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Quasiparticle density of state in d-wave superconducting superlattice systems
Physica C 390 (2003) 167–174 www.elsevier.com/locate/physc
Quasiparticle density of state in d-wave superconducting superlattice systems Bor-Luen Huang a, Chung-Yu Mou a
a,b,*
Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan b National Center for Theoretical Sciences, P.O. Box 2-131, Hsinchu, Taiwan
Received 11 July 2002; received in revised form 19 December 2002; accepted 13 January 2003
Abstract The effect of the superlattice structure along the c-axis on the single particle density of high temperature superconductors is investigated by considering superlattice systems that consists of metals/insulators and d-wave superconductors (NS/IS superlattices). It is shown that the position of quasiparticle peak is sensitive to how one models the c-axis tunneling. For the NS superlattice, the position of the quasiparticle peak can be considerably smaller than the gap value, while for the IS superlattice, the quasiparticle peak always remains at the bulk value so that the whole system behaves like a single superconductor. Furthermore, in the presence of a small current, the density of state at zero energy becomes non-zero for the NS superlattice while it sticks to zero for the IS superlattice. This results in different behaviors in the specific heat measurements, providing a possible way to differentiate between these two models. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 74.20.)z; 74.80.)g; 74.25.Fy; 74.50.+r Keywords: Superlattice; Superconductor; Density of state
1. Introduction The artificial prepared superlattice designed by conventional superconductors and normal metals have been under intense study by both theoretical and experimental works [1]. After the discovery of the layered structure of the high critical temperature superconductors (HTSC), the interest in superlattice systems has been further boosted up
*
Corresponding author. Address: Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan. E-mail address: [email protected] (C.-Y. Mou).
[2–6]. The layered structure along the c-axis and particularly the existence of closely packed CuO2 planes make HTSC a natural-occurring superconducting superlattice system. After the intense work for so many years, it now becomes clear that in the overdoped region, the normal state of high Tc superconductors is metallic like for both c-axis and in-plane directions [7]. In particular, the splitting due to hopping between adjacent CuO2 planes in a unit cell are seen directly in recent photoemission experiments [8]. While these works imply that single particle hopping definitely occurs along the c-direction, it is not clear how exactly one should model the
0921-4534/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4534(03)00698-1
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transport properties of the high Tc cuprates along the c-axis in the superconducting phase and how the superlattice structure affects the single particle transportation along the c-direction. This splitting implies that neighboring CuO2 planes in a unit cell are strongly coupled by a large hopping integral. Therefore, one can model it by a single superconducting layer, which is coupled to the superconducting layers in adjacent unit cells via an effective hopping integral teff . This is also consistent with the fact that the coherence length along the cdirection is about the width of each superconducting layer. The problem is then how one should model layers between adjacent superconducting layers. It was pointed in Ref. [3] that there exist two distinct limits, depending on the ratio teff to Tc . In the limit when teff is much smaller than Tc , the system behaves more like a superconducting superlattice. On the other hand, if teff is close to Tc in magnitude, different superconducting layers are coupled strongly and the system behaves as a single anisotropic superconductor. In this work, we shall work in the first limit. As we shall show, it turns out that even in this limit, the density of state (DOS) for single particle can still resemble that of a single bulk superconductor. This will depend on how one model layers between adjacent superconductors. To model the interlayers between the superconducting CuO2 layers, we first realize that the experimentally observed metallic behavior in the ab-direction may be entirely due to the CuO2 planes. Therefore we shall consider several concrete models. The simplest models are to model the c-axis structure as a NS or a IS superlattices as both of them can give rise metallic behaviors in cand ab-direction in the normal state. Possible modification will be considered too. For instance, the interlayer could be insulating like in ab-directions but metallic like in the c-direction. This leads us to consider a NS superlattice with a large mass anisotropy (effective mass in the c-axis effective mass in ab-directions) in the metal layers so that electrons essentially hop along c-direction in the interlayers. We shall calculate the DOS for single particles for these two models. This will be useful for specific heat measurements or transport measurements with high resistance contacts. We shall
include the d-wave nature and consider general Fermi surface topology and mismatches. Our results will be useful for artificial superlattices such as the YBa2 Cu3 O7d /PrBa2 Cu3 O7d superlattice system [5]. At phenomenological level, they also supply signatures to differentiate between different models, providing insights to how one should model superlattice structure along the caxis. For instance, our results indicate that for a NS superlattice, if the mass anisotropy is small, the superlattice structure will induce subgap structure in the DOS. For large mass anisotropy in the metal, however, the bulk d-wave-like DOS is reproduced with the quasiparticle peak shifted to a smaller value, in contrast to a IS superlattice where the quasiparticle peak remains at the gap value. The NS superlattice model provides a possible explanation of why some measurements of the gap size along the c-direction give smaller values [5,9]. Our results also indicate that one can further differentiate the NS superlattice from the IS superlattice by applying a small current along c-axis. In the former case, the DOS at zero energy becomes finite while for the later, it sticks to zero. This results in different behaviors in the specific heat measurement: c T for the NS superlattice but c T 2 for the IS superlattice. This paper is organized as follows: In Section 2, we outline our theoretical formulations and models. Numerical results are presented and discussed in Section 3. In Section 4, we summarize and conclude.
