2 n =xnc +xnd then gives xn Tc 1 teff ltr p : l Tc 1 2 1 teff ltr
3
Typical values of xn(Tc)/ l(Tc) = 0.3±0.5 are observed in our samples, revealing that the barriers are near the clean limit at Tc but crossover to the dirty limit at lower temperatures. Using this ratio multiplied by L/xn(Tc) obtained in Section II.A allows a calculation of L/l(Tc)1 3±6,
Microscopic barrier properties in Josephson junctions
281
showing that the physical length is typically 3±6 mean-free paths long at Tc. These are summarized in Table 1. Using (1), one can calculate the ratio of the Fermi surface area SF in the irradiated material to that of the unirradiated material S0F in terms of L/l(Tc), the junction RnA product (at Tc), and the known properties of unirradiated YBCO such as resistivity slope @r0/@T 1 0.83 mO/cm (here the superscript ``naught'' indicates unirradiated material). For these barriers, this ratio SF/ S0F10.05±0.13; that is, the Fermi surface area has been reduced an order of magnitude from the unirradiated ®lm. This appears reasonable for YBCO that has been pushed to near the metal± insulator transition, as discussed below. Using the ratio of Fermi surface areas, the ratio of the mean-free paths before and after irradiation can be calculated (typically l/l010.2±0.4 at Tc), ®nally yielding the ratio of resistivities r/r0=S0Fl0/SFl120±55 at Tc for these barriers. Given these ratios, values for the barrier material's mean-free path l(Tc)1 1±2.5 nm, resistivity r(Tc)1 1.5±4 mO-cm, normal coherence length xn(Tc)1 0.6±0.7 nm, and physical length L 1 6±8 nm are estimated using known values for unirradiated YBCO r0(Tc)1 75 mO-cm and l0(Tc)16 nm [17]. DISCUSSION
A. Fermi surface area and the metal±insulator transition Optimal doping in HTS materials (de®ned as highest Tc) is found almost universally to occur at hole concentrations of 0.16 per Cu±O2 plane per unit cell away from half-®lling [18]. At this doping level the Fermi surface is large, roughly cylindrical and centered around (p, p) points in the Brillouin zone. Though the details of the Fermi surface evolution with doping are still under scrutiny [19±22], photoemission evidence suggests that as the insulting phase is approached, pockets form around (p/2, p/2) by which point the Fermi surface area is drastically reduced. This behavior is consistent with the above calculated reduction of Fermi surface area by an order of magnitude in the barrier. The usual criteria kFl 0 1 at the metal±insulator (MI) transition can be expressed in terms of SF as SFlc0/4p2 0 1, where c0 is the c-axis lattice constant. Using the values generated from the Boltzmann analysis gives SFlc0/4p2 11±4 for these eight barriers (01.1 for the junction of Figs 1 and 2), as shown in Table 1. This oers further evidence that the barriers are approaching such an MI transition at the crossover temperature. The metallic behavior of these barriers, crossing over into semiconducting behavior at a temperature 0 Tc/2 has been observed in underdoped HTS materials [23] with similar kFl. Large resistivities and near-unity kFl values of these barriers scribed at low temperatures suggest higher defect densities than comparable room-temperature irradiation [24]. B. Limits of normal properties of irradiated YBCO In conventional proximity-coupled junctions, IcRn is primarily determined through the exponential dependence on the reduced length L/xn, as well as the suppression of the gap at the SN interface: IcRn A D2i(L/xn)eÿL/xn. The discussion above shows that conventional SNS theory is sucient to describe the behavior of these e-beam junctions, without modi®cation for the possible d-wave symmetry of the order parameter in the electrodes. This is likely a result of the long barrier (L/xn(Tc) >> 1), as eects of d-wave symmetry are theoretically predicted to appear only at shorter reduced lengths [25, 26]. Consequently, to probe this eect the reduced length should be made as small as possible. In addition, applications of Josephson junctions almost universally bene®t from a high IcRn product to improve performance, speed or sensitivity. Since the physical extent of the irradiated region L 1 5 nm estimated above by the Boltzmann analysis is consistent with Monte Carlo simulations of spreading of a 200 keV electron beam in a 50 nm ®lm of YBCO on a LAO substrate, L is quite possibly already at the limits of the fabrication technology. Thus one is brought immediately to the issue of increasing the normal coherence length in the interlayer as a means of increasing IcRn. This raises a more general question: what is the limit of normal coherence length in a metal which is produced solely by local weakening of a single, homogenous superconductor? Electron irradiation weakens superconductivity and modi®es the normal state properties of YBCO through increased impurity scattering from in-plane defects
