Surface states and tunneling spectroscopy of high-Tcsuperconductors

Superlattices and Microstructures, Vol. 25, No. 5/6, 1999 Article No. spmi.1999.0717 Available online at http://www.idealibrary.com on

Surface states and tunneling spectroscopy of high-Tc superconductors S. K ASHIWAYA Electrotechnical Laboratory, Umezono, Tsukuba, Ibaraki 305-8568, Japan Ginzton Laboratory, Stanford University, Stanford, CA 94305-4085, U.S.A. Y. TANAKA Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan (Received 24 March 1999) Recent studies of high-Tc superconductors have clarified new aspects of tunneling spectroscopy. The unconventional pairing states, i.e. d-wave symmetry in these materials have been established through various measurements. Differently from isotropic s-wave superconductors, d-wave pairing states have an internal phase of the pair potential. The internal phase modifies the surface states due to the interference effect of the quasiparticles. Along these lines, a novel formula of tunneling spectroscopy has been presented that fully takes into account of the anisotropy of the pair potential. The most essential difference of this formula from conventional ones is that it suggests the phase-sensitive capability of tunneling spectroscopy. The formula suggests that the symmetry of the pair potential is determined by the orientational dependence measurements of tunneling spectroscopy. Along these lines, several experiments have been performed on high-Tc superconductors. The observation of the zero-bias conductance peaks (ZBCP) on YBa2 Cu3 O7−δ strongly suggests the dx 2 −y 2 -wave pairing states of hole-doped high-Tc superconductors. On the other hand, the absence of ZBCP on (electron-doped) Nd1.85 Ce0.15 CuO4−δ indicates that the pair potential of this material is a nodeless state. In this paper, recent developments of tunneling spectroscopy for anisotropic superconductors are reviewed both on theoretical and experimental aspects. c 1999 Academic Press

Key words: tunneling spectroscopy, pairing symmetry zero-bias peak, surface states.

1. Introduction High capabilities of tunneling spectroscopy for studies of superconductors have been established through a large number of theories and experiments [1, 2]. The wide applicability of this method is based on the equivalency of tunneling conductance spectra with the density of states (DOS) of samples at the low temperature and at the high barrier (tunneling) limit. On the other hand, the unconventional pairing states of d-wave symmetry in high-Tc superconductors have been established through various measurements including several phase-sensitive tests [3, 4]. As for tunneling spectroscopy, a large number of trials have been made to detect the pairing symmetries of these materials. Unfortunately, many of them have failed except for V-shaped gap observations on c-axis oriented surfaces [5–7]. Otherwise, i.e. when the tunneling direction is in the ab-plane, not only gap structures but also zero-bias conductance peaks (ZBCP) have been widely 0749–6036/99/051099 + 16 $30.00/0

c 1999 Academic Press

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A

H δ(x) Energy 1(θ)

Electron injection E

x B

Insulator ELQ Normal reflection

θS

b

a

Andreev reflection θN

HLQ

Injected electron Normal metal

dx2–y2– wave superconductor

Anisotropic superconductor

Fig. 1. A, Schematic illustration of the tunneling junction. For the pair potential, Heviside step functional form is assumed. B, Schematic illustration of the elastic scattering at the surface. In the figure, the anisotropic pair potential of dx 2 −y 2 -wave symmetry is shown. Due to the translational symmetry, momentum parallel to the interface is conserved.

reported [8–10]. Since these results cannot be understood in terms of the classical tunneling theory on superconductors, the origin of the ZBCP has long been attributed to some extrinsic effects, such as the spin-flip tunneling via magnetic impurities [11]. An important clue to reveal the origin of the ZBCP has been presented by Hu who discussed the boundary condition on dx y -wave superconductor surfaces (equivalent to dx 2 −y 2 -wave with π/4 misorientation) [12]. On these surfaces, injected and reflected quasiparticles feel different sign of the pair potentials to each other due to the sign change of the internal pair potential. The sign change results in a interference effect of quasiparticle states, and the zero-energy states (ZES) are induced at the surface. The effect of the pair-potential phase on quasiparticle states is well understood in terms of the quantized bound states (so called Andreev bound states) in potential wells [13]. By developing this idea, a novel theory of tunneling spectroscopy is constructed which correctly takes into account of the anisotropy of the pair potential [14–17]. The essential difference of these theories from conventional ones is that they claim the phase-sensitive capability of tunneling spectroscopy. It has also been verified that conductance spectrum converges to surface DOS at the tunneling limit [16]. Motivated by these works, microscopic theories of the surface bound states have also been presented using Green’s function methods [18, 20, 21]. Detailed calculations have pointed out several possibilities of additional effects occurring on surfaces of anisotropic superconductors; they are the inducement or smearing of the ZES due to the roughness [18, 22–24], and the spatial dependence of the pair potential including the spatial variation of the pairing symmetry [18, 19, 25–28]. Trials to check the validity of above concepts have been accomplished by scanning tunneling spectroscopy (STS), and thin film junctions. The observation of ZBCP on hole-doped superconductors have been reported