2. Theoretical formulations and models We shall start by considering a NS superlattice. The c-axis will be chosen as z-axis. Such configuration with a spherical Fermi surface and s-wave pairing was previously considered by Hahn [2] to model transport along the c-axis for YBCO. To account for the c-axis transport of high Tc cuprates, we shall extend it to include d-wave nature and use a more general Fermi surface. In particular, during our calculation, we shall also briefly address the effect due to a particular form for the c-axis hopping tc ¼ t? cos2 ð2/Þ. This particular form of hopping is suggested from the measure-
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ment of angle-resolved photoemission (ARPES) and band theoretical calculation [8,10]. Here / is the angle tan1 ðpy =px Þ. R 3 In general, the DOS gðEÞ ¼ ððd3 qÞ=ð2pÞ Þ dðE ðqÞÞ can be cast in the form Z dqx dqy oj gðEÞ ¼ ; ð2pÞ3 o ¼E where j is the Bloch wave number along c-axis and is a function of qx , qy , and E. If we further parameterize qx ¼ kF sinðhÞ cosðuÞ and qy ¼ kF sinðhÞ sinðuÞ, we get Z k 2 sinðhÞ cosðhÞdh du ; ð1Þ gðEÞ ¼ n1 F ð2pÞ3 where n1 joj=oj¼E is the effective 1D DOS for given qx and qy . For the system of superconducting superlattice, a suitable framework for calculating jðE; qx ; qy Þ is the Bogoliubov de-Genne (BdG) equation which can be written as b uðrÞ uðrÞ H D ¼ E : ð2Þ b vðrÞ vðrÞ D H b ¼ nq ðlF V Þ with nq being the kinetic Here H energy given by ( 2 h h2 2 2 2 in S; 0 ðqx þ qy Þ þ 2m qz S nq ¼ 2m ð3Þ h2 2 2 h2 2 ðqx þ qy Þ þ 2mN qz in N: 2m V is an effective potential which will take a finite value only when we consider the IS superlattice later. D ¼ D0 cosð2uÞ is the energy gap which will be set to be zero for non-superconducting cells. The neglect of proximity effect is a good approximation when one considers the effect of Fermi energy mismatch [11]. Since the system is translationally invariant in x- and y-direction, qx and qy are conserved so that we can write uðrÞ ¼ uðzÞ expiðqx xþqy yÞ and vðrÞ ¼ vðzÞ expiðqx xþqy yÞ . Eq. (2) then reduces to one dimensional equations with the boundary conditions [12] uN ð0Þ ¼ uS ð0Þ; vN ð0Þ ¼ vS ð0Þ; mN 0 mN 0 u0N ð0Þ ¼ u ð0Þ; v0N ð0Þ ¼ v ð0Þ; mS S mS S uS ðbÞ ¼ kuN ðaÞ; vS ðbÞ ¼ kvN ðaÞ; mN 0 mN 0 u ðbÞ ¼ ku0N ðaÞ; v ðbÞ ¼ kv0N ðaÞ; mS S mS S
ð4Þ
169
where a is the width of the non-superconducting cell (N cell) while b is that for the superconducting cells (S cell). The parameter k can be expressed in terms of the Bloch wave vector: expðijdÞ with d a þ b. Our goal is to solve j. This is proceeded by first writing uðzÞ and vðzÞ in terms of suitable combinations of plane waves: expðik Þ or expðip Þ with k (p ) being the quasiparticle (þ)/ quasihole ()) momentum in the N cell (S cell). Using the boundary conditions to eliminate the combination coefficients [12], we find that k satisfies D0 þ D1 k þ D2 k2 þ D3 k3 þ D4 k4 ¼ 0;
ð5Þ
where Di are all real. The fact that jkj ¼ 1 further implies D0 ¼ D4 and D1 ¼ D3 . Therefore, j is determined by the ratios D1 =D0 and D2 =D0 0
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 2 1 D1 D1 D2 cosðjdÞ ¼ @ 4 þ 8A ; 4 D0 D0 D0 ð6Þ where indicates electron-like and hole-like quasiparticles respectively. Note that in the above derivation, we have made use of the Andreev approximation in which excitation energies are assumed to be small in comparison to the Fermi energy lF [12]. For a fixed / and h, we find D1 ¼ 4½cosð2F AX Þ cosð2F cBXm ÞC1 D0 sinð2F AX Þ sinð2F cBXm ÞC2 ; D2 ¼ 4C22 þ ðg2 1Þ½1 þ cosð4F AX Þ D0
ð7Þ
þ cosð4F cBXm Þ 2C3 þ ðg2 þ 1Þ cosð4F AX Þ cosð4F cBXm Þ 2g sinð4F AX Þ sinð4F cBXm Þ: Here notations are defined as: X ¼ cosðhÞ, Xm ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 ððm=m0 Þ sin2 hÞ, g ððXm =cX Þ þ ðcX =Xm ÞÞ=2, F ffiffiffiffiffiffiffiffiffiffiffiffiffiffi lF =D ffi 0 , A a=nN , B b=nN and c p mS =mN . We have made use of the length scale pffiffiffiffiffiffiffiffiffiffi nN h2 kF =ðD0 mmN Þ. The functions Ci are given by
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c/ B cos Xm A c/ B g sin sin ; / X Xm A c/ B C2 ¼ g cos cos X Xm ð8Þ A c/ B sin sin ; / X Xm c B / C3 ¼ cos2 ½1 þ cosð4F AX Þ Xm A þ cos2 ½1 þ cosð4F cBXm Þ X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for 2 P w2 , where E=D0 , / 2 w2 and w ¼ D=D0 . While for 2 < w2 , one simply replaces pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos (sin) by cosh (sinh), and / by w2 2 . Eqs. (6)–(8) constitute a generalization of previous results obtain in Ref. [12]. The mismatch in Fermi wave vectors is included explicitly via the mass anisotropy parameters: m=m0 and mS =mN . It is easy to see that when m ¼ m0 and mS ¼ mN , one has g ¼ c ¼ 1 and Xm ¼ X and thus C1 ¼ C2 CðÞ. One then recovers previous results [2,12] C1 ¼ cos
A X
cos ½ðj kF X Þd ¼ CðÞ:
ð9Þ
In the next section we shall include more features such as applying a small current in the current model.
