Tunneling spectra of strongly coupled superconductors: Role of dimensionality
aa r X i v : . [ c ond - m a t . s up r- c on ] J u l Tunneling spectra of strongly coupled superconductors: Role of dimensionality
C. Berthod
DPMC-MaNEP, Universit´e de Gen`eve, 24 quai Ernest-Ansermet, 1211 Gen`eve 4, Switzerland (Dated: February 15, 2017)We investigate numerically the signatures of collective modes in the tunneling spectra of superconductors.The larger strength of the signatures observed in the high- T c superconductors, as compared to classical low- T c materials, is explained by the low dimensionality of these layered compounds. We also show that the strong-coupling structures are dips (zeros in the d I / dV spectrum) in d -wave superconductors, rather than the steps(peaks in d I / dV ) observed in classical s -wave superconductors. Finally we question the usefulness of effectivedensity of states models for the analysis of tunneling data in d -wave superconductors. PACS numbers: 74.55.+v, 74.72.–h
I. INTRODUCTION
Many experiments have shown that the electrons in cupratehigh- T c superconductors (HTS) are significantly renormalizedby the interaction with collective modes. This renormal-ization appears in photoemission measurements as velocitychanges in the quasi-particle dispersion (the “kinks”) accom-panied by a drop of the quasi-particle life-time. In tunnel-ing, the renormalization shows up as a depression, or “dip”,in the dI / dV curve with the associated nearby accumula-tion of spectral weight (the “hump”). Similar signatures ob-served by tunneling spectroscopy in classical superconductorswere successfully explained by the strong-coupling theory ofsuperconductivity.
There are, however, two striking differ-ences between the structures observed in the cuprates and inlow- T c metals such as Pb or Hg. The dip in the cuprates iselectron-hole asymmetric, being strongest at negative bias,while no such asymmetry is seen in lead. The electron-holeasymmetry of the dip is due to the electron-hole asymmetryof the underlying electronic density of states (DOS). Sec-ond, while the structures are subtle in low- T c materials—theyinduce a change smaller than 5% in the tunneling spectrum—the cuprate dip is generally a strong effect which, for instance,can reach 20% in optimally doped Bi Sr Ca Cu O + d (Bi-2223). It is tempting to attribute this difference of intensi-ties to a difference in the overall coupling strength, as sug-gested by the largely different T c values. However, a compari-son of the effective masses indicates that the couplings are notvery different in Pb where m ∗ / m = . m ∗ / m varies between 1 . Here we show that the large magnitude of thedip feature results from the low dimensionality of the materi-als and the associated singularities in the electronic DOS.Tunneling experiments in strongly coupled classical super-conductors have been interpreted using a formalism that ne-glects the momentum dependence of the Eliashberg functionsand of the tunneling matrix element, and further assumes thatthe normal-state DOS N ( w ) is constant over the energy rangeof interest. The tunneling conductance, then, only dependson the gap function D ( w ) , whose energy variation reflects thesingularities of the phonon spectrum. The dimensional-ity of the materials does not enter in this formalism. The ef-fect of a non-constant N ( w ) on the gap function has beendiscussed in the context of the A15 compounds. However the direct effect of a rapidly varying N ( w ) on the tunnel-ing conductance became apparent only recently in the high- T c compounds, and requires to go beyond the formalisms ofRefs 13 and 16. In particular, one can no longer assume thatthe tunneling conductance is proportional to the product of thenormal-state DOS by the “effective superconducting DOS” Re (cid:2) | w | / p w − D ( w ) (cid:3) , so that nothing justifies a priori tonormalize the low-temperature tunneling conductance by