Fermi surface investigation in the scanning tunneling microscopy of Bi2Sr2CaCu2O8

Physica C 416 (2004) 75–84 www.elsevier.com/locate/physc

Fermi surface investigation in the scanning tunneling microscopy of Bi2Sr2CaCu2O8 K.K. Voo b

a,*

, W.C. Wu b, H.Y. Chen c, C.Y. Mou

a,d

a Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan, ROC Department of Physics, 88, Sec. 4, Ting-Chou Rd., National Taiwan Normal University, Taipei 11650, Taiwan, ROC c Texas Center for Superconductivity and Department of Physics, University of Houston, Houston, TX 77204, USA d Physics Division, National Center for Theoretical Sciences, P.O. Box 2-131, Hsinchu 300, Taiwan, ROC

Received 8 March 2004; received in revised form 24 June 2004; accepted 9 September 2004 Available online 22 October 2004

Abstract Within the ideal Fermi liquid picture, the impurity-induced spatial modulation of local density of states (LDOS) in the d-wave superconductor Bi2Sr2CaCu2O8 is investigated at different superconducting (SC) gap sizes. These LDOS spectra are related to the finite-temperature dI/dV spectra in scanning tunneling microscopy (STM), when the Fermi distribution factor is deconvoluted away from dI/dV. We find stripe-like structures even in the zero gap case due to a local-nesting mechanism. This mechanism is different from the octet-scattering mechanism in the d-wave SC (dSC) state proposed by McElroy et al. [K. McElroy, R.W. Simmonds, J.E. Hoffman, D.H. Lee, J. Orenstein, H. Eisaki, S. Uchida, J.C. Davis, Nature 422 (2003) 592]. The zero gap LDOS is related to the normal state dI/dV. The zero gap spectra when Fourier-transformed into the reciprocal space, can reveal the information of the entire Fermi surface at a single measuring bias voltage, in contrast to the point-wise tracing out proposed by McElroy et al. This may serve as another way to check the reality of Landau quasiparticles in the normal state. We have also re-visited the octet-scattering mechanism in the dSC state and pointed out that, due to the Umklapp symmetry, there are additional peaks in the reciprocal space that experimentally yet to be found.
2004 Elsevier B.V. All rights reserved. PACS: 74.70.Pq; 74.20.Rp; 74.25.Ld Keywords: High-Tc superconductors; Scanning tunneling microscopy

1. Introduction *

Corresponding author. Tel.: +886 3 571 2121x56176; fax: +886 3 572 5230. E-mail address: [email protected] (K.K. Voo).

A fundamental question on the high-Tc cuprates remains after more than fifteen years of the discovery of the material. It is still not clear

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2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.09.007

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whether the cuprates are systems of Fermi liquid (FL), non-FL with exotic orders such as the stripes [1], or systems with more intricate co-existence of different states of matter [2]. In fact, this question arises in both their SC and normal states. Moreover, it is not unusual that different experimental probes give different implications. The experimental findings are not yet converged. A substantial progress in the STM measurement has made another route to this problem. Not only STM looks directly into the real space, but also it can be readily connected to the reciprocal space, so-called the Fourier-transformed STM (FT-STM). Data of the low (and fixed) temperature STM on Bi2Sr2CaCu2O8 (BSCCO) was claimed to be an excellent manifestation of the FL behavior [3,4]. But, since the cuprates are such involved systems, one should be more careful to nail down the conclusion. Whether the observed STM modulation [5] is solely the Friedel stripe arising from the quasiparticle (QP) interference, or the Zaanen–Kivelson stripe [1] coexisting with the Friedel stripe, is actually an issue still in debate (see the contradictory data of Refs. [6,7]) 1. Even if the stripes can be attributed to the QP interference alone, it is still crucial to ask how well do the quasiparticles behave, how is the extension of the ‘‘Fermi-arc’’, and up to what temperatures do they survive [8]? So far there has been few finite-temperature and normal-state STM studies on the cuprates [9]. These are the major concerns of the present paper. Based on the FL scenario, a simplified model named the ‘‘octet’’ scattering model [5] has successfully ascribed the experimentally observed FT-STM peaks to the quantum interference of the QPs. Later specific single-impurity scattering calculations [10,3,4] also supported that. The occurrence of FT-STM peaks and their evolution with the bias change give informations of the Fermi surface (FS) of the measured system which are