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in a large number of papers (see references in Ref. [29]). Recent detailed studies on YBa2 Cu3 O7−δ (YBCO) thin films show that the appearance of ZBCP is consistent with dx 2 −y 2 -wave symmetry of the pair potential [30, 31]. On the other hand, tunneling spectroscopy on electron-doped Nd1.85 Ce0.15 CuO4−δ (NCCO) shows the absence of ZBCP for (110), (100) and (001) orientations, which is consistent with a nodeless pairing state [32, 33]. In this paper, recent studies of tunneling spectroscopy and surface states on high-Tc superconductors are reviewed. In Section 2, the theoretical aspects of tunneling spectroscopy are explained. We will present the formulation of tunneling spectroscopy and the relation between tunneling conductance spectrum and the surface DOS. Several possible effects predicted to occur at the surface of high-Tc superconductors are also presented. In Section 3, experimental results on tunneling spectroscopy are given for YBCO and NCCO. The consistency between theory and experiments are discussed. Finally, we will conclude in Section 4.

2. Theory of tunneling spectroscopy Let us start from the tunneling spectroscopy formula in the case of s-wave superconductors. There have been presented mainly two different formulation of the tunneling conductance spectrum. One is based on the transfer Hamiltonian method [34] which applies a time-dependent perturbation method to calculate the conductance spectrum. When the bias voltage of V is applied to the tunneling junction, tunneling current I is given by Z ∞ I (eV ) ∝ d E ρL (E)ρR (E + eV )[ f (E) − f (E + eV )], (2.1) −∞

where ρL (E), ρR (E), f (E) are DOS of left and right electrodes, and the Fermi distribution function, respectively. In the case of a normal/insulator/superconductor (NIS) junction, ρL (E) is assumed to be flat, then the conductance spectrum reduces to   Z ∞ −∂ σR (eV ) ∝ d EρBCS (E) f (E + eV ) , (2.2) ∂eV −∞ where ρBCS (E) is the BCS DOS with pair potential amplitude of 1s given by " # |E| ρBCS (E) = Re p . E 2 − |1s |2

(2.3)

This relation is only valid for the high-barrier limit (tunneling limit) because it includes only the lowest perturbation term. Therefore, this formula failed to explain the origin of excess current in low-resistance junctions. Later on, another formulation has been presented based on the scattering theory which is valid for arbitrary barrier height case (referred to as BTK formula) [35]. The essential concept of this method is quite similar to Landauer formula, in the sense that the conductance is expressed in terms of the reflection amplitudes for the electron injection [36]. The extension made by BTK is the inclusion of the energy-dependent probability amplitude for the Andreev reflection processes. Since the Andreev reflection process corresponds to the second-order tunneling term, its probability amplitude is suppressed as the tunneling probability increases. At the tunneling limit, the BTK formula gives the equivalent result with that by the transfer Hamiltonian method. Next we move to anisotropic superconductor cases. In order to correctly introduce the anisotropy of the pair potential, the BTK formula is extended to two-dimensional case [14, 15]. The model used in this formulation is shown in Fig. 1A. Even if we assume three-dimensional cases or the tunneling barrier with a finite thickness, the formulation does not essentially change (see [16]). A spin-singlet superconductor with a cylindrical Fermi-surface (quasi-2D) is assumed in the following discussion. For the c-axis direction, a slight modulation of the Fermi surface is assumed, which allows the electron conduction for this direction. The superconductor has an anisotropic pair potential in k-space represented by 1(k). We neglect the spatial

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Normalized conductance

σT (eV) 3

2

(110) (001)

1 (100)