thenormal-state conductance as was done with low- T c supercon-ductors.Among the new approaches introduced to study strong-coupling effects in HTS, some have focused on generalizingthe classical formalism to the case of d -wave pairing, still overlooking the dimensionality. Other models are strictlytwo dimensional (2D) and pay attention to the full electrondispersion, taking into account the singularities of N ( w ) . Most of these studies assume that the collective moderesponsible for the strong-coupling signatures is the sharp ( p , p ) spin resonance common to all cuprates near 30–50 meV(Ref. 26), but a phonon scenario was also put forward. Inthe present work, we extend these approaches to three dimen-sions (3D) by means of an additional hopping t z describingthe dispersion along the c axis, and we study the evolution ofthe strong-coupling features in the tunneling spectrum alongthe 2D to 3D transition on increasing t z . For simplicity werestrict to the spin-resonance scenario; the electron-phononmodel can be treated along the same lines, and both mod-els lead to the same main conclusions. The model we useis described in Sec. II, results are presented and discussed inSec. III, and Sec. IV is devoted to investigating the validityand usefulness of the effective superconducting DOS concept. II. MODEL AND METHOD
Following previous works we assume that the differ-ential conductance measured by a scanning tunneling micro-scope (STM) is proportional to the thermally-broadened lo-cal density of states (LDOS) at the tip apex, and that fur-thermore the energy dependence of the LDOS just outside thesample follows the energy dependence of the bulk DOS N ( w ) : dIdV (cid:181) Z d w [ − f ′ ( w − eV )] N ( w ) , (1)where f ′ is the derivative of the Fermi function. In STM ex-periments, various sources of noise may contribute to broaden N ( w ) further; when comparing theory and experiment weshall take these into account by a phenomenological Gaussianbroadening.In the superconducting state, the interaction with longitudi-nal spin fluctuations is described by the 2 × S ( kkk , w ) = − N (cid:229) qqq b (cid:229) i W n g c s ( qqq , i W n ) × ˆ G ( kkk − qqq , i w n − i W n ) (cid:12)(cid:12) i w n → w + i + (2)with c s the h S z S z i spin susceptibility, g = √ hJ / J the spin-spin interaction energy, and ˆ G the 2 × N vectors qqq ≡ ( qqq k , q z ) in the three-dimensional Brillouinzone and the even Matsubara frequencies i W n = n p / b with b = ( k B T ) − . Equation (2) gives the lowest-order term of anexpansion in J . We follow Ref. 7 and use for c s a model in-spired by neutron-scattering experiments on the high- T c com-pounds. In this model, c s has no q z dependence and is theproduct of a Lorentzian peak centered at ( p , p ) in the 2D Bril-louin zone and another Lorentzian peak centered at the reso-nance energy W sr . The widths of the peaks are D q in momen-tum space and G sr in energy. This simple separable form of c s allows to evaluate analytically the frequency sum in Eq. (2),and to perform analytically the continuation from the odd fre-quencies i w n = ( n + ) p / b to the real-frequency axis. Theremaining momentum integral is a convolution which can beefficiently performed using fast Fourier transforms. Also, theabsence of q z dependence in c s implies that the self-energydoes not depend on k z . We use a BCS Green’s function broad-ened by a small phenomenological scattering rate G ,ˆ G ( kkk , i w n ) = [ i w n + i G ( i w n )] ˆ t + x kkk ˆ t + D kkk k ˆ t [ i w n + i G ( i w n )] − x kkk − D kkk k , (3)where G ( i w n ) = G sign ( Im i w n ) , ˆ t i are the Pauli matrices withˆ t the identity, and D kkk k = D ( cos k x − cos k y ) / d -wavegap which we assume k z independent for simplicity. The addi-tional effects resulting from a possible weak modulation of the gap along k z will be discussed toward the end of Sec. III.We do not address here the origin of the pairing leading to