1 We think that the possible non-dispersive FT-STM peaks reported in Ref. [7] might be due to the bilayer splitting of the electronic bands in the BSCCO compounds. The bilayer splitting [
ðcos k x  cos k y Þ2 ] is more prominent at near the antinodes, and may give rise to splitted FT-STM peaks, while some of them could be weakly dispersive at higher biases.

consistent with previous results from the angle-resolved photoemission spectroscopy (ARPES). This provide a strong identification of a Friedel stripe rather than the Zaanen–Kivelson stripe. But nevertheless, some weak non-dispersive peaks, which could be due to a coexisting Zaanen–Kivelson stripe, may also exist [6,7] 1. It is important to investigate how far the FL picture can be promoted, especially when the temperature is raised and the system enters into the normal state. We thus have performed similar FL-based calculations at different gap sizes, which are related to different temperatures to investigate the Friedel stripes. This is for the reference of future experiments to see how the cuprates are deviated from a ‘‘standard’’ FL picture. It is found that stripe-like structure exists even in the normal state. This may be counter-intuitive in the octet model, since the octets should vanish in the normal state. We argue that apart from the octets, a localnesting property of the FS (especially important in the normal state) can also give rise to a sizable joint-DOS, and hence sizable QP scattering. In Section 2, we formulate the problem as the scattering from a single impurity. In Section 3, we present the calculated spectra at different temperatures and the essential features are highlighted. In Section 4, the physical origin of the normal-state spectra are illucidated. It is noted that in the normal state, the information of the entire FS can be revealed from the data from a single measuring bias. This is in contrast to the pointwise tracing out in the dSC state. Section 5 is a closing section. In the Appendix A, we make a comment to the present FT-STM data, pointing out that within the understanding of the octet model some peaks are missing.

2. Formalism Since the single-impurity scattering model was proved to be an excellent start for understanding the STM features in the low-temperature dSC state, we proceed to study the finite-temperature case based on it. We will be interested in those regions away from the impurity neighborhood. We consider the following Hamiltonian

K.K. Voo et al. / Physica C 416 (2004) 75–84

H ¼ H 0 þ H I;

ð1Þ

where H0 is the usual BCS mean-field Hamiltonian i X Xh H0 ¼ nk cykr ckr þ Dk cyk" cyk# þ H:c: ð2Þ k;r

k

hi;ji

  þ V 0 cy0" c0" þ cy0# c0# :

ð3Þ

Parameter dt is the deviation of local hopping, dD is the deviation of local pairing potential, and V0 is the on-site impurity potential. This is a simplified model but nevertheless should be enough for our discussion. It is convenient to apply the Nambu representation for the reciprocal and real-space operators: " # " # c ck" bk b j j" ; C and C ð4Þ y ck# cyj# which are related to each other via Fourier transformation X b j: b k ¼ p1ffiffiffiffi eikrj C ð5Þ C N j Here the sum is over all the lattice sites rj. Define a 2 · 2 energy matrix

 nk Dk ^ek ; ð6Þ Dk nk thus H0 ¼

X

b y ^ek C b k: C k

ð7Þ

k

Similarly X y b ^ b HI ¼ C i uij C j þ H:c;

function. Define a single-particle GreenÕs function matrix D E b y ð0Þ ; b i ðsÞ C b i ; rj ; sÞ  Ts C ð9Þ Gðr j the LDOS is then given by

with nk and Dk are the band dispersion relative to chemical potential and the SC gap function, respectively, and HI is the part associated with an impurity at site 0, i X Xh HI ¼ dtij cyir cjr þ dDij cyi" cyj# þ H:c: hi;ji;r