0 0

2

1 Normalized energy, eV/1d

tip (001)

a

tip

Cu–O plane b

(100) dx 2–y2 – wave superconductor

(110) tip

Fig. 2. Calculated results of dx 2 −y 2 -wave superconductors for (100), (110) and (001) orientations. A zero-bias conductance peak is expected for (110) orientation.

dependence of the pair potential for simplicity (for spatially varying pair potential cases, see Ref. [29]). To deal with the tunneling junctions formed between normal metals and high-Tc superconductors, the Fermi wavenumber in the normal metal kFN and the superconductor kFS are assumed to be different. The ratio κr is defined as κr ≡ kFS /kFN . The momentum conservation law for the y-direction is kFS sin θS = kFN sin θN , with −π/2 < θS < π/2 {see Fig. 1B}. We assume that quasiparticles are injected from normal side with angle θN to the interface. Taking account of the fact that electron-like quasiparticles (ELQ) and hole-like quasiparticles (HLQ) feel different pair potentials to each other (represented by 1+ and 1− , respectively), the tunneling current is calculated from Z ∞ Z −θc I (eV ) ∝ dE dθN [1 + |a(E, θN )|2 − |b(E, θN )|2 ][ f (E + eV ) − f (E)], (2.4) −∞

−θc

where |a(E, θN )|2 and |b(E, θN )|2 represent Andreev reflection and normal reflection amplitudes, respectively. When κr ≤ 1, θC is π/2, otherwise θC is sin−1 (1/κr ) Using the conductance when the whole junction is in the normal states represented by σN the normalized conductance σT (eV ) is given by h i R∞ R θc ∂ f (E+eV ) d E dθ σ cos θ σ (E, θ ) − N N N R N −∞ −θc ∂eV , (2.5) σT (eV ) = R π/2 −π/2 dθN σN cos θN σR (E, θN ) =

1 + σN |0+ |2 + (σN − 1)|0+ 0− |2 , |1 + (σN − 1)0+ 0− exp(iφ− − iφ+ )|2

(2.6)

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Normalized conductance

2.0 1.5 1.0 0.5 0.0

STS of YBCO (110)

32

96

Displacement (nm)

64

128

160 –80 –60 –40 –20

0

20

40

60

80

8 6

2.5 2.0 1.5 1.0 0.5 0.0

Peak height

4 2

Tip trace

0 0

50

100

Peak height

Tip trace (nm)

Bias voltage (mV)

150

Displacement (nm) Fig. 3. Spatially continuous observation of the ZBCP on YBCO (110) by STM/STS. In the lower part of the figure, tip traces before and after spectroscopy and spatial variation of the peak heights are shown. Similar STS data have been reproducibly observed at most of the area on high-quality YBCO (110) films.

φ± = arg[1± ], λr = κr

cos θS , cos θN

E − ± , |1± | Z Z0 = , cos θN

0± =

4λr , (1 + λr )2 + 4Z 02 mH Z= 2 . ~ kFN σN =

∗ . The function The conductance σR (E, θN ) for negative E(E < 0) is calculated by replacing 0± with −1/ 0± σN cos θN gives the probability distribution p(θS ) of tunneling electrons. In the above formulation, we have used a property that σN is symmetric to θN . It is important to note that this formula includes not only the amplitudes but also the phases of the pair potentials. This fact indicates that the tunneling spectrum is phasesensitive. The effect of the phase has long been overlooked, because most theories were based on isotropic s-wave superconductors, and failed to include two pair potentials, i.e. 1+ and 1− .

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3

ZBCP 2

Surface

Normalized conductance

E

SBS 1(x) ZES

x

1 Second peak 0 0

1

2

Normalized voltage, eV/ 1d Fig. 4. Theoretical fitting of tunneling spectra on YBCO (110). The solid line represents experimental data, and the dotted line represents theoretical curve which assumes dx 2 −y 2 -wave symmetry, 1d = 16.5 meV, and 0G = 0.151d . The small peak at eV ≈ 1d corresponds to the SBS. The inset shows a schematic diagram on the concept of the second bound state. The observation of this state reflects that the pair potential amplitude is suppressed near (110) surface. Actually, this peak does not appear in theoretical curve of Fig. 2 which assumes spatially constant pair potential.