3. Numerical analysis The NS superlattice. Eq. (6) represents the function jðqx ; qy ; EÞ which has to be integrated to get the DOS. This is done via numerical integration. We start by first discussing symmetry properties of the superconducting superlattice. First, the BdG equations have the particle–hole symmetry [13]. That is, if ðu; vÞT is the eigenvector for T energy E, ðv ; u Þ is the eigenvector for the eigenvalue E. Therefore, the DOS is an even function of E. This, however, has nothing to do with the relation between the DOS of electron-like and hole-like quasiparticles. In general, there is no symmetry between electron-like and hole-like quasiparticles. Only in the homogeneous case
(m ¼ m0 and mN ¼ mS ), we see that in Eq. (9), sign does not change the dispersion relation. In fact, both electron-like and hole-like quasiparticles have the same 1D DOS oC 1 1 ð10Þ n1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi : d 1 C o In this equation, C has to be smaller than one. Suppose that C ¼ 1 happens at ¼ 0 , we obtain when ! þ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C0 ðÞj¼0 1 ; ð11Þ n1 pffiffiffiffiffiffiffiffiffiffiffiffi d 0 2 which displays a BCS-like square rootpsingularity ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for a fixed /. When A ¼ 0, CðÞ ¼ cosð 2 w2 B= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X Þ for 2 P w2 and CðÞ ¼ coshð 2 w2 B=X Þ for 2 < w2 . Hence, for given h and /, we obtain 1 for 2 P cos2 2/; n1 ¼ nN X / ð12Þ 0 for 0 < 2 < cos2 2/: Note that this is not exactly the same as the bulk swave DOS due to the presence of the factor 1=nN X which describes the number of times for folding into the reduced Brillouin zone. After integration over / and h, it gives rise to the bulk d-wave DOS with the quasiparticle peak at ¼ 1. Once A becomes non-zero, the system contains non-superpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi conducting layers so that coshð 2 w2 B=X Þ is reduced by cosðA=X Þ. For small A, it shifts from 0 ¼ jwj to 0 < jwj. Therefore, the DOS along each direction / behaves as bulk s-wave like with reduced gap value. This moves the quasiparticle peak into the subgap region as demonstrated in Fig. 1. Note that there are numerical noises between the peak and zero energy, caused by the singularity of s-wave like behavior in n1 . The physics for reduced gap can be understood from different point of view. Consider an isolated SNS structure. The quasiparticle in the N cell can form so-called Andreev bound states whose energies can be evaluated via semi-classical quantization condition (the total phase for the quasiparticle moving in one round in the N cell is 2np) [14]: 2 tan1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! A w2 2 d/ þ 2 ¼ 2np: X
ð13Þ
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Large mass anisotropy limit. As we mention, another possible way to model the c-axis is to force that electrons can only hop along the c-axis between CuO2 layers. This can be implemented by setting m=m0 to be very large. As a result, h must be small enough so that Xm would not become an imaginary number. Therefore in the limit of infinite m=m0 , we can simply set X ¼ 1 and g ¼ ðc þ ð1=cÞÞ=2 The DOS then becomes Z 2 kF d/ oj gðEÞ ¼ : 3 ð2pÞ oE X ¼1
Fig. 1. 3D DOS for A ¼ 0:0, 0.1, 0.2 and B ¼ 1. The inset is a similar plot with reduced B: A ¼ 0:1, B ¼ 0:9, which is almost the same as that for A ¼ 0:1, B ¼ 1:0.
Here the first term is the phase change for the process when a quasiparticle is Andreev reflected into a quasihole or vice versa, d/ is the phase difference between two superconductors and 2ðA=X Þ is the phase accumulation for traveling within the N cell. Obviously, it permits solutions with < 1. These solutions represents localized states in the N cell. When one assembles many SNS structure together, each Andreev bound state percolates and become a band. Since their energies are less than the bulk gap value, they contribute significant portion in the DOS, resulting in moving the quasiparticle peak into the subgap region.
When mS ¼ mN and thus c ¼ 1, the system behaves essentially the same as a single 2D-like superconductor. When c is not equal to one (i.e., hopping amplitudes are different in the S cell and the N cell), we have to use Eq. (6). In this case, the expression ðD1 =D20 Þ 4ðD2 =D0 Þ þ 8 has to be positive in order that j is real. This, in turn, implies that the DOS will be different for electron-like and hole-like quasiparticles. In Fig. 2, we show our the numerical results for different c. Here the parameters are chosen in accordance with the YBCO samples: nN nab 1:5 nm, a 0:85 nm, and b 0:38 nm. In our notations, these correspond to A ¼ 0:57 and B ¼ 0:21. We see that electron-like and hole-like quasiparticles have different DOS. At low energy section, the DOS for the electron-like quasiparticle behaves similar to the bulk d-wave DOS, except that the quasiparticle peak moves to smaller energy. If we take D0 20–30 meV, from the dash line (mS :mN ¼ 1:2 for YBCO) in Fig. 2,
Fig. 2. The DOS for different c are shown for electron-like (left) and hole-like (right) quasiparticles in the limit of infinite m=m0 . The parameters are chosen to be close to that for YBCO are A ¼ 0:57, B ¼ 0:21 and F ¼ 10.
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the quasiparticle peak of DOS is around 4–6 meV, which is close to what experiments have seen, Dc ¼ 5 0:5 meV [5]. Apparently, reducing c pushes the quasiparticle peak further into the subgap region. This can be understood by using Eq. (13) as follows: Reduction of c is equivalent to increase mN , hence nN is reduced. For a fixed a, it thus increases A. The third term of Eq. (13) then implies the reduction of and thus the position of the peak. Another feature requiring explanation pffiffiffiffiffiffiffi is the appearance of the second peak (c ¼ 0:9) in the hole-like spectrum. For this purpose, we plot out n1 versus for different /Õs. This is shown in Fig. 3. Note that we have used the negative energy axis to represent the spectrum for the hole-like quasiparticle. One can see that there appears to have two kinds of gaps. As / approaches p=4, those gaps that get closed are the superconducting gaps (they may be repeated due to the superlattice structure), while if their gap sizes remain finite, they are the band gaps. We see that the gap near ¼ 0 is the superconducting gap, while the second peak is seen to be due to the edge of the band gap around 0:4. The positions of the superconducting gaps are the same for both electron-like and hole-like
quasiparticles becausethey are created by destroying the same Cooper pairs. The positions for the band gaps, however, need not to be the same as they see different effective potentials due to the mass anisotropy. IS superlattice. We now model interlayers between superconducting layers by insulators. For this purpose, a large potential VI is introduced to force metal cells become insulating cells (I cell). To simplify our discussion, the homogeneous case which m ¼ m0 ¼ mN ¼ mS is considered. Following the same procedure that leads to Eq. (6), we obtain a similar expression for j and Di except now the wave number in the I cell is purely imaginary. Fig. 4 shows our numerical results. The numerical noise becomes more severe due to the exponential decay factors in the I cell. We see that except that values of the DOS get reduced, both the electron-like and hole-like particles are bulk d-wave like with peaks right at the bulk gap value. This can be easily understood by noting that there is no Andreev bound state that could arise in the I cell, hence no states can survive below the gap for any fixed /. Current-carrying state. Here we propose another way to differentiate the NS and IS superlattices by
Fig. 3. The 1D DOS n1 for different /Õs (increasing from top to bottom: 0–0:25p). pThe ffiffiffiffiffiffiffi parameters are A ¼ 0:57, B ¼ 0:21, F ¼ 10, and c ¼ 0:9. The negative energy represents the spectrum for the hole-like quasiparticles.