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(which act as pair-breaking) and a reduction of the Fermi surface area. (This is in contrast to the case where the normal layer is deposited separately from the superconducting electrodes, in which arti®cial interfaces occur and the normal layer can have radically dierent normal properties from those of the superconductor.) We show below for the case in which weakening occurs via a pair-breaking mechanism, xn at Tc is limited to less than 11.5 the superconducting coherence length xs(0) of the original superconductor. Lower limits for xn and l are determined by the unit cell size, below which a description in terms of metallic conduction is not appropriate and these quantities are no longer meaningful. (There may be more esoteric considerations involved which prevent an insulating transition, however, in the general case). The upper limit of the quasiparticle mean-free path in a weakened superconductor is naturally given by that of the original superconductor, as the introduction of pair-breaking defects can only serve to increase the impurity scattering. In order to place an upper limit on the normal coherence length, we ®rst analyze separately the clean (xn << l) and dirty (xn >> l) limits and then combine them using a formula valid for arbitrary l. In the clean limit, the ratio of the normal coherence length at Tc to the coherence length of the original superconductor at T = 0 is given by xnc Tc hF pD 0 F D 0 3 F 0 1 ; 2pkB Tc h0F xs 0 F 2kB TC 2 0F
4
with nF the Fermi velocity in the normal material, n0F that of the original superconductor, and assuming 2D(0)/kBTc 1 6 in optimally-doped YBCO [27]. This is the proper form for xs since the ÿ1 ÿ1 ÿ1 Pippard coherence length is given by xÿ1 s =x0 +l 1 x0 and l diverges at low temperatures [28]. Thus in the clean limit, xnc can be at most about twice xs, or lower if the Fermi velocity is reduced during the modi®cation. In the dirty limit, the ratio appears 1=2 xnd Tc hFl pD 0 F 3p D 0 1=2 : xs 0 2 2 h=t 4pkB Tc 0F h0F
5
We can identify h/t 0 a, the pair-breaking energy according to the Abrikosov±Gorkov theory [29±32] here evaluated at Tc. Application of the pair-breaking theory to irradiated HTS materials has been studied in detail [33, 34, 8]. For the case in which the material is converted completely normal (TcN=0), 2a/D(0) 1 1, and the last term in parenthesis in Equation (5) is p 3p=2 1 2.2. If the material retains a ®nite TcN (lower irradiation dose), 0<2a/D(0)<1, and the p term in parenthesis is 3p=2. Thus F xnd Tc F 3p D 0 1=2 0 2:2 0 : 6 xs 0 2 2a F F As discussed previously, in the general case the interpolation formula for the normal coherence length valid for arbitrary l can be used to express the ratio xn/xs(0). Then xn/xs(0) is a maximum when 2a/D(0) 4 0: " 2 #ÿ1=2 F F xn Tc 4 2a 0:21 0 1:5 0 : 7 xs 0 9 D F F Thus the upper limit on xn/xs(0) occurs when nF/n0F=1 and 2a/D(0) is small; in this case xn(Tc) can be a factor of 1.5 larger than xs(0). This might pertain, in the case of selective oxygen disordering of YBCO, at very light ¯uences and within a small temperature range near Tc. In general, when Tc is signi®cantly reduced (i.e., high ¯uences) xn(Tc) will be smaller than xs(0), consistent with the numerical results obtained in the analysis here. Equation (7) applies in the general case where the normal material is produced by modi®cation of the superconductor, and the modi®cation mechanism can only weaken the superconductivity by reducing the Fermi surface area and/or increasing pair-breaking.
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283
CONCLUSION
The results deduced from the Boltzmann analysis paint a consistent picture of the properties of the irradiated barrier material, and show that these high-¯uence barriers are near the metal± insulator transition with signi®cantly reduced Fermi surface area and increased resistivity. Combining this data with the discussion of limiting normal properties of irradiated YBCO allows a discussion of tailoring the microscopic properties of the normal barrier. For example, as the ¯uence is decreased from the high levels studied here, the mean-free path can be varied between l(Tc) 11.5 nm to that of the initial ®lm l0(Tc)1 6 nm. Similarly, the normal coherence length xn can be varied between 11.5 xs(0) 1 2.5 nm at light ¯uences to 10.6 nm as seen here. Increasing xn may decrease L/xn and yield higher IcRn products, and decreasing L/l may promote ballistic instead of diusive transport. This may also open the possibility of observing the eects of the d-wave order parameter of the electrodes on the SNS junction properties. It seems likely that suppression of the gap at the SN interface Di is ®xed primarily by the graded interface produced by the Gaussian beam pro®le, since the width of the graded region should be much less than xs in order to maximize de Gennes' extrapolation length b and hence maximize Di. Unless there exists some saturation or threshold eect for e-beam induced modi®cation, or unless recovery of damage by annealing has a dose dependence, it seems unlikely that the interface can be sharpened signi®cantly. Thus a reduced Di due to the graded nature of the interface may be an inherent limitation, and it seems reasonable that attempts to increase IcRn will likely be achieved through control of xn, l and L, and not Di. These improvements may also be more pronounced in HTS materials with intrinsically longer coherence lengths (BSCCO and TBCCO), since xn(Tc) is fundamentally limited by xs(0) in the original ®lm. In the opposite direction, higher ¯uences may push the barrier beyond the MI transition and allow the possibility of coherent tunneling across an insulating barrier. Observation of such will likely require more stringent control of the length L than is seen in the junctions analyzed here.
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