The most important situation in actual experiments is the tunneling limit. In this limit, we can easily verify that σR (E, θN ) is reduced to surface DOS of isolated superconductors. We will define a function ρκ (E, θ), ρκ (E, θ ) =

|1+ 1− | + (E − + )(E − − ) exp(iφ− − iφ+ ) . |1+ 1− | − (E − + )(E − − ) exp(iφ− − iφ+ )

(2.7)

The real part of ρκ (E, θ ) gives the angle resolved local DOS at the surface. The relation between conductance spectrum and DOS is given by lim σR (eV , θN ) = Re[ρκ (E, θS )],

H→∞

(2.8)

where E = eV applies. Finally, we get a convenient expression for the normalized conductance spectrum described by n o (E+eV ) p(θS )Re[ρκ (E − i0G , θS )] − ∂ f ∂(eV ) R . (2.9) σT (eV , 0G ) = dθS p(θS ) Here, 0G is a smearing factor due to the lifetime broadening effect [37]. These equations are easily extended to superconductor/insulator/superconductor (SIS) junctions. The quasiparticle tunneling current in this situation is given by Z ∞ Z π/2 I (eV ) ∝ dθ d E Re[ρkL (E − i0GL , θ)]Re[ρkR (E + eV − i0GR , θ)][ f (E) − f (E + eV )], R

−π/2

dθS

R∞

−∞ d E

−∞

(2.10) ρkR,L

i0GR,L

where and correspond to the DOS and smearing factors in right (left) electrodes, respectively. Calculated results of this equation are shown to give consistent result with experimental properties of YBCO/ Pb junctions [38]. For more details, see other publications [39–41]. The physical implication of the tunneling spectroscopy formula and calculated results for various pairing symmetries have already been described in other publications [16, 29]. One of the examples of calculated

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STS of YBCO (100)

2.0

1.0 0.5

Normalized conductance

1.5

0.0 0.9 Displacement (nm)

1.8 2.7 3.5 4.4 –80

–40

0

40

80

Bias voltage (mV)

Tip trace (nm)

3 0 –3 –6 –9 0

1

2

3

4

Displacement (nm) Fig. 5. STS observation of spatially uniform gap structure on YBCO (100). Such a kind of the observation belongs to an exceptional data. In the lower part of the figure, tip traces before and after the spectroscopy are shown. Although the tip traces indicates that the surface is not flat, the effective surface can be regarded as (100) in these regions.

result for dx 2 −y 2 -wave is shown in Fig. 2. We can clearly understand the strong orientational dependence of tunneling conductance spectra. Especially, ZBCP appears depending on the orientation of the junction. The results for various symmetries are summarized as follows. (i) For the c-axis tunneling case, 1+ = 1− applied for all θN . In this case, the conductance spectrum converges to the bulk DOS at the tunneling limit. (ii) For the ab-plane tunneling case the relation of 1+ and 1− depends on the orientation of the crystal. When the phase difference between 1+ and 1− is just π , the ZBCP come out in conductance spectra. For dx 2 −y 2 -wave superconductors, ZBCP are expected for (110) but not for (100). On the other hand, for dx y -wave, ZBCP are expected for (100) but not for (110). For extend s-wave case, if we assume cos 4θ dependence of the pair potential, ZBCP are not expected both on (100) and (110). However, ZBCP should be observed in wide range of ab-plane tunneling if the surface has misorientation. (iii) When the pair potential has an imaginary component (for example, s ± id-wave), the conductance peaks appear at some finite voltage between zero and Max{|1(θs )|}.