Fig. 4. The DOS for the IS superlattice. Here A ¼ 0:57, B ¼ 0:21, and VI is set to 1:2F with F ¼ 20. The solid line is electron-like, while the dash line is hole-like. For comparison, we also show the DOS of bulk d-wave superconductor (open circles).
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applying a small current. A similar problem was studied by Tanaka [15] in which the superconductors are s-wave. Here we shall extend it to include the d-wave nature and effect of Fermisurface mismatch via including mass anisotropy. We start from the simplest case: m ¼ m0 ¼ mN ¼ mS . The current can be imposed by forcing a phase difference / between nearest neighbor S cells: Dðz þ dÞ ¼ DðzÞei/ . Note that in principle, to conserve the current, there will be supercurrents in each S cell so that the phase would not be a constant across each S cell. We will, however, bypass this problem by focusing on the supercurrent (/=a) and assuming the current is conserved. This will be correct in the limit of small width (b) for the S cell. The phase difference in adjacent S cells also implies a new Bloch condition: uS ðz þ dÞ ¼ uS ðzÞk expði/=2Þ and vS ðz þ dÞ ¼ vS ðzÞk expði/=2Þ. This, however, can be then solved by re-absorbing the phase into the wave functions of the quasiparticles. In the N cell, we redefine the new wave functions as: uN ðzÞ ! uN ðzÞ expði/z= 2aÞ and vN ðzÞ ! vN ðzÞ expði/z=2aÞ. As a result, the wave vectors of electron-like and hole-like quasiparticles are shifted as follows: / k~ ¼ k : 2a
173
Fig. 5. The DOS for a NS superlattice carrying current along c-axis direction. The mass ratios are m=m0 ¼ 1 and c ¼ mpNffiffiffiffiffiffi =mffi S (except for the first one, c ¼ 1:0, the remainings have c ¼ 0:5). Other parameters are A ¼ 0:57, B ¼ 0:21, and ¼ 10.
ð14Þ
Eq. (9) is the same except that C has to be replaced by ! A / / B C ¼ cos cos þ 2 X X ! A / / B sin sin þ : ð15Þ 2 / X X , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos (sin) are replaced by Here again for 2 < w2p cosh (sinh), and / by w2 2 . In the large mass anisotropy limit, the result is also the same as before with the same replacement: A ! A þ ð/=2Þ. In Fig. 5, we shows the DOS with current using the material parameters for YBCO. We see that the mass anisotropy due to c moves the peak of DOS into the subgap region, but the current further moves the peak into lower energy region. Most importantly, the DOS has a finite value at zero energy. These behaviors can be
Fig. 6. Illustration of how the shift of the DOS is induced. The dash line is the energy profile for the case without applying a current, while the solid illustrates that when a current is present, the quasiparticle effectively sees higher energy channels in the S cell.
understood via the illustration of Fig. 6. Due to the shift of momentum, the energies of quasiparticle in the metal increase for a given k : E ¼ 2 h2 kk2 =2m þ h2 ðk þ /=2aÞ =2m l. As a result, the energies of the tunneling channel in the S cell seen by the quasiparticles also increase effectively. This results in the shift of the DOS observed in Fig. 5. The above argument relies heavily on the concept of Andreev bound states. Therefore, for the IS superlattice, we expect no shift happens (except possible small shift due to existence of small supercurrents in the S cell), hence the DOS for the IS superlattice remains
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bulk-like with vanishing DOS at zero energy. This results in different behaviors in the specific heat measurements, providing a possible way to differentiate between NS and IS superlattices.
Acknowledgements This work is supported by the NSC of Taiwan. We thank Profs. H.H. Lin, T.M. Hong and C.S. Chu for useful discussions.
4. Conclusions References In summary, we have investigated the effect of the superlattice structure along the c-axis on the single particle density of HTSC by considering the NS and IS superlattices. We find that the position of quasiparticle peak depends on how one models the c-axis tunneling. For the NS superlattice, the position of the quasiparticle peak can be considerably smaller than the gap value, while for the IS superlattice, the quasiparticle peak always remains at the bulk value so that the whole system behaves like a single superconductor. As evidences are emerging that at high dopings the cuprates does behave like 3D metals, these results are illuminating. In particular, they imply that in the overdoped region, the gap value extracted from measurements using the DOS might not be the real gap value of the CuO2 plane. Since the real gap value also decreases as doping increases, the observed reduction of the gap value in the overdoped region might be a combined result of both effects. The NS and IS superlattice models can be further differentiated by applying a small current along c-axis. This would result in different dependences on temperature in the specific heat measurements, proving a possible signature to look for in experiments.