theBCS gap D kkk k . With the high- T c compounds in mind, we con-sider a one-band model of quasi-2D electrons with a normal-state dispersion, x kkk = x kkk k + t z cos ( k z c ) , (4)where x kkk k = e kkk k − m , m being the chemical potential and e kkk k a five-neighbor tight-binding model on the square lat-tice ( a ≡ e kkk k = t ( cos k x + cos k y ) + t cos k x cos k y + t ( cos 2 k x + cos 2 k y ) + t ( cos 2 k x cos k y + cos k x cos 2 k y ) + t cos2 k x cos 2 k y .The momentum dependence of the self-energy in Eq. (2) isnot small (see, e.g., Fig. 1 of Ref. 7 and Fig. 2 below). This is a major difference with respect to the electron-phonon modelsdescribing low- T c three-dimensional metals, where the mo-mentum dependence of the self-energy can be neglected. Thecalculation of the DOS is therefore much more demandingsince two three-dimensional momentum integrations must beperformed for every energy w . The DOS is given by N ( w ) = N (cid:229) kkk (cid:0) − p (cid:1) Im ˆ G ( kkk , w ) , (5)where ˆ G is the first component of the matrix ˆ G ( kkk , w ) =[ ˆ G − ( kkk , w ) − ˆ S ( kkk , w )] − . In Eq. (5), the k z integration canbe performed analytically (see Appendix) but not in Eq. (2).In order to achieve a good accuracy when computing the DOS N ( w ) , we evaluate the self-energy using a 2048 × × G = . G is increasedto 2 meV, which allows to decrease the mesh size to 1024 × × c -axis hopping energy t z . In HTS, t z is notlarger than a few meV, and setting it to zero seems appropri-ate to discuss tunneling data. Indeed, in the 2D limit the modelwas found to fit the experimental data for optimally doped Bi-2223 very well. Here we take the parameters determinedfrom one such fit as a starting point, and we vary t z to demon-strate the role of the dimensionality on the tunneling spec-trum. The band parameters t ... are − −
16, 8, and − m = −
200 meV. Thegap magnitude is D =
46 meV. The spin-resonance energyis W sr =
34 meV, its energy width G sr = D q = . a − . Finally the coupling strength is g =
775 meV, which implies a quasi-particle residue Z = . m ∗ / m = .
32 at the nodal point ofthe Fermi surface. The temperature is set to T = t z = III. RESULTS
The evolution of N ( w ) with increasing t z is displayed inFig. 1a. In the 2D limit, the DOS shows sharp and particle-hole asymmetric coherence peaks, strong and asymmetricdips, as well as humps and shoulders where the spectralweight expelled from the dips is accumulated. This produces,in particular, a characteristic widening of the coherence peaksbasis, which becomes triangular. The particle-hole asymme-tries reflect the particle-hole asymmetry of the correspondingbare DOS N ( w ) shown in Fig. 1b, whose Van Hove singu-larity (VHS) lies slightly below E F at −
16 meV: on the onehand, the spectral weight of the VHS goes to a larger de-gree into the negative-energy coherence peak, and on the otherhand the enhancement of the scattering rate due to the VHSis stronger at negative energy, explaining the stronger dip at w < This can also be seen in Fig. 2 where the electron-scattering rate − Im ˆ S ( kkk , w ) is displayed for the nodal and PSfrag replacements (a) (b) (c) D O S ( s t a t e s / e V / ce ll ) Energy ω (meV) d I / d V ( a r b . un i t s ) Bias V (mV)
000 02468 − − −
100 100100100 t z (meV)01050100200 2D2D 3D3D 4 t z N ( ω ) N ( ω ) FIG. 1. (a) Density of states N ( w ) for superconducting electrons cou-pled to a ( p , p ) spin resonance, as a function of the c -axis hopping t z . For t z =
0, the system is two dimensional while for t z =
200 meV,it is three dimensional. (b) Normal-state bare DOS N ( w ) for thesame t z values. (c) Tunneling conductance for the same t z values.The temperature is T = t z = anti nodal points of the Fermi surface. The scattering rate van-ishes for | w | < W sr and has a pronounced, particle-hole asym-metric maximum near | w | = W sr + D (more precisely between W sr + D and W sr + [ x ( p , ) + D ] / ). It is also clear from thefigure that the energy of the scattering-rate peak shows no dis-persion with momentum but its intensity is strongly momen-tum dependent and larger by a factor ∼ . t z >
