Magnetic Field Dependence of Electronic Specific Heat in

The specific heat of electron-doped Pr1.85Ce0.15CuO4 single crystals is reported for the temperature range 2 10 K and magnetic field range 0 10 T. A non-linear magnetic field dependence is observed for the field range 0 2 T. Our data supports a model with lines of nodes in the gap function of these superconductors. Theoretical calculations of the electronic specific heat for dirty d-wave, clean d-wave, and s-wave symmetries are compared to our data. PACS no.s: 74.25.Bt, 74.20.Rp, 74.25.Jb Typeset using REVTEX 1 The order parameter (gap) symmetry of the high-Tc cuprate superconductors(HTSC) is an important parameter in attempting to understand the pairing mechanism in these materials. For hole-doped cuprates experimental evidence strongly favors d-wave symmetry [1,2]. Surprisingly, early experiments on electron-doped(n-type) Nd1.85Ce0.15CuO4 (NCCO) suggested a s-wave symmetry. Recent penetration depth [3], tri-crystal [4], photoemission [5], Raman scattering [6] and point contact tunneling experiments [7] on NCCO and Pr2−xCexCuO4 (PCCO) favor a d-wave symmetry. In addition to these measurements which show s-wave or d-wave symmetry, there are penetration depth [8] and point contact tunnelling [7] experiments that have shown evidence for a change in the order parameter as the doping changes from under-doped(d-wave) to over-doped(s-wave). However, since these prior measurements on the n-type cuprates are surface sensitive there is a need for bulk measurements (e.g. specific heat) to convincingly determine the pairing symmetry, as was the case for the p-type cuprates [9–12]. The specific heat is sensitive to low temperature electronic excitations. Different gap symmetries have different density of electronic states close to the Fermi level. Conventional low-Tc superconductors show a s-wave gap symmetry in which the electronic specific heat has an exponential temperature dependence, Cel ∝ T e , where ∆ is the energy gap [13]. For a clean d-wave superconductor electronic excitations exist even at the lowest temperatures. The electronic DOS is predicted to have a linear energy dependence close to Fermi level, and this shows up in the electronic specific heat as Cel ∝ T 2 [14]. In the mixed state, there are two types of quasiparticle excitations in the bulk of the superconductor: bound states inside the vortex cores, and extended states outside the vortex cores. In conventional s-wave superconductors, the in-core bound states dominate the quasiparticle excitations, therefore the electronic specific heat is proportional to the number of vortices. The number of vortices is linear in field, therefore the electronic specific heat is also linear in field. [15]. In a superconductor with lines of nodes(e.g. d-wave symmetry), the extended quasiparticles dominate the excitation spectrum in the clean limit. It has been shown that the electronic specific heat has a √ H dependence in the clean limit [16] 2 at T=0. For non-zero temperatures there is a minimum field that depends on temperature after which the √ H dependence should be observed. In the dirty limit the energy scale related to impurity bandwidth (or impurity scattering rate) is much larger than the energy scale related to the Doppler shift due to magnetic field(the dominant mechanism for the clean d-wave case), and much less than the superconducting gap maximum. In this limit, i.e. kBT << (H/Hc2)∆0 << h̄γ0 << ∆0, where ∆0 is the gap maximum and h̄γ0 is the impurity band width, the magnetic field dependence deviates from √ H, and an H log(H) like dependence is predicted below a certain field H*, which depends on temperature and impurity concentration in the sample [17]. In this paper we present the first magnetic field dependent specific heat measurements on n-type cuprates which probe the symmetry of the superconducting state. The electronic specific heat has been observed to have a non-linear magnetic field dependence. The theoretical model for a clean d-wave symmetry fits reasonably well to our data, however there are deviations from this type of field dependence below H∗=0.6 T (Fig.3). We find that a H log(H) type dependence gives a better fit to our data over the whole range, which means our data can better be described by a dirty d-wave symmetry. It is important to emphasize that the main point of this work is to address the question of s-wave vs d-wave, rather than clean d-wave vs dirty d-wave. The specific heat data was obtained in the temperature range 2 10 K and the magnetic field range 0 10 T using the relaxation method [18]. The measurements were repeated in two systems, a home-made setup and a Quantum Design PPMS with some modifications on the sample holder to remove the field dependence of the original chip. The addenda consists of a sapphire substrate with a thermometer and heater, and Wakefield thermal compound to hold the PCCO crystal. The addenda was measured separately and was found to have no magnetic field dependence within the resolution of our experiment(±2.5%). The experiment was done on several optimally doped Pr1.85Ce0.15CuO4 single crystals (the mass of the crystals was 3-5 mg). The sample heat capacity is approximately equal to two times that of the addenda at T = 2 K, and equal to that of the addenda at T = 10 K. The crystals 3 were grown by the directional solidification technique [19]. The samples were characterized with a SQUID magnetometer and found to be fully superconducting, with similar transition temperatures Tc=22 K±2 K. The specific heat of a d-wave superconductor usually has the following main contributions: the electronic contribution, which could have the form γT or γT 2 depending on the field and temperature range the measurement is done, the phonon contribution, which at the temperature range of our experiment can be written as βT , and a Schottky contribution, which is caused by spin-1/2 paramagnetic impurities [20]. Furthermore γ = γ(0) + γ(H), where γ(H) gives the field dependent part of the electronic specific heat coefficient, and γ(0)T is the residual linear temperature dependent part of the electronic specific heat. γ(0) is sample dependent, and its origin is not completely understood. [9–11]. Non-electronic two-level systems away from the copper-oxide planes are one of the possible candidates for the origin of this term [17]. This term has been found in all hole-doped samples studied [9–11]. Fig.1 shows temperature dependence of the specific of PCCO heat at four different fields, 0 T, 1 T, 2 T, and 10 T applied perpendicular to the ab-plane of the crystal. The field range 0-2 tesla is the relevant field range to extract the gap symmetry information [16], and at H=10 T the sample is completely in the normal state(Hc2 = 8T at T=2K). Driving the sample to the normal state enables us to extract an important parameter, γn = 6.7 ± 0.5 mJ/mole-K, which is needed to compare our data to theoretical predictions quantitatively. A global fit which assumes the phonon coefficient, β, constant for all fields and γ variable gives a γ(0) = 1.4± 0.2 mJ/mole-K. This value of γ(0) is consistent with the values found for γ(0) in the hole-doped superconductors(γ(0) ≈ 1 mJ/mole-K for YBCO [9–11]). The fact that we do not have any Schottky upturn at low temperatures for any field shows that our sample is free from a detectable level of magnetic impurities. From the slope of the lines, obtained through a global fit, β = 0.29± 0.01 mJ/mole K, and a Debye temperature ΘD = 362 ± 4 K has been extracted. These values are in reasonable agreement with the other published data in the literature(β = 0.244 mJ/mole K, and ΘD = 382 K [21]). Since the phonon specific heat is field independent and there is no Schottky contribution 4 to the specific heat, subtracting the zero field specific heat from the specific heat at other fields gives the field dependent part of the electronic specific heat. Fig.2 shows the field dependent part of the electronic specific heat,γ(H)T, vs magnetic field at 3.4 K in the field range 0 8 T. Fig.3 shows theoretical fits to the 3.4 K data in the field range 0 2 T. The clean d-wave fit is calculated using the equation: [11]