[1] E.L. Ivchenko, G.E. Pikus, Superlattices and Other Heterostructures-Symmetry and Optical Phenomena, second ed., Springer, Germany, 1997. [2] A. Hahn, Physica B 165&166 (1990) 1065. [3] L.N. Bulaevskii, R. Rammal, Phys. Rev. B 44 (1991) 9768. [4] A. Crisan, G. Balestrino, S. Lavanga, P.G. Medaglia, E. Milani, Physcia C 313 (1999) 70. [5] J.C. Martinez et al., cond-mat/002111; J.C. Martinez et al., Phys. Rev. B 61 (2000) 9162; M. Varela et al., Phys. Rev. Lett. 83 (1999) 3936; D.H. Lowndes et al., Phys. Rev. Lett. 65 (1990) 1160. [6] S. Komiya, Y. Ando, X.F. Sun, A.N. Lavrov, cond-mat/ 0203107. [7] For a review, see S.L. Cooper, K.E. Gray, in: D.M. Ginsberg (Ed.), Physical Properties of High Temperature Superconductors IV, World Scientific, Singapore, 1994. [8] For recent progress, see Y.-D. Chuang et al., cont-mat/ 0107002. [9] H.-W. Cheng, Ph.D. thesis, National Tsing Hua University, 2000. [10] See also T. Xiang, W.N. Hardy, Phys. Rev. B 63 (2001) 024560. [11] H. Plehn, O.-J. Wacker, R. K€ ummel, Phys. Rev. B 49 (1994) 12140. [12] A.P. Van Gelder, Phys. Rev. 181 (1969) 787. [13] P.G. de Gennes, Superconductivity of Metals and Alloys, Benjamin, New York, 1966. [14] S. Kashiwaya, Y. Tanaka, M. Koyanagi, K. Kajimura, Phys. Rev. B 53 (1996) 2667. [15] Y. Tanaka, M. Tsukada, Phys. Rev. B 44 (1991) 7578.
Quasiparticle density of state in d-wave superconducting superlattice systems Bor-Luen Huang a, Chung-Yu Mou a
a,b,*
Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan b National Center for Theoretical Sciences, P.O. Box 2-131, Hsinchu, Taiwan
Received 11 July 2002; received in revised form 19 December 2002; accepted 13 January 2003
Abstract The effect of the superlattice structure along the c-axis on the single particle density of high temperature superconductors is investigated by considering superlattice systems that consists of metals/insulators and d-wave superconductors (NS/IS superlattices). It is shown that the position of quasiparticle peak is sensitive to how one models the c-axis tunneling. For the NS superlattice, the position of the quasiparticle peak can be considerably smaller than the gap value, while for the IS superlattice, the quasiparticle peak always remains at the bulk value so that the whole system behaves like a single superconductor. Furthermore, in the presence of a small current, the density of state at zero energy becomes non-zero for the NS superlattice while it sticks to zero for the IS superlattice. This results in different behaviors in the specific heat measurements, providing a possible way to differentiate between these two models. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 74.20.)z; 74.80.)g; 74.25.Fy; 74.50.+r Keywords: Superlattice; Superconductor; Density of state
1. Introduction The artificial prepared superlattice designed by conventional superconductors and normal metals have been under intense study by both theoretical and experimental works [1]. After the discovery of the layered structure of the high critical temperature superconductors (HTSC), the interest in superlattice systems has been further boosted up
*
Corresponding author. Address: Department of Physics, National Tsing Hua University, Hsinchu 30043, Taiwan. E-mail address: [email protected] (C.-Y. Mou).
[2–6]. The layered structure along the c-axis and particularly the existence of closely packed CuO2 planes make HTSC a natural-occurring superconducting superlattice system. After the intense work for so many years, it now becomes clear that in the overdoped region, the normal state of high Tc superconductors is metallic like for both c-axis and in-plane directions [7]. In particular, the splitting due to hopping between adjacent CuO2 planes in a unit cell are seen directly in recent photoemission experiments [8]. While these works imply that single particle hopping definitely occurs along the c-direction, it is not clear how exactly one should model the
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transport properties of the high Tc cuprates along the c-axis in the superconducting phase and how the superlattice structure affects the single particle transportation along the c-direction. This splitting implies that neighboring CuO2 planes in a unit cell are strongly coupled by a large hopping integral. Therefore, one can model it by a single superconducting layer, which is coupled to the superconducting layers in adjacent unit cells via an effective hopping integral teff . This is also consistent with the fact that the coherence length along the cdirection is about the width of each superconducting layer. The problem is then how one should model layers between adjacent superconducting layers. It was pointed in Ref. [3] that there exist two distinct limits, depending on the ratio teff to Tc . In the limit when teff is much smaller than Tc , the system behaves more like a superconducting superlattice. On the other hand, if teff is close to Tc in magnitude, different superconducting layers are coupled strongly and the system behaves as a single anisotropic superconductor. In this work, we shall work in the first limit. As we shall show, it turns out that even in this limit, the density of state (DOS) for single particle can still resemble that of a single bulk superconductor. This will depend on how one model layers between adjacent superconductors. To model the interlayers between the superconducting CuO2 layers, we first realize that the experimentally observed metallic behavior in the ab-direction may be entirely due to the CuO2 planes. Therefore we shall consider several concrete models. The simplest models are to model the c-axis structure as a NS or a IS superlattices as both of them can give rise metallic behaviors in cand ab-direction in the normal state. Possible modification will be considered too. For instance, the interlayer could be insulating like in ab-directions but metallic like in the c-direction. This leads us to consider a NS superlattice with a large mass anisotropy (effective mass in the c-axis effective mass in ab-directions) in the metal layers so that electrons essentially hop along c-direction in the interlayers. We shall calculate the DOS for single particles for these two models. This will be useful for specific heat measurements or transport measurements with high resistance contacts. We shall
include the d-wave nature and consider general Fermi surface topology and mismatches. Our results will be useful for artificial superlattices such as the YBa2 Cu3 O7d /PrBa2 Cu3 O7d superlattice system [5]. At phenomenological level, they also supply signatures to differentiate between different models, providing insights to how one should model superlattice structure along the caxis. For instance, our results indicate that for a NS superlattice, if the mass anisotropy is small, the superlattice structure will induce subgap structure in the DOS. For large mass anisotropy in the metal, however, the bulk d-wave-like DOS is reproduced with the quasiparticle peak shifted to a smaller value, in contrast to a IS superlattice where the quasiparticle peak remains at the gap value. The NS superlattice model provides a possible explanation of why some measurements of the gap size along the c-direction give smaller values [5,9]. Our results also indicate that one can further differentiate the NS superlattice from the IS superlattice by applying a small current along c-axis. In the former case, the DOS at zero energy becomes finite while for the later, it sticks to zero. This results in different behaviors in the specific heat measurement: c T for the NS superlattice but c T 2 for the IS superlattice. This paper is organized as follows: In Section 2, we outline our theoretical formulations and models. Numerical results are presented and discussed in Section 3. In Section 4, we summarize and conclude.