0, the logarithmic divergence in N ( w ) is cut onthe scale of 4 t z due to dispersion along the c axis (Fig. 1b).No significant change in either N ( w ) or dI / dV is observedfor t z =
10 meV. This value is an upper bound for the c -axishopping energy in the cuprates, and the relative insensitivityof the DOS to a small c -axis dispersion justifies the use oftwo-dimensional models for these systems. At larger t z val-ues, however, the suppression of the divergence in N ( w ) in-duces a drop of the coherence peaks in N ( w ) and dI / dV . Thisis a direct effect of dimensionality on the tunneling spectrum,which was overlooked in the conventional strong-coupling ap-proaches of Refs 13 and 16. Simultaneously the peak in thescattering rate is also suppressed with increasing t z (Fig. 2),leading to a weakening of the dip feature in N ( w ) and dI / dV .This is an indirect effect of dimensionality, that is only re-vealed in the strong-coupling signatures.As Fig. 2 shows, increasing the dimension not only sup-presses the peak at W sr + D in the scattering rate but it alsoreduces its momentum dependence. In 2D, this peak arisesbecause the qqq sum in Eq. (2) is dominated by the saddle-point region near kkk M ≡ ( p , ) and kkk M ′ ≡ ( , p ) , where thespectral weight of the BCS Green’s function is largest— i.e. , kkk k − qqq k ≈ kkk M , M ′ . Hence the peak energy is determinedchiefly by the BCS excitation energy at kkk M , M ′ , shifted by W sr
00 0
PSfrag replacements (a) (b) − I m ˆ Σ ( k , ω )( m e V ) ω (meV) ω (meV) − −
100 100100 100200300400500600 t z (meV)01050100200 2D2D 3D3D Ω sr Ω sr + ∆ ΓΓ MM M ′ M ′ FIG. 2. Scattering rate − Im ˆ S as a function of energy for severalvalues of t z at (a) the nodal point kkk k ≈ ( . , . ) p / a and (b) theanti nodal point kkk k ≈ ( , . ) p / a of the Fermi surface shown in theinsets. Curves are offset vertically for clarity. The dashed verticallines delimit the energy range | w | < W sr where inelastic scatteringby the spin resonance is forbidden. The dotted vertical lines indicate ± ( W sr + D ) . due to the convolution with the spin susceptibility, and thepeak intensity is controlled by the momentum dependence of c s (cid:0) kkk k − kkk M , M ′ , W sr (cid:1) , which is at maximum for kkk k = kkk M ′ , M .The situation changes in 3D because the anti-nodal regions nolonger dominate the spectral weight, as illustrated in Fig. 3.This figure displays the partial BCS density of states, i.e. , thepart of the BCS DOS originating from states close to the ( p , ) and equivalent points. While in the 2D limit, a region cover-ing just 14% of the zone around ( p , ) provides 56% of thespectral weight for energies between W sr and W sr + D , itscontribution is reduced to 21% in the 3D limit. Hence thescattering rate in 3D is nearly momentum independent and al-most constant above W sr + D . Finally, the 2D to 3D transition PSfrag replacements
2D 3D P a r t i a l D O S ( s t a t e s / e V / ce ll ) ω (meV) ω (meV)
00 02468 − −
100 100100 t z = 0 t z = 200 meV Γ MM ′ FIG. 3. Partial BCS density of states in two (2D) and three (3D)dimensions. The thin solid lines show the contribution coming fromthe ( p , ) region of the two-dimensional Brillouin zone, shaded inblack in the inset while the dashed lines show the contribution of theremainder of the zone (shaded in gray). The thick line is the totalBCS DOS. also suppresses the particle-hole asymmetry of the scatteringrate. This again results from the disappearance of particle-hole asymmetry in the underlying bare DOS (Fig. 1b) andin the corresponding BCS DOS (Fig. 3). Thus the k z dis-persion simultaneously defeats four players who contribute tomake the strong-coupling signatures in the 2D high- T c super-conductors distinctly different from those in 3D metals: theVan Hove singularity, the particle-hole asymmetry, the mo-mentum dependence, and the strong scattering enhancementat | w | ≈ W sr + D , especially near ( p , ) .In the curves of Figs. 1a and 1c corresponding to the 3Dlimit, the strong-coupling signatures are barely visible. Theirmagnitude is ∼ ∼