77

ð8Þ

i;j

where the matrix elements ^ uij are to be given later. Since we are interested in the real-space STM which measures the LDOS, one needs to know the real-space, equal-site, single-particle GreenÕs

1 Im½G11 ðr; r; ixn ! x þ i0þ Þ 2p  G22 ðr; r; ixn ! x  i0þ Þ;

Dðr; xÞ ¼ 

ð10Þ

where Gab is an element of the 2 · 2 GreenÕs function matrix in (9), being Fourier-transformed to the Matsubara-frequency space Z b b i ; rj ; ixn Þ ¼ b i ; rj ; sÞ: Gðr dseixn s Gðr ð11Þ 0

b in (9) is expanded to the first order in The full G HI given by (8), b i ;rj ;sÞ ¼ G b 0 ðri ;rj ;sÞ Gðr * + Z b X y y b b b b C k ðs1 Þ^ þ ds1 Ts C i ðsÞ uk‘ C ‘ ðs1 Þ C j ð0Þ 0

k;‘ 2

þ Oð^ u Þ;

ð12Þ 0

0

b ðri ; rj ; sÞ G b ðri  rj ; sÞ is the mean field where G b ‘‘non-interacting’’ G when HI = 0. The terms included in HI in (3) are given explicitly here. For BSCCO, we consider a square lattice. (Throughout this paper, lattice constant a 1 for simplicity.) In the case of a single, extended and weak impurity, we consider local deviations dt1 and dD1 which couple the impurity site and its nearest neighbors, and dt2 and dD2 which couple the impurityÕs nearest neighbors and its next nearest neighbors. Consequently, there are 17 non-vanishing ^u matrices in (8):



 0 dD1 V0 dt1 ^u0;0 ¼ ; ^u0;^x ¼ ; dD1 dt1

0 V 0  dt1 dD1 ^u0;^y ¼ ; dD1 dt1

 dD2 dt2 ^u^x;2^x ¼ ^u^x;2^x ¼ ^u^y ;2^y ¼ ^u^y ;2^y ¼ ; dD2 dt2 ^u^x;^x^y ¼ ^u^x;^x^y ¼ ^u^y ;^y ^x  dt2 dD2 ¼ ^u^y ;^y ^x ¼ : dD2 dt2 ð13Þ

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Hamiltonians similar to this have been successfully used by Tang and Flatte´ [10] to explain the resonant STM spectra for Ni doped BSCCO, and by Wang and Lee [3] and Zhang and Ting [4] to explain the energy-dependent modulation of the FT-STM spectra on SC BSCCO at T = 0. Since we are considering the weak impurity scatb in (12) is only calculated up to the tering limit, G first order of ^ u (the Born limit). The first order term has already included the essential interference effect of the QPs. A strong impurity is expected to give new features such as a bound state, only at the immediate neighborhood of the impurity. Eq. (12) is then reduced to b i ; rj ; ixn Þ ¼ G b 0 ðri  rj ; ixn Þ Gðr X 0 b ðri  rk ; ixn Þ^ b 0 ðr‘  rj ; ixn Þ: G þ uk‘ G k;‘

ð14Þ

The first term on the right is translational invariant, and spatial modulation of the LDOS at a fixed frequency comes from the second term. The current I(r, V) that tunnels into the STM tip at site r and bias voltage V is proportional to dxD(r, x)[f(x)  f(x  eV)] (assuming that the LDOS of the tip is constant), where D(r, x) is the LDOS of the system given previously in (10), f is the Fermi function, and e (>0) is the electron charge. The resulting differential conductance, which is the usual spectrum discussed by STM, is thus dI(r, V)/dV/   dxD(r, x)df(x  eV)/dV. At temperatures T  x and ejVj, the derivative of the Fermi function df/dV is essentially a d-function and hence the dI(r, V)/dV of STM is directly proportional to the LDOS, D(r, eV). In general, df/dV needs to be deconvoluted from any finitetemperature dI(r, V)/dV before the intrinsic D(r, x) can be revealed. But in the case when D(r, x) depends only weakly on x, D(r, x) can be pulled out from the convolution and hence dI(r, V)/dV/D(r, eV) dxdf(x  eV)/dV/D(r, eV), i.e., the spatial modulation of D(r, eV) is not much smeared by temperature but still proportional to dI(r, V)/dV. The latter is actually the case of zero SC gap as we will see shortly. In an ideal FL and elastic impurity model, the temperature affects the intrinsic LDOS D(r, x)