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In the above, we have assumed perfectly flat interfaces and spatially constant pair potential to get the analytical expression. However, in order to apply the theories to real situation, several additional effects should be taken into account; they are (i) roughness at the surface, and (ii) the spatial dependence of the pair potential. In the following, the theoretical aspects of these effects are simply explained. The roughness at the surface induces the random reflection. This effect has been theoretically studied both by the continuum model [18, 22, 23] and by the extended Hubbard model [24]. Especially, the atomic-scale spatial variation of the conductance spectrum can be calculated by the extended Hubbard model. The results give useful information to compare scanning tunneling microscopy (STM) and STS results with theoretical calculation. Summaries of these theories are: (i) the roughness at the surface smear the zero-energy peaks, (ii) even on (100)-oriented surfaces of dx 2 −y 2 -wave superconductors, if the defects or steps exists, ZBCP are locally induced around these structures, (iii) the roughness induces quantum interference effects in the quasiparticle spectrum [24]. The concept of the interference effect is important for the analysis of the high-Tc superconductors because of their short coherent lengths. In the extreme case, even on (110)-oriented surfaces of dx 2 −y 2 -wave superconductors, ZBCP have been verified to disappear depending on the microscopic profile of the surface. The spatial dependence of the pair potential has also been analyzed by the continuum models [18, 25, 26] and by the lattice models [27, 28]. Three important effects are mainly obtained in these theories. First, the amplitude of the pair potential is suppressed near the surface depending on the orientation. This effect can be observed as the second bound state (SBS) formation at the (110)-oriented surfaces of dx 2 −y 2 -wave superconductors. Second, the pairing symmetry of the pair potential is locally modified near the surfaces. This effect arises the possibility of spontaneous broken time-reversal symmetry (BTRS) [42]. The possible subdominant pairing symmetries are is-wave and idx y -wave. In accordance with the inducement of imaginary components, the ZBCP split into two peaks. The amplitude of splitting corresponds to the amplitude of the imaginary components that becomes larger as the temperature is lowered. In continuum models, the amplitude of the subdominant component depends on the strength of the hidden channel (pairing interaction). However, in the t-J models, similar results are obtained without assuming explicitly such channels [27, 28]. Third, the subdominant components are expected to be stabilized by applied magnetic field. The response of the peak splitting to the field is estimated to show a nonlinear feature, which is distinct from the linear behavior due to Zeeman splitting [11, 25].

3. Experimental All our measurements described below were carried out by STM/STS at 4.2 K. After the surface preparation, the samples were mounted on the STM unit quickly in the air, however, the surfaces were exposed to the air about 1 hour before cooling down to low temperatures. Mechanically sharpened Pt and Pt-Ir wires were used for STM tips. High energy resolution of our LT-STM system has been demonstrated by the observation of superconducting gap in NbN films. Details about our LT-STM system are described in Ref. [43]. The main advantages of the spectroscopy by STS compared to that by thin film junctions are; (i) the spatially resolved capability, (ii) the potential for the adjustable measurements by changing the biasing condition. These features are important in order to avoid the influences of the nonequilibrium effect, the higher-order tunneling terms, and the impurity effects. The conductance spectrum shown below were obtained under tip-biased conditions with the typical resistance of 1 ∼ 0.1 G. This high resistance measurements are quite important to keep the junction under the tunneling limit condition. 3.1. YBCO The YBCO samples used here are highly oriented thin films fabricated by sputtering methods on MgO substrates. We selected (110), (100) and (001) orientations to detect the effect of the internal phase of the

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STS of YBCO (100)

2.0

1.0 a

0.5 0.0

b

c

Normalized conductance

1.5

d e f

50 –50 0 Bias voltage (mV)

Height (nm)

–100 3 2 1 0 –1 –2 –3 –4

100

Tip trace f a

b d

e

c

0

2

4

6

8

Displacement (nm) Fig. 6. Typical STS observation of YBCO (100). The spectra show strong spatial dependence. At point A, the spectrum has a semiconducting feature. It changes into s-wave like gap structure at point B, then ZBCP gradually grows from c to f. In the lower part of the figure, tip traces before and after spectroscopy are shown. The original STS data include 51 spectra. Only six spectra are shown here for clarify.

pair potential (the results on (001) are not shown here). The film qualities were carefully evaluated by using temperature dependence of resistivity (R-T), X-ray diffraction (XRD), reflection high energy electron diffraction (RHEED), atomic force microscopy (AFM), and STM. By comparing these data, we can certify that the surfaces observed by STS have really specified orientations. Details about the fabrication processes and their qualities are described elsewhere [30]. Figure 3 shows the results of STS on (110)-oriented films. The spectra are obtained at 51 positions with equal intervals along a scan line. In the lower part of the figure, tip traces before and after the spectroscopy and spatial dependence of the peak height are also shown. Although the peak height is slightly changing spatially, the ZBCP are observed almost uniformly. This feature is inconsistent with those expected from magnetic impurity model which assumes spatially isolated magnetic impurities. We have measured seven samples of (110) films. The total number of tunneling spectrum data obtained on these samples were nearly 1500. Although gap structures are locally observed depending on the sample, 61% region in the whole span area showed ZBCP (18%:gap structures, 21%: other features like

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STS of YBCO (100)

1.0 0.5

Normalized conductance

1.5

0.0 Displacement (nm)

2 3 5 7 8 –40

–20

0

20

40

Tip trace (nm)

Bias voltage 2 0 –2 –4 –6 –8 –10 0

2

4

6

8

Displacement (nm)

[t]

Fig. 7. Spatial dependence of the conductance spectra on YBCO (100) by STM/STS at 4.2 K without an applied magnetic field (Pt–Ir tip). The ZBCP splitting are observed with spatial variation. Such feature is observed only in small area (less than 10%) on the whole surface.