5% value observedin Pb. The origin of this difference lies in the gap symmetry.In d -wave superconductors, the coherence peaks in the BCSDOS are weak logarithmic singularities while in s -wave su-perconductors, they are strong square-root divergences. Thestrength of the scattering-rate peak at W sr + D , and conse-quently the strength of the dip in the DOS and tunnelingspectrum, are determined by the strength of the coherencepeaks in the BCS DOS, as is clear from Eq. (2). In the caseof a d -wave superconductor, the coherence peaks are cut in3D as compared to 2D (see Fig. 3) in the same way as thelogarithmic VHS in Fig. 1b, resulting in the suppression ofthe scattering-rate enhancement at W sr + D in Fig. 2. (Notethat, roughly speaking, the scattering rate is proportional tothe BCS DOS shifted in energy by ± W sr .) The suppressionof the BCS coherence peaks with increasing dimension alsooccurs in s -wave superconductors, but with one difference:if, on the one hand, the part of the coherence-peak spectralweight coming from the VHS gets suppressed, on the otherhand, the square-root gap-edge singularities persist in any di-mension. Therefore, in s -wave superconductors the strong-coupling signatures remain clearly visible in 3D. This is il-lustrated in Fig. 4a. The 2D and 3D DOS curves of Fig. 1aare compared to the curves obtained for the corresponding s -wave model, i.e. , with all parameters unchanged except thegap which is replaced by D kkk k ≡ D =
46 meV. The changesare quite dramatic. The first effect to notice is a drastic reduc-tion in the peak-to-peak gap D p in the s -wave case: a conse-quence of the pair-breaking nature of the coupling Eq. (2) inthe s -wave channel. Still, the strong-coupling signaturesappear at the same energy W sr + D =
80 meV in both d and s wave, due to our choice of the lowest-order model ˆ S (cid:181) c s ˆ G in Eq. (2). The second observation is that the strong-couplingsignatures look like steps in the s -wave DOS, like in the clas-sical superconductors, reflecting the asymmetric shape ofthe BCS s -wave coherence peaks. In contrast, the signaturesappear as local minima in the d -wave DOS, because the co-herence peaks of the d -wave BCS DOS are nearly symmetricabout their maximum. In short, the strong-coupling featuresgive an “inverted image” of the BCS coherence peaks. An in-teresting consequence follows: while in s -wave superconduc-tors, the strong-coupling structures correspond to peaks in thesecond-derivative d I / dV spectrum, for a d -wave gap theycorrespond to zeros in the d I / dV spectrum, as demonstratedin Fig. 4b. This conclusion applies equally to phonon mod-els and calls for a reinterpretation of cuprate d I / dV data in PSfrag replacements (a)(b)
2D 3D D O S ( s t a t e s / e V / ce ll ) d I / d V ( a r b . un i t s ) ω (meV) ω (meV) Bias voltage V (mV) − − − − − − − −
60 100100 t z = 0 t z = 200 meV s -wave s -wave d -wave d -waveΩ sr + ∆ Ω sr + ∆ FIG. 4. (a) Comparison of the d -wave and s -wave DOS for t z = t z =
200 meV (3D). The thick lines show N ( w ) as in Fig. 1a.The thin lines show N ( w ) computed with an s -wave gap of the samemagnitude D =