only via the change of the SC gap D(T). The temperature dependence of the gap in BSCCO has been directly measured, and a reasonable interpolation is D(T) = D(0)[1  (T/Tc)3]1/2 with D(0) = 44 meV. In our calculation, we have studied the cases of D(T) = 44, 26, 10, and 0 meV that correspond to T = 0, 0.85Tc, 0.98Tc, and Tc, respectively. It is noted that the FT-STM spectrum discussed in Ref. [5] is the Fourier transform of the spatial modulation of dI/dV. In this paper we have studied the spatial modulation of the LDOS due to the second term in Eq. (14). We have used a 800 · 800 square lattice with the impurity at the center. We have chosen a simple but reasonable impurity potential, 2dt1 = 4dt2 = 2dD1 = 4dD2 = V0 and assume that these scales are small and in the perturbative limit. For nk, we use a tight-binding band, nk = t1(cos kx + cos ky) / 2 + t2 cos kx cos ky + t3(cos2kx + cos 2ky)/2 + t4(cos 2 kxcos ky + cos kx cos 2 ky) / 2 + t5 cos 2kx cos 2ky  l, with t15 = 0.60, 0.16, 0.05, 0.11, 0.05 eV and chemical potential l = 0.12 eV, appropriate for an optimally-doped BSCCO [11]. The SC gap is taken to be Dk = D(T)(cos kx  cos ky)/2. Besides, we have introduced an artificial broadening c = 2 meV to the GreenÕs function, such that eV + i 0+ is replaced by eV + ic.

3. Local density of states spectra In Fig. 1, we present the real-space LDOS spectra at different SC gap sizes for two different negative frequencies. The case of positive frequencies will not be discussed as spectra at x are qualitatively the same as that at x. The gap magnitudes are taken from D(T = 0) = 44 meV to D(Tc) = 0 meV to simulate the transition from the SC to the normal state. At a distance of several lattice constants away from the impurity, oscillating Friedel stripes are seen in all cases, even in the case of normal states [D(T) = 0]. Spectra with a similar ratio of jxj/D(T) share a similar behavior, such as those in Fig. 1(b) and (c) with jxj/D(T) = 25/ 44, 15/26
0.57. On the other hand, spectra at zero D(T) is robust at the change of x. Comparing the relative intensities of the modulations, we see

K.K. Voo et al. / Physica C 416 (2004) 75–84

Fig. 1. Superconducting gap size dependence of the LDOS modulation surrounding the impurity. All panels are plotted within a square region, rx, ry = 20–20, with the impurity at the center. The left and right columns show the spectra at frequencies x = 15 and 25 meV, respectively, and from top to bottom the gap magnitude D(T) = 44, 26, 10, and 0 meV, respectively. For better visualization, spectra in different panels are plotted in different intensity windows from x (deepest blue) to +x (deepest red) as specified by the color scale at the bottom. The individual value of x is given at the lower-right corner of the individual panel. Those regions of the deepest blue/red have intensities below/above the window. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