2. Theoretical formulations and models We shall start by considering a NS superlattice. The c-axis will be chosen as z-axis. Such configuration with a spherical Fermi surface and s-wave pairing was previously considered by Hahn [2] to model transport along the c-axis for YBCO. To account for the c-axis transport of high Tc cuprates, we shall extend it to include d-wave nature and use a more general Fermi surface. In particular, during our calculation, we shall also briefly address the effect due to a particular form for the c-axis hopping tc ¼ t? cos2 ð2/Þ. This particular form of hopping is suggested from the measure-
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ment of angle-resolved photoemission (ARPES) and band theoretical calculation [8,10]. Here / is the angle tan1 ðpy =px Þ. R 3 In general, the DOS gðEÞ ¼ ððd3 qÞ=ð2pÞ Þ dðE ðqÞÞ can be cast in the form Z dqx dqy oj gðEÞ ¼ ; ð2pÞ3 o ¼E where j is the Bloch wave number along c-axis and is a function of qx , qy , and E. If we further parameterize qx ¼ kF sinðhÞ cosðuÞ and qy ¼ kF sinðhÞ sinðuÞ, we get Z k 2 sinðhÞ cosðhÞdh du ; ð1Þ gðEÞ ¼ n1 F ð2pÞ3 where n1 joj=oj¼E is the effective 1D DOS for given qx and qy . For the system of superconducting superlattice, a suitable framework for calculating jðE; qx ; qy Þ is the Bogoliubov de-Genne (BdG) equation which can be written as b uðrÞ uðrÞ H D ¼ E : ð2Þ b vðrÞ vðrÞ D H b ¼ nq ðlF V Þ with nq being the kinetic Here H energy given by ( 2 h h2 2 2 2 in S; 0 ðqx þ qy Þ þ 2m qz S nq ¼ 2m ð3Þ h2 2 2 h2 2 ðqx þ qy Þ þ 2mN qz in N: 2m V is an effective potential which will take a finite value only when we consider the IS superlattice later. D ¼ D0 cosð2uÞ is the energy gap which will be set to be zero for non-superconducting cells. The neglect of proximity effect is a good approximation when one considers the effect of Fermi energy mismatch [11]. Since the system is translationally invariant in x- and y-direction, qx and qy are conserved so that we can write uðrÞ ¼ uðzÞ expiðqx xþqy yÞ and vðrÞ ¼ vðzÞ expiðqx xþqy yÞ . Eq. (2) then reduces to one dimensional equations with the boundary conditions [12] uN ð0Þ ¼ uS ð0Þ; vN ð0Þ ¼ vS ð0Þ; mN 0 mN 0 u0N ð0Þ ¼ u ð0Þ; v0N ð0Þ ¼ v ð0Þ; mS S mS S uS ðbÞ ¼ kuN ðaÞ; vS ðbÞ ¼ kvN ðaÞ; mN 0 mN 0 u ðbÞ ¼ ku0N ðaÞ; v ðbÞ ¼ kv0N ðaÞ; mS S mS S
ð4Þ
169
where a is the width of the non-superconducting cell (N cell) while b is that for the superconducting cells (S cell). The parameter k can be expressed in terms of the Bloch wave vector: expðijdÞ with d a þ b. Our goal is to solve j. This is proceeded by first writing uðzÞ and vðzÞ in terms of suitable combinations of plane waves: expðik Þ or expðip Þ with k (p ) being the quasiparticle (þ)/ quasihole ()) momentum in the N cell (S cell). Using the boundary conditions to eliminate the combination coefficients [12], we find that k satisfies D0 þ D1 k þ D2 k2 þ D3 k3 þ D4 k4 ¼ 0;
ð5Þ
where Di are all real. The fact that jkj ¼ 1 further implies D0 ¼ D4 and D1 ¼ D3 . Therefore, j is determined by the ratios D1 =D0 and D2 =D0 0
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 2 1 D1 D1 D2 cosðjdÞ ¼ @ 4 þ 8A ; 4 D0 D0 D0 ð6Þ where indicates electron-like and hole-like quasiparticles respectively. Note that in the above derivation, we have made use of the Andreev approximation in which excitation energies are assumed to be small in comparison to the Fermi energy lF [12]. For a fixed / and h, we find D1 ¼ 4½cosð2F AX Þ cosð2F cBXm ÞC1 D0 sinð2F AX Þ sinð2F cBXm ÞC2 ; D2 ¼ 4C22 þ ðg2 1Þ½1 þ cosð4F AX Þ D0
ð7Þ
þ cosð4F cBXm Þ 2C3 þ ðg2 þ 1Þ cosð4F AX Þ cosð4F cBXm Þ 2g sinð4F AX Þ sinð4F cBXm Þ: Here notations are defined as: X ¼ cosðhÞ, Xm ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q 1 ððm=m0 Þ sin2 hÞ, g ððXm =cX Þ þ ðcX =Xm ÞÞ=2, F ffiffiffiffiffiffiffiffiffiffiffiffiffiffi lF =D ffi 0 , A a=nN , B b=nN and c p mS =mN . We have made use of the length scale pffiffiffiffiffiffiffiffiffiffi nN h2 kF =ðD0 mmN Þ. The functions Ci are given by
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c/ B cos Xm A c/ B g sin sin ; / X Xm A c/ B C2 ¼ g cos cos X Xm ð8Þ A c/ B sin sin ; / X Xm c B / C3 ¼ cos2 ½1 þ cosð4F AX Þ Xm A þ cos2 ½1 þ cosð4F cBXm Þ X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for 2 P w2 , where E=D0 , / 2 w2 and w ¼ D=D0 . While for 2 < w2 , one simply replaces pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos (sin) by cosh (sinh), and / by w2 2 . Eqs. (6)–(8) constitute a generalization of previous results obtain in Ref. [12]. The mismatch in Fermi wave vectors is included explicitly via the mass anisotropy parameters: m=m0 and mS =mN . It is easy to see that when m ¼ m0 and mS ¼ mN , one has g ¼ c ¼ 1 and Xm ¼ X and thus C1 ¼ C2 CðÞ. One then recovers previous results [2,12] C1 ¼ cos
A X
cos ½ðj kF X Þd ¼ CðÞ:
ð9Þ
In the next section we shall include more features such as applying a small current in the current model.