46 meV, and all other parameters unchanged. (b)Second-derivative tunneling spectrum, d I / dV , in the region of thestrong-coupling signature at negative bias. In the s -wave case, thereis a peak in d I / dV close to the energy W sr + D while in the d -wavecase, there is a sign change in 2D and no clear signature in 3D. which peaks were assigned to phonon modes. Finally, onesees from Fig. 4 that in 3D the signatures remain strong for an s -wave gap, for the reason explained above, while they havealmost disappeared in the d -wave case.The previous discussion underlines the role of the BCScoherence peaks in the formation, strength, and shape ofthe strong-coupling signatures. More generally, for suchsignatures to occur there must be divergences (or at leastpronounced maxima) in the non-interacting DOS. Peaks inthe “bosonic” spectrum are not sufficient, although they arenecessary. Indeed, phonon structures are absent from thenormal-state spectra of classical superconductors becausethe normal-state DOS is flat, in spite of the facts that thephonon spectrum and the electron-phonon coupling do notchange significantly at T c . In contrast, the normal-state DOSof 2D high- T c superconductors exhibits structures, either thepseudogap or the bare VHS. One can therefore expect tosee strong-coupling features in the normal-state spectra ofHTS, provided that the peaks in the bosonic spectrum sub-sist above T c . Figure 5 (thin lines) shows the normal-stateDOS implied by setting D = ± W sr and x ( p , ) − W sr = −
50 meV while nothing but very weakstructures subsist in 3D, signaling the onset of scattering at ± W sr . Unfortunately it turns out that in the HTS the spin reso-nance is absent above T c —or at least below the backgroundlevel of neutron-scattering experiments. The normal stateof Bi-2223 has not been investigated by neutron scatteringso far but we may borrow information from the much stud-
PSfrag replacements t z = 0 t z = 200 meV D O S ( s t a t e s / e V / ce ll ) ω (meV) ω (meV)
000 0 11 2234 − −
100 100100
FIG. 5. Normal-state DOS. The thin lines show the T = D =
0; the thick lines show the T =
200 K DOS for D = G sr =
14 meV (see text). All other parameters are as in Fig. 1a. ied YBa Cu O + x system (Y-123). In Y-123, the normal-statespin susceptibility preserves its separable form with indepen-dent momentum and energy variations. It is still centered at ( p , p ) with a broad maximum at a characteristic temperature-dependent frequency W sf ≈ W sr . For the purpose of illus-trating the effect of a broad spin-fluctuations continuum onthe normal-state tunneling spectrum, it is sufficient to use thesame model as in the superconducting state but with the newparameter G sr =
14 meV. The resulting DOS calculated at T =
200 K is shown by the thick lines in Fig. 5. The strong-coupling signatures are almost washed out in 2D and com-pletely in 3D. This is not due to the thermal broadening ofEq. (1), not included in the DOS N ( w ) , but mostly to the in-trinsic temperature dependence of the self-energy in Eq. (2),and, to a lesser extent, to the broader spin response. Hence, ifstructures due to interaction with spin fluctuations are unlikelyto show up in the normal state of HTS, those associated withthe interaction with phonons may well be observable if thecoupling is strong enough since this coupling will not changeappreciably at T c .In the present study, we have overlooked a possible k z de-pendence of the BCS gap, retaining only the k z dependenceof the bare dispersion. A weak modulation of the BCS gapalong k z is expected in 3D systems. As shown in Ref. 36,such a modulation has the effect of cutting the logarithmic co-herence peaks on the scale of 2 D z , with D z the amplitude ofthe gap modulation. This is similar to the effect of t z on theBCS coherence peaks, which are cut on a scale correspond-ing to the gap variation along the warped 3D Fermi surface,namely, ∼ D t z / t , as seen in Fig. 3. The expected effect of D z on the scattering rate is also an additional broadening ontop of the one produced by t z , ˆ G being replaced by its k z av-erage in Eq. (2). Therefore, we expect that the gap modulationalong k z will contribute to suppress the coherence peaks andthe strong-coupling features even further with increasing t z , ascompared to the results in Fig. 1.Our results can be summarized as follows. The formationof clear strong-coupling structures