3 Normalized conductance

NCCO (001)

2 (110) (100)

1

0 –20

–10

0

10

20

Bias voltage (mV) Fig. 8. Typical conductance spectra obtained on the surfaces of NCCO for(100), (110), and (001) orientations.

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Normalized conduc

STS of NCCO (110)

1.2 1.0 0.8 0.6

tance

0.4

21.1 28.2

t (nm)

cemen

14.1

Displa

0.2 0.0 7.0

Tip trace (nm)

35.2 –30 –20 –10 0 10 Bias voltage (mV)

20

0

30

30

7 5 3 2 0 10

20

40

Displacement (nm) Fig. 9. A experimental result of STS data on an (110)-oriented surface of NCCO. The figure shows successive 50 points spectroscopy measurement obtained along a line of 35 nm. Almost spatially uniform gap structures are observed. In the lower part of the figure, the tip traces before and after the spectroscopy measurement are also shown.

semiconducting). It is noted that the ratio of ZBCP increases as the quality of the film observed by RHEED becomes higher. Especially, in the case of our best quality film (Tc of 88 K with sharp transition and showing clear RHEED spots), 98% of the spectra showed ZBCP. This fact suggests that ZBCP is the intrinsic property for (110)-oriented surfaces. Another important feature of tunneling spectra on (110) is the observation of second peak. Figure 4 shows detailed comparison between theoretical and experimental curves. For the pair potential, we assume 1(θs ) = 1d cos 2θs .

(3.1)

The self-consistently determined pair potential has been used for the theoretical calculation. In addition to the large peak at zero bias due to the ZES, a small peak can be seen near 16.5 meV. This peak correspond to the SBS reflecting the suppression of the pair potential amplitude near the surfaces (see the inset of Fig. 4) [20]. The good agreement of these two curves suggests the high reliability of the experimental spectra and the theoretical calculation. On the other hand, results on (100) films are rather complicated. Theoretically, spatially uniform gap structure should be observed only if: (i) the misorientation of the crystal is zero, (ii) the surface is atomically flat. These conditions are hard to be realized in the actual samples. One of the exceptional example of spatially uniform gap structure is shown in Fig. 5. However, the tip trace shown in the lower part of the figure is not always flat. This indicates that the effective surface does not always correspond to the tip trace in nm length scale (the tip trace is largely influenced by the degradation and the work function distribution). Most of data on (100) showed strong spatial dependence with the appearance and disappearance of the ZBCP.

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uctance

Normalized cond

2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

STS of NCCO (100)

14.1 21.1

nm)

28.2

ement(

Displac

7.0

35.2 –30

–20

–10

10

0

20

30

Tip trace (nm)

Bias voltage (mV) 6 4 2 0 –2 –4 –6 0

10

20

30

40

Displacement (nm)

Fig. 10. A experimental result of STS data on a (100)-oriented surface of NCCO. The figure shows successive 50 points spectroscopy measurement obtained along a line of 35 nm. Although the gap structures are uniformly observed, some of the spectra show distorted background. They are obtained near the steps on the surface as can be seen from the tip trace in the lower part of the figure.