3. Numerical analysis The NS superlattice. Eq. (6) represents the function jðqx ; qy ; EÞ which has to be integrated to get the DOS. This is done via numerical integration. We start by first discussing symmetry properties of the superconducting superlattice. First, the BdG equations have the particle–hole symmetry [13]. That is, if ðu; vÞT is the eigenvector for T energy E, ðv ; u Þ is the eigenvector for the eigenvalue E. Therefore, the DOS is an even function of E. This, however, has nothing to do with the relation between the DOS of electron-like and hole-like quasiparticles. In general, there is no symmetry between electron-like and hole-like quasiparticles. Only in the homogeneous case
(m ¼ m0 and mN ¼ mS ), we see that in Eq. (9), sign does not change the dispersion relation. In fact, both electron-like and hole-like quasiparticles have the same 1D DOS oC 1 1 ð10Þ n1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi : d 1 C o In this equation, C has to be smaller than one. Suppose that C ¼ 1 happens at ¼ 0 , we obtain when ! þ 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C0 ðÞj¼0 1 ; ð11Þ n1 pffiffiffiffiffiffiffiffiffiffiffiffi d 0 2 which displays a BCS-like square rootpsingularity ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi for a fixed /. When A ¼ 0, CðÞ ¼ cosð 2 w2 B= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X Þ for 2 P w2 and CðÞ ¼ coshð 2 w2 B=X Þ for 2 < w2 . Hence, for given h and /, we obtain 1 for 2 P cos2 2/; n1 ¼ nN X / ð12Þ 0 for 0 < 2 < cos2 2/: Note that this is not exactly the same as the bulk swave DOS due to the presence of the factor 1=nN X which describes the number of times for folding into the reduced Brillouin zone. After integration over / and h, it gives rise to the bulk d-wave DOS with the quasiparticle peak at ¼ 1. Once A becomes non-zero, the system contains non-superpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi conducting layers so that coshð 2 w2 B=X Þ is reduced by cosðA=X Þ. For small A, it shifts from 0 ¼ jwj to 0 < jwj. Therefore, the DOS along each direction / behaves as bulk s-wave like with reduced gap value. This moves the quasiparticle peak into the subgap region as demonstrated in Fig. 1. Note that there are numerical noises between the peak and zero energy, caused by the singularity of s-wave like behavior in n1 . The physics for reduced gap can be understood from different point of view. Consider an isolated SNS structure. The quasiparticle in the N cell can form so-called Andreev bound states whose energies can be evaluated via semi-classical quantization condition (the total phase for the quasiparticle moving in one round in the N cell is 2np) [14]: 2 tan1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! A w2 2 d/ þ 2 ¼ 2np: X
ð13Þ
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Large mass anisotropy limit. As we mention, another possible way to model the c-axis is to force that electrons can only hop along the c-axis between CuO2 layers. This can be implemented by setting m=m0 to be very large. As a result, h must be small enough so that Xm would not become an imaginary number. Therefore in the limit of infinite m=m0 , we can simply set X ¼ 1 and g ¼ ðc þ ð1=cÞÞ=2 The DOS then becomes Z 2 kF d/ oj gðEÞ ¼ : 3 ð2pÞ oE X ¼1
Fig. 1. 3D DOS for A ¼ 0:0, 0.1, 0.2 and B ¼ 1. The inset is a similar plot with reduced B: A ¼ 0:1, B ¼ 0:9, which is almost the same as that for A ¼ 0:1, B ¼ 1:0.
Here the first term is the phase change for the process when a quasiparticle is Andreev reflected into a quasihole or vice versa, d/ is the phase difference between two superconductors and 2ðA=X Þ is the phase accumulation for traveling within the N cell. Obviously, it permits solutions with < 1. These solutions represents localized states in the N cell. When one assembles many SNS structure together, each Andreev bound state percolates and become a band. Since their energies are less than the bulk gap value, they contribute significant portion in the DOS, resulting in moving the quasiparticle peak into the subgap region.
When mS ¼ mN and thus c ¼ 1, the system behaves essentially the same as a single 2D-like superconductor. When c is not equal to one (i.e., hopping amplitudes are different in the S cell and the N cell), we have to use Eq. (6). In this case, the expression ðD1 =D20 Þ 4ðD2 =D0 Þ þ 8 has to be positive in order that j is real. This, in turn, implies that the DOS will be different for electron-like and hole-like quasiparticles. In Fig. 2, we show our the numerical results for different c. Here the parameters are chosen in accordance with the YBCO samples: nN nab 1:5 nm, a 0:85 nm, and b 0:38 nm. In our notations, these correspond to A ¼ 0:57 and B ¼ 0:21. We see that electron-like and hole-like quasiparticles have different DOS. At low energy section, the DOS for the electron-like quasiparticle behaves similar to the bulk d-wave DOS, except that the quasiparticle peak moves to smaller energy. If we take D0 20–30 meV, from the dash line (mS :mN ¼ 1:2 for YBCO) in Fig. 2,
Fig. 2. The DOS for different c are shown for electron-like (left) and hole-like (right) quasiparticles in the limit of infinite m=m0 . The parameters are chosen to be close to that for YBCO are A ¼ 0:57, B ¼ 0:21 and F ¼ 10.