79

that the strongest modulations (at a fixed x) appear at temperatures near Tc. An important feature to note is that the different ripples live in well-separated patches and have little overlaps. The Fourier-transforms of the above LDOS spectra (FT-LDOS) are given in Fig. 2. Roughly speaking, there are two regimes for the spectra in the D(T) 5 0 SC state, as classified by the ratio jxj/D(T). When jxj/D(T) < 1, there are local peaky structures that have their locations described by the octet model (see Fig. 2(a)–(c)), which is in agreement with previous studies [3,4]. Previous studies discussed the cases with different x and a fixed D(T)  D(0), while in our case D(T) is varied (by changing the temperature). Comparing Fig. 2(b) and (c) which have a similar ratio jxj/ D(T)
0.57, one sees that the essential features of the spectra depend mainly on the ratio jxj/ D(T), as noted before in the discussion of the real-space spectra. In Fig. 3, a closer look of the locations of the interference peaks in Fig. 2(a)– (c) is given. When jxj/D(T) > 1 (see Fig. 2(d)–(f)), some extra peaky structures beyond the description of the octet model appear near q = 0. These new structures are not expected to be seen in real BSCCO compound because they are related to the maximum gap part of the STM spectra, which is highly inhomogeneous in space even in the absence of the impurity. As D(T) is further decreased, the strong peak at q = 0 is suppressed and vanishes at entering the normal state. In the normal state, the FT-LDOS spectra are reduced to some neat ridges instead of peaks, and they are rather robust against the change of x (see Fig. 2(g) and (h)). As the interference peaks in the dSC states (in the regime of jxj/D(T) < 1) were well documented in the literature [3,4], we will only give a supplementary comment on it in the Appendix A. In Section 4 we will pay special attention to the normalstate spectra.

4. Origin of the normal-state spectra In this section, we show that the normal-state FT-LDOS spectra have an intimate relationship with the underlying FS.

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K.K. Voo et al. / Physica C 416 (2004) 75–84 3.0

(c)

2.5

FT-LDOS (a.u.)

(a) 2.0

(b)

1.5 1.0 0.5

0.0 (0.56π, 0.56π)

(0,0)

(0.8π,0)

Fig. 3. This figure shows that the locations of the interference peaks (indicated by arrows) in the FT-LDOS spectra depend mainly on the ratio jxj/D(T). Spectra in Figs. 2(a)–(c) which have jxj/D(T) = 0.34, 0.57 and 0.58, respectively, are scanned along q = (0.56p, 0.56p) ! (0, 0) ! (0.8p, 0). It is seen that Fig. 2(b) and (c) which has similar jxj/D(T), also has similar peak positions and intensities. For easier comparison, the intensity of the spectrum in Fig. 2(a) is enlarged by a factor of 2.

Fig. 2. Superconducting gap size dependence of the FT-LDOS spectra (corresponding to the real-space results in Fig. 1) in the first Brillouin zone qx, qy = p
p. Left and right columns are for x = 15 and 25 meV, respectively. From top to bottom, D(T) = 44, 26, 10, and 0 meV, respectively. The spectrum in each panel is plotted in an intensity window from xl to xu. The values of xl and xu are specified at the bottom of each panel, and the color scales are shown at the most bottom.

The pronounced feature of the FT-LDOS in the dSC state is the x dependence of the peak

locations. As long as the ratio jxj/D(T) is small enough, peak locations are more or less as described by the octet model [3]––a model that assumes the dominant QP scattering comes between regions of high DOS on the FS, the octets. The octets are regions with the smallest velocity, which appear at the tips of the banana-shaped 2 constant energy contour n2k þ j Dk ðT Þj ¼ x2 . The agreement of the low-temperature experimental data with the picture was claimed to be good [5]. Later specific single-impurity scattering calculations [3,4] (similar to the one we did in Section 2) also supported this picture. In the normal state, such large DOS octets no longer exist. However, we point out that there exists a different mechanism which can also cause a substantial jointDOS, and leads to distinguished structures in the q-space. The upper panel of Fig. 4 shows a typical normal-state FT-LDOS spectrum in an extended Brillouin zone (BZ). The underlying FS is also shown in the lower panel. Comparing the panels, it is readily seen that the ‘‘ridges’’ in the spectrum are of the same shape as the FS, but have twice the

K.K. Voo et al. / Physica C 416 (2004) 75–84

81

‘‘local-nesting’’. 2 As a result, stripes in the realspace are still understood as due to the scattering on the FS. In the local-nesting picture, it is seen that structures in the FT-LDOS near zero wave vector comes from QP scattering between the Fermi surface near (0, ±p) and (±p, 0). In the normal state, the scattering occurs all over the Fermi surface and hence the stripes in real-space are smoothly deformed in contrast to those in the dSC state. The stripes are formed by the interference of quasiparticles coming from different directions vF, hitting the impurity, and bounced back with some transition wave vector k joining low-energy regions in the reciprocal space. As the coming-in and bouncing-back directions are related in some way, within the same patch of real-space there will be no two differently oriented stripes. Therefore to explain the ‘‘checkerboard’’ pattern (overlapping stripes) [5] in the experiment using this picture, one should consider the existence of a dilute concentration of impurities in the system [14] 3.