in the tunneling conduc-tance requires two ingredients: (A) at least one peak in thespectrum of collective excitations and (B) at least one peakin the non-interacting or superconducting DOS. In classicalsuperconductors, (A) is provided by optical phonons and (B)is the asymmetric square-root singularity at the edge of the s -wave gap: strong-coupling features are asymmetric steps— peaks in the d I / dV curve—and dimensionality plays no bigrole because (A) and (B) are present in any dimension. Inthe normal state, there is no signature because (B) is absent.In high- T c layered superconductors, (A) is provided by thespin resonance and (B) has two sources: (B1) the logarith-mic Van Hove singularity in the bare DOS; (B2) the symmet-ric logarithmic singularities at the edge of the d -wave gap.Strong-coupling signatures appear as local minima—zeros inthe d I / dV curve—but they vanish with increasing dimen-sionality from 2D to 3D because (B1) and (B2) both get sup-pressed by the c -axis dispersion. In the normal state of two-dimensional HTS, (B2) is absent, leaving aside the questionof the pseudogap but (B1) remains and strong-coupling sig-natures are thus expected unless (A) disappears at T c . This isthe case for the spin resonance but certainly not for phonons,leaving open the possibility that phonon structures might beobservable in the normal-state tunneling spectra. IV. DOS AND EFFECTIVE DOS
The conventional theory of electron tunneling intosuperconductors leads to an equation identical to Eq. (1)for the tunneling conductance, except that the DOS N ( w ) is replaced by an “effective tunneling DOS” N T ( w ) = N ( ) Re (cid:2) | w | / p w − D ( w ) (cid:3) . N ( ) is the normal-state DOSat zero energy— N ( w ) ≡ N ( ) is assumed—and D ( w ) isthe gap function. The latter must be understood as a Fermi-surface average of weakly momentum-dependent quantities, D ( w ) = h F ( kkk , w ) / Z ( kkk , w ) i FS with F and Z the Eliashbergpairing and renormalization functions. In the notation ofEq. (2), they read Z ( kkk , w ) = − [ ˆ S ( kkk , w ) + ˆ S ( kkk , w )] / ( w ) and, for an s -wave gap D , F ( kkk , w ) = D + ˆ S ( kkk , w ) . Ina d -wave superconductor, the Fermi-surface average of thegap D kkk vanishes, and so does the average of the off-diagonalself-energy since ˆ S ( kkk , w ) (cid:181) D kkk . The effective tunnelingDOS concept is logically generalized by writing N T ( w ) = N ( ) Re (cid:10) | w | / p w − [ D kkk f ( w )] (cid:11) FS with f ( w ) = (cid:28) + ˆ S ( kkk , w ) / D kkk − [ ˆ S ( kkk , w ) + ˆ S ( kkk , w )] / ( w ) (cid:29) FS . (6)This form of N T ( w ) is an even function of w , and cannot fitthe particle-hole asymmetric spectra in HTS. Therefore, a fur-ther generalization of the effective tunneling DOS has beennecessary, namely, N T ( w ) = N ( w ) Re * | w | p w − [ D kkk f ( w )] + FS (7)which suggests that the “true” superconducting DOS can beobtained by dividing the tunneling spectrum in the supercon-ducting state by the spectrum in the normal state. Equation (7) is very convenient, but lacks a formal justi-fication. Our model offers the opportunity to investigate theusefulness of Eq. (7), by comparing numerically the actualtunneling DOS N ( w ) of Eq. (5) with the effective tunneling PSfrag replacements 2D3D t z = 0 t z = 200 meV φ ( ω ) − φ ( ω ) − D O S ( s t a t e s / e V / ce ll ) D O S ( s t a t e s / e V / ce ll ) ω (meV) ω (meV) − − − − − N ( ω ) N T ( ω ) FIG. 6. (Left panels) Real part (solid lines) and imaginary part(dashed lines) of the pairing function defined in Eq. (6) for t z = t z =
200 meV (3D). (Right panels) Comparison of the ef-fective tunneling DOS N T ( w ) of Eq. (7) with the actual DOS N ( w ) of Eq. (5). The dashed lines show N T ( w ) / N ( w ) . In all graphs, thedotted vertical lines mark the energy ± ( W sr + D ) . DOS N T ( w ) . For the practical evaluation of N T ( w ) , we de-fine the Fermi-surface average as h · · · i FS ≡ (cid:229) kkk A ( kkk , ) ( · · · ) (cid:229) kkk A ( kkk , ) (8)with A ( kkk , ) the zero-energy spectral function in the ab-sence of pairing: A ( kkk , ) = ( − / p ) Im ˆ G ( kkk , ) | D kkk = . Withthis definition, the average is performed on the renormalized Fermi surface, defined by x kkk + Re ˆ S ( kkk , ) =