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the quasiparticle peak of DOS is around 4–6 meV, which is close to what experiments have seen, Dc ¼ 5 0:5 meV [5]. Apparently, reducing c pushes the quasiparticle peak further into the subgap region. This can be understood by using Eq. (13) as follows: Reduction of c is equivalent to increase mN , hence nN is reduced. For a fixed a, it thus increases A. The third term of Eq. (13) then implies the reduction of and thus the position of the peak. Another feature requiring explanation pffiffiffiffiffiffiffi is the appearance of the second peak (c ¼ 0:9) in the hole-like spectrum. For this purpose, we plot out n1 versus for different /Õs. This is shown in Fig. 3. Note that we have used the negative energy axis to represent the spectrum for the hole-like quasiparticle. One can see that there appears to have two kinds of gaps. As / approaches p=4, those gaps that get closed are the superconducting gaps (they may be repeated due to the superlattice structure), while if their gap sizes remain finite, they are the band gaps. We see that the gap near ¼ 0 is the superconducting gap, while the second peak is seen to be due to the edge of the band gap around 0:4. The positions of the superconducting gaps are the same for both electron-like and hole-like
quasiparticles becausethey are created by destroying the same Cooper pairs. The positions for the band gaps, however, need not to be the same as they see different effective potentials due to the mass anisotropy. IS superlattice. We now model interlayers between superconducting layers by insulators. For this purpose, a large potential VI is introduced to force metal cells become insulating cells (I cell). To simplify our discussion, the homogeneous case which m ¼ m0 ¼ mN ¼ mS is considered. Following the same procedure that leads to Eq. (6), we obtain a similar expression for j and Di except now the wave number in the I cell is purely imaginary. Fig. 4 shows our numerical results. The numerical noise becomes more severe due to the exponential decay factors in the I cell. We see that except that values of the DOS get reduced, both the electron-like and hole-like particles are bulk d-wave like with peaks right at the bulk gap value. This can be easily understood by noting that there is no Andreev bound state that could arise in the I cell, hence no states can survive below the gap for any fixed /. Current-carrying state. Here we propose another way to differentiate the NS and IS superlattices by
Fig. 3. The 1D DOS n1 for different /Õs (increasing from top to bottom: 0–0:25p). pThe ffiffiffiffiffiffiffi parameters are A ¼ 0:57, B ¼ 0:21, F ¼ 10, and c ¼ 0:9. The negative energy represents the spectrum for the hole-like quasiparticles.
Fig. 4. The DOS for the IS superlattice. Here A ¼ 0:57, B ¼ 0:21, and VI is set to 1:2F with F ¼ 20. The solid line is electron-like, while the dash line is hole-like. For comparison, we also show the DOS of bulk d-wave superconductor (open circles).
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applying a small current. A similar problem was studied by Tanaka [15] in which the superconductors are s-wave. Here we shall extend it to include the d-wave nature and effect of Fermisurface mismatch via including mass anisotropy. We start from the simplest case: m ¼ m0 ¼ mN ¼ mS . The current can be imposed by forcing a phase difference / between nearest neighbor S cells: Dðz þ dÞ ¼ DðzÞei/ . Note that in principle, to conserve the current, there will be supercurrents in each S cell so that the phase would not be a constant across each S cell. We will, however, bypass this problem by focusing on the supercurrent (/=a) and assuming the current is conserved. This will be correct in the limit of small width (b) for the S cell. The phase difference in adjacent S cells also implies a new Bloch condition: uS ðz þ dÞ ¼ uS ðzÞk expði/=2Þ and vS ðz þ dÞ ¼ vS ðzÞk expði/=2Þ. This, however, can be then solved by re-absorbing the phase into the wave functions of the quasiparticles. In the N cell, we redefine the new wave functions as: uN ðzÞ ! uN ðzÞ expði/z= 2aÞ and vN ðzÞ ! vN ðzÞ expði/z=2aÞ. As a result, the wave vectors of electron-like and hole-like quasiparticles are shifted as follows: / k~ ¼ k : 2a
173
Fig. 5. The DOS for a NS superlattice carrying current along c-axis direction. The mass ratios are m=m0 ¼ 1 and c ¼ mpNffiffiffiffiffiffi =mffi S (except for the first one, c ¼ 1:0, the remainings have c ¼ 0:5). Other parameters are A ¼ 0:57, B ¼ 0:21, and ¼ 10.
ð14Þ
Eq. (9) is the same except that C has to be replaced by ! A / / B C ¼ cos cos þ 2 X X ! A / / B sin sin þ : ð15Þ 2 / X X , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos (sin) are replaced by Here again for 2 < w2p cosh (sinh), and / by w2 2 . In the large mass anisotropy limit, the result is also the same as before with the same replacement: A ! A þ ð/=2Þ. In Fig. 5, we shows the DOS with current using the material parameters for YBCO. We see that the mass anisotropy due to c moves the peak of DOS into the subgap region, but the current further moves the peak into lower energy region. Most importantly, the DOS has a finite value at zero energy. These behaviors can be
Fig. 6. Illustration of how the shift of the DOS is induced. The dash line is the energy profile for the case without applying a current, while the solid illustrates that when a current is present, the quasiparticle effectively sees higher energy channels in the S cell.
understood via the illustration of Fig. 6. Due to the shift of momentum, the energies of quasiparticle in the metal increase for a given k : E ¼ 2 h2 kk2 =2m þ h2 ðk þ /=2aÞ =2m l. As a result, the energies of the tunneling channel in the S cell seen by the quasiparticles also increase effectively. This results in the shift of the DOS observed in Fig. 5. The above argument relies heavily on the concept of Andreev bound states. Therefore, for the IS superlattice, we expect no shift happens (except possible small shift due to existence of small supercurrents in the S cell), hence the DOS for the IS superlattice remains
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bulk-like with vanishing DOS at zero energy. This results in different behaviors in the specific heat measurements, providing a possible way to differentiate between NS and IS superlattices.
Acknowledgements This work is supported by the NSC of Taiwan. We thank Profs. H.H. Lin, T.M. Hong and C.S. Chu for useful discussions.
4. Conclusions References In summary, we have investigated the effect of the superlattice structure along the c-axis on the single particle density of HTSC by considering the NS and IS superlattices. We find that the position of quasiparticle peak depends on how one models the c-axis tunneling. For the NS superlattice, the position of the quasiparticle peak can be considerably smaller than the gap value, while for the IS superlattice, the quasiparticle peak always remains at the bulk value so that the whole system behaves like a single superconductor. As evidences are emerging that at high dopings the cuprates does behave like 3D metals, these results are illuminating. In particular, they imply that in the overdoped region, the gap value extracted from measurements using the DOS might not be the real gap value of the CuO2 plane. Since the real gap value also decreases as doping increases, the observed reduction of the gap value in the overdoped region might be a combined result of both effects. The NS and IS superlattice models can be further differentiated by applying a small current along c-axis. This would result in different dependences on temperature in the specific heat measurements, proving a possible signature to look for in experiments.
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