5. Concluding remarks

Fig. 4. Upper panel: The FT-LDOS spectrum of a normal-state system at a x = 23 meV. The ridge-like structure is a typical feature of all normal-state spectra. Lower panel: Several of the wave vectors on the above ridges are shown in the extended Brillouin zones. They are wave vectors having a local-nesting property, i.e., they joint locally parallel parts of the Fermi surface (as illustrated by pairs of red lines). (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

Our perturbative approach is not a concern regarding the validity of our discussion since it is not our purpose to investigate the non-perturbative local bound states at near the impurity. We are only interested in regions remote from the impurity where there are only interference between scattered quasiparticles, and our approach has included the interference effect. Within the framework of an ideal FL, we have studied the LDOS around an elastic impurity, from the dSC to the normal state. On the connection of 2

size, and differently oriented branches overlapped together. The occurrence of the ridges can be understood from the scattering wave vectors drawn in the lower panel. Those wave vectors which are pivoted at q = (mp, np), n, m 2 integer, are special in the sense that they joint locally parallel segments of the FS, i.e., they possess a weak

Such local-nesting effect is similar to the mechanism leads to the incommensurate peaks observed by inelastic neutron scattering (INS) in LSCO or YBCO [12]. In the dSC state, localnesting effect in the INS is transfered-energy dependent and gives rise to transfered-energy dependent peak locations. 3 In a real material, the real-space spectra may be seen as some crossingly superposed stripes due to individual impurities, showing some ‘‘checkerboard’’ pattern. It was first pointed out by us in cond-mat/0302473 [13] that the existence of dilute impurities is crucial in relating this kind of calculation to the experimentally observed checkerboard pattern.

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our LDOS to the dI/dV in the experimental STM, there are two more factors to consider, the (extrinsic) temperature smearing (discussed in Section 2) and the (intrinsic) self-energy of the QPs. The LDOS can be deconvoluted from the dI/dV since the Fermi distribution function is a known function. When the LDOS depends only weakly on frequency, like the case of the normal state discussed in Section 2, its spatial modulation is indeed proportional to the finite-temperature dI/dV at before deconvolution. The weak dependence of LDOS on frequency is a direct reflect of the lack of a characteristic energy scale in the scattering process. If the self-energy effect is included, the frequency dependence will no longer be weak since the self-energy may contain some energy scales. On the self-energy effect, it is interesting to note that there is an apparent contradiction between data from the STM and ARPES [15], already in the translational invariant low-energy LDOS. Generally speaking, the ARPES data around the diagonal or nodal-gap region show a MarginalFermi-Liquid [16] 4 behavior, which can be summarized by a phenomenological self-energy. That self-energy contains the effects of inelastic scattering at different temperatures on the QPs. When this self-energy is used to calculate the LDOS, the low-frequency part of the obtained LDOS is found not proportional to the frequency [15], even at low temperatures. Furthermore, if we disregard this fact and proceed to calculate the spatial modulation of LDOS due to the second term in Eq. (14), the modulation is found to be very much blurred and many of the peaks in the FT-LDOS are gone. Conceivably, within the QP scattering scenario, the QPs just have to have a very long lifetime for the interference pattern to form. On the other hand, the FT-STM experiments in the lowtemperature dSC state [5] presents a compelling evidence on the existence of well-defined QPs around the diagonal wave vector region, in the sense that the obtained Fermi surface agrees well with that by the ARPES. Nevertheless, we have