0, rather than thebare Fermi surface x kkk =
0. Furthermore, each state gets cor-rectly weighted if the spectral weight is unevenly distributedalong the Fermi surface.A comparison of N ( w ) and N T ( w ) is displayed in Fig. 6,where the pairing function f ( w ) is also shown. In two dimen-sions, the real part of f ( w ) has a maximum at w = W sr + D ,where its imaginary part shows a rapid variation. This is analogous to the behavior reported in Ref. 13. The result-ing N T ( w ) also shows a behavior similar to the one foundin Ref. 13: N T ( w ) is larger than the BCS density of statesat energies smaller than W sr + D and drops below the BCSDOS at W sr + D . The actual DOS N ( w ) , however, behavesdifferently: it is smaller than the BCS DOS between the co-herence peak and some energy above the dip minimum (seealso Fig. 1 of Ref. 8). Thus, although the positions of thestrong-coupling features are identical in N T ( w ) and N ( w ) ,their shape is markedly different in 2D d -wave superconduc-tors. In 3D, the difference between N T ( w ) and N ( w ) is lesssevere than in 2D, and both curves show very weak signatures,although those in N ( w ) are slightly stronger. Finally, the N T ( w ) curves show structures which are absent in the N ( w ) curves. In 2D, a peak at w = −
16 meV = x ( p , ) appears due tothe VHS in N ( w ) ; this peak is unphysical because in the ac-tual energy spectrum, the VHS is pushed to − [ x ( p , ) + D ] / .In 3D, N T ( w ) has a structure near −
100 meV, which alsocomes from the bare DOS N ( w ) as can be seen in Fig. 1b.In the actual spectrum, this structure is suppressed due to thepersistence of a large scattering rate at energies much higherthan the threshold W sr (see Fig. 2). These problems illustratethe limitations of the simple product Ansatz Eq. (7) for ana-lyzing the tunneling spectrum of d -wave superconductors. ACKNOWLEDGMENTS
I thank John Zasadzinski for useful discussions. This workwas supported by the Swiss National Science Foundationthrough Division II and MaNEP.
APPENDIX: ANALYTICAL k z INTEGRATION
If the Nambu self-energy has no k z dependence, the k z sumin Eq. (5) can be performed analytically. This is the case in ourmodel defined in Eq. (2). Solving Dyson’s equation ˆ G ( kkk , w ) =[ ˆ G − ( kkk , w ) − ˆ S ( kkk , w )] − with ˆ G given by Eq. (3), we findˆ G ( kkk , w ) = w + i G − x kkk − ˆ S ( kkk , w ) − [ D kkk k + ˆ S ( kkk , w )] (cid:14) [ w + i G + x kkk − ˆ S ( kkk , w )] . (9)Since ˆ S does not depend on k z (although it does depend on t z ), the k z dependence only comes from x kkk in Eq. (4) and we canmake it explicit by rewritingˆ G ( kkk , w ) = z − t z cos ( k z c ) − z (cid:14) [ z + t z cos ( k z c )]= (cid:18) + zl (cid:19) h + l − t z cos ( k z c ) + (cid:18) − zl (cid:19) h − l − t z cos ( k z c ) . (10)In Eq. (10), the quantities z , z , z , z , l , and h are all functions of kkk k and w but not of k z . Explicitly, z = w + i G − x kkk k − ˆ S ( kkk k , w ) , z = w + i G + x kkk k − ˆ S ( kkk k , w ) , z = D kkk k + ˆ S ( kkk k , w ) , z = ( z + z ) / l = q z − z , and h = ( z − z ) / k z integration can then be performed by means of theidentity,˜ D t ( z ) ≡ p Z p − p dxz − t cos x = √ z − t √ z + t (11) and yields N ( w ) = N k (cid:229) kkk k (cid:0) − p (cid:1) Im (cid:26) (cid:18) + zl (cid:19) ˜ D t z ( h + l )+ (cid:18) − zl (cid:19) ˜ D t z ( h − l ) (cid:27) , (12)where N k is the number of kkk k points in the 2D Brillouin zone. A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod. Phys. ,473 (2003). J. C. Campuzano, M. Norman, and M. Randeria, in
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