Figure 6 shows one of such STS data. The spectrum varies from semiconductor-like feature to s-wave like gap structure, distorted spectra, and a zero-bias conductance peak. By comparing the tip traces shown in the lower part of the figure, it is clear that the spectra gradually change at each terrace. Actually, the total number of tunneling spectra obtained on 13 samples of (100) is more than 2000. In the whole span area, although 34% area showed gap structures, 41% region showed ZBCP. In our measurements, some STS data on YBCO (110) and (100) showed the splitting of the ZBCP even without the applied magnetic field (for magnetic field response, see Ref. [15]). Similar feature has already been pointed out by using macroscopic YBCO tunnel junctions [44]. In our cases, since STS has spatially resolved capability, we can see that the amplitude of the splitting is fluctuating spatially (see Fig. 7 and Ref. [45]). The ratio of the area showing this kind of behavior is utmost 10% in the whole scan. The splitting amplitude varies in the length order of the coherent length. In this sense, the above feature is reasonably analyzed in terms of the inducement of the BTRS states coexisting with the predominant bulk d-wave. However, the splitting amplitude 1pp (4.2 K) measured by peak-to-peak values are up to 20 mV, which are somewhat larger than those of the theoretical predictions. Also, as far as we know, the observations of the peak splitting are restricted to YBCO surfaces. Hence, we are trying to find out other possibilities to explain the origin of peak splitting by introducing the role of chain site and the antiferromagnetic effects. The negative result for the BTRS states is also presented by the other measurements [46]. The absence of ZBCP on ab-plane tunneling junctions have been reported in several papers [47, 48]. On the other hand, relatively recent papers reporting on this orientation present the presence of ZBCP on various hole-doped high-Tc superconductors [15, 30, 31, 44, 45, 49–51]. Since the surface electronic states

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are sensitive to the sample preparation, we believe, the ZBCP becomes observable with the improvement of the measurement technology. 3.2. NCCO For most of the high-Tc superconductors, their carriers are holes, which indicates that they are hole-doped superconductors. On the other hand, a family of superconducting materials R2−x Mx CuO4−δ (R = Nd, Pr, Sm; M = Ce, Th) and Sr0.9 La0.1 CuO2 belong to exceptions. The Hall coefficients of these materials have negative values, which implies that their carriers are electrons. Unlikely to the hole-doped high-Tc superconductors, strong trends for conventional behaviors have been presented for NCCO based on various experimental methods such as penetration depth measurements [52]. At this stage, tunneling spectroscopy measurements of NCCO are meaningful in two different aspects; one is to make secure the validity of our tunneling formula by checking the consistency with other measurement methods, and the other is to determine the pairing symmetry in NCCO by applying phase-sensitive capability of tunneling spectroscopy. The single crystals of NCCO are used in this measurement. They were grown by traveling-solvent floatingzone (TSFZ) method. After the growth, the samples were annealed in an oxygen with a pressure of 10−4 atmosphere at 1000 ◦ C for four days. We used two crystals with their Tc values are about 17.5 K evaluated by magnetization measurements. Details about the crystal growth conditions are described in Ref. [53]. The orientations of the crystals were identified by the Laue diffraction patterns, and the surfaces with specified orientations were prepared by the mechanical cleavages in the air. Figure 8 shows orientational dependent tunneling conductance spectra. Completely different from holedoped cases, the conductance spectra have V-shaped gap structures independent of the orientation of the sample. Less dependence of the gap structure on the tunneling directions seems to reflect rather isotropic electronic structure in NCCO. The amplitudes 1pp (4.2 K) are in the range from 3.5 to 5 meV. If we assume BCS-like temperature dependence of the gap amplitude, measured 21(0)/kB Tc values are in the range from 4 to 7. The most important feature to note is the absence of the ZBCP. This feature is inconsistent both with dx 2 −y 2 -wave and with dx y -wave. The possibility of d-wave symmetry is also rejected from the STS measurements in the ab-plane tunneling configuration. Figures 9 and 10 show the results of STS on the surfaces of (110) and (100) orientations, respectively. These figures show successive 50 points spectroscopy data obtained along lines of 35 nm. In the lower parts of these figures, the tip traces before and after the spectroscopy measurements are also shown. The shift of the tip traces indicates the existence of a drift of the tip position. It is apparent that the gap structures are insensitive to the surface morphology. As described in Section 2, the robustness of the gap structure on the surface roughness in NCCO suggests that the pair potential in NCCO has no sign change at everywhere in k-space, and clearly reject the possibilities of d-wave symmetry. On the other hand, the background (outside the gap) of the spectra on (100) surface have strong spatial dependence. Around the large steps, the spectra show strong asymmetries. This feature reflects the existence of the interference effect of the wavefunctions due to the random scattering at the surface [24]. Although the experimental data strongly suggest a nodeless pairing state, we cannot easily conclude that NCCO has conventional s-wave pairing state. As is clear from the spectra in the figures, the gap structures are completely different from those expected for conventional BCS-superconductor. The conductance at the zerobias level is not zero but about half of its background normal conductance. The gap structures are V-shaped forms. The peak structures around 1 are not present in some spectra. These features are commonly observed on the surfaces of NCCO, and quite similar to those observed on YBCO surfaces. To see the gap structure of NCCO more clearly, Fig. 11 shows the comparison between theoretical curve and an experimental spectrum on the (100)-oriented surface. Reasonable fitting is obtained by assuming anisotropic (extended) s-wave symmetry. The dotted curve in Fig. 11 is obtained by assuming 1(θs ) = 1S + 1ex cos(4θs )