4

There is also a possibility that some of the broadness of the quasiparticle peak in the ARPES is due to unresolved bilayer splitting. See, e.g., [17].

presented a study based on well-defined QP as seen through the eye of the STM. In the literature, it was proposed that temperature could be a detrimental factor to the quasiparticles [8]. If this is true, the normal-state stripe described in this paper is expected to vanish. Stripes of a different origin may exist [1], but they are likely to differ in many aspects, such as the orientation or temperature dependence of the stripes. In our case, the real-space normal-state stripes 3 and the corresponding ridges in the FT-LDOS spectra reflect the information of the Fermi surface. If there are destructions of the QPs at some Fermi surface segments, the corresponding segments of the ridges in the FT-LDOS spectrum should also be destroyed [18]. The presence of a pseudogap in the normal state may also be signified as some missing segments of the ridges. Therefore this study shows another way to monitor the QPs on the Fermi surface. On the observability of the normal state interference stripes, its modulation magnitude depends on the strength of the impurity scattering which is not known at the moment. But since it is comparable to that in the dSC state, which is already observed, we believe that it can be observed if it exists. Owing to its weak dependence on frequency, we might be able to see it in dI/dV before the deconvolution is carried out. Our purpose in this paper is not to provide a comprehensive description of the current STM data. Rather, we wish to show how an ideal FL will behave in terms of the LDOS modulation upon impurity scattering. It might be useful for the reference of future experiments to see how the real system is deviated from an ideal FL. In view of the possible contradiction between the STM and ARPES, making closer comparison of data from both measurements is meaningful.

Acknowledgments This work is supported by the National Science Council of Taiwan under the Grant no. 92-2112M-003-009 (WCW) and 92-2811-M-007-027 (KKV and CYM). HYC thanks the support of

K.K. Voo et al. / Physica C 416 (2004) 75–84

National Center for Theoretical Sciences (Physics Division) of Taiwan during his visit.

Appendix A. Umklapp symmetry In this Appendix, a brief comment is given to the experimental data in Ref. [5]. If one considers the octet model in the dSC state more carefully, one will see the existence of additional peaks, which are not yet observed in the current STM measurement. In Fig. 5, for a certain bias voltage, we have illustratively shown a few direct and Umklapp

(-1.2)

(1.2)

(a)

q4'

q5'

(1.1)

ky(π)

(-1.1)

scattering wave vectors in the octet model. 5 In addition to the scattering vectors within the first Brillouin zone (direct), there are also vectors connecting octets in different Brillouin zones (Umklapp). When all these vectors are taken into account, there should exist FT-STM peaks as shown in the lower panel of the figure. As an example, a new peak at the vector q05 is seen along the qy axis (when the bias voltage exceeds some value). Peak at q5 was reported by Ref. [5], but the corresponding q05 peak was not. More disturbingly, there also exists a q04 peak which may come close to q2,6 peaks (or vice versa speaking). It may seem at odds that a mathematical Fourier transform of experimental data on a discrete lattice does not automatically possess the Umklapp symmetry. The reason may be due to the fact that practical STM actually probes a space more continuous than the underlying lattice. It scans in steps smaller than the lattice constants and then identifies the locations of individual atoms by looking at the modulation profile!

References q4

(-1.0)

(1.0)

q5

(-1.-1)

[1] [2] [3] [4] [5]

(1.-1)

kx(π) 2.0

[6]

(b)

qy(π)

q5

[7]

q4

1.0

0.0 -2.0

83

q5' -1.0

q2.6 0.0

q'4 1.0

2.0

qx(π) Fig. 5. (a) For illustration, a few of the scattering wave vectors in the octet model (the octets are indicated as circles) are shown. Wave vectors discussed by McElroy et al. [5] are shown in blue, while the Umklapp wave vectors (which are not discussed in Ref. [5]) are in red. (b) Locations of all the wave vectors connecting all the octets. The Umklapp wave vectors are in red. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)

[8] [9] [10] [11] [12] [13]

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5 For easy comparison, our notations are chosen in a way corresponding to that in Ref. [5].

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