(3.2)

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Normalized conductance

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1

0 –10

0

10

Bias voltage (mV) Fig. 11. A comparison between theory and experimental spectra. The solid line represents an experimental spectrum obtained on the (100)-oriented surface of NCCO normalized by a parabolic background. The dotted line represents a theoretical curve for anisotropic s-wave symmetry.

with 1S = 2.2 meV, 1ex = 1.5 meV, and 0G = 0.2(1S + 1ex ). This form of the pair potential has no sign change for all directions. It is interesting that rather large value of 0G is required to get reasonable fitting with theory as in the case of YBCO. In the case of conventional BCS superconductors, even in the case of dirty superconductors, we have never observed tunneling spectra with such a large 0G . This fact may reject the possibility of the usual impurity effect for the origin of the large 0G . We guess three possible origins for the large 0G : (i) the strong electron correlation effect, (ii) the existence of unpaired quasiparticles, (iii) surface degradation. These possibilities will be tested in future works. The final problem is what is the symmetry of the pair potential in NCCO. In our previous paper, we concluded that anisotropic s-wave is more reasonable (see discussion in Ref. [33]). Actually, most of the previous papers of tunneling spectroscopy on NCCO have reported the absence of ZBCP ([32, 54] and references in [33]). On the other hand, only one paper exceptionally reports the observation of ZBCP [55]. If we assume that all these measurements are correctly observing the surface states, these data are reasonably interpreted that the appearance of the nodes of pair potential depends on the doping rate. This may suggest that symmetry of NCCO is extended s-wave rather than anisotropic s-wave. However, to get the definitive conclusion, more detailed experiments are required to check the reliability of the ZBCP observation in NCCO. We are also checking the consistency of present data with novel pairing states proposed by Ogata [56], and Kuroki and Aoki [57].

4. Conclusion In this paper, tunneling spectroscopy and surface states on anisotropic superconductors have been discussed both on theoretical and on experimental aspects. The most important concept of this study is that the translational symmetry in the bulk is lost at the surfaces. According to this, various new features which are different from bulk states arise near the surfaces. Recent detailed theoretical studies suggest that these features include important information on the electronic states. In this sense, detailed studies of the surface states are quite important to reveal the pairing mechanism of high-Tc superconductors. One of the most drastic effects at the surface is the formation of bound states. To analyze the influence of the surface states on tunneling effects, a novel formula on tunneling spectroscopy have been presented. The formula predicts the strong orientational dependence of conductance spectra. The surface bound states is also shown to have large influences on the Josephson effect [58].

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Experimental studies of tunneling spectroscopy have been accomplished on YBCO and NCCO surfaces for various surface orientations. The results on YBCO show orientational observation of ZBCP, which is consistent with a dx 2 −y 2 -wave. On the other hand, the orientational dependent measurements on NCCO show the lack of ZBCP, which is consistent with the nodeless pairing state. Two possibilities can be considered for the present experimental result. One is that the superconducting electronic states of cuprates show antisymmetric behaviors for the conversion of the carrier type between hole doping and electron doping, and the other is that the nodes of the pair potential in NCCO is happen to be hidden by some special reasons independent of the carrier type (such as described by Ogata-state and Kuroki-state [56, 57]). In future works, by using in situ sample preparation technique, the correlation between topography and spectroscopy should be studied in the microscopic scale on various material surfaces. Acknowledgements—The authors would like to thank M. Koyanagi, K. Kajimura, N. Terada, T. Ito, K. Oka, H. Takashima, S. Ueno, L. Alff, M. Ogata and K. Kuroki, M. R. Beasley, H. Takayanagi and I. Iguchi for valuable discussions. This work has been partially supported by the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST).

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