B. E. Smith, K. Cho, W. A. Coniglio, I. Mihut and C. C. Agosta Department of Physics, Clark University, Worcester, MA, S. W. Tozer, T. P. Murphy and E. C. Palm The National High Magnetic Field Laboratory, Tallahassee, FL J. A. Schlueter Materials Science Division, Argonne National Laboratory, Argonne, IL Abstract We have calculated the Pauli paramagnetic limit (H p ) for different quasi 2D superconductors using a semi-empirical method. We then compared the calculated Pauli paramagnetic limits to penetration depth data obtained using a tunnel diode oscillator technique at low temperatures in a swept applied magnetic field. The organic superconductors examined are layered such that their behavior is dependent on their orientation to the applied magnetic field. In order to eliminate the effect of vortex dynamics, we examined data taken with the conducting layers oriented parallel to the applied magnetic field. For one of these materials,  -(BEDT-TTF) 2 Cu(NCS) 2, we find that eliminating vortex effects leaves us with one remaining feature in the data that may correspond to H p. We also find that the material  ''-(BEDT-TTF) 2 SF 5 CH 2 CF 2 SO 3 exhibits a change in slope for temperature versus upper critical field when the upper critical field exceeds the calculated H p. In addition, many of the examined quasi 2D superconductors, including the above organic superconductors and CeCoIn 5, exhibit upper critical fields that exceed their calculated H p suggesting some type of non-conventional superconductivity. Introduction to Superconductivity When certain conducting materials are cooled below a critical temperature (T C ) they exhibit a property known as superconductivity. Superconductors are perfect electrical conductors. Normal conductors experience loss during the transmission of electricity due to resistance. Superconductors do not. This leads to many possible industrial applications for superconducting materials. We endeavor to understand the physics behind many of these materials and gain a more complete understanding of superconductivity. One way to study the transition of a superconductor from its superconducting state to its normal state is to change the temperature from below T C to above T C. Another way to study this transition is to keep the superconducting material below T C and apply a strong external magnetic field. Once the applied magnetic field exceeds the upper critical field (H C2 ) the material becomes a normal conductor again. Elimination of Vortex Effects:  -(ET) 2 Cu(NCS) 2 All of the materials we will be looking at are layered quasi-2D samples. The picture to the right shows one of these materials,  -(ET) 2 Cu(NCS) 2, oriented with its conducting BEDT-TTF layers (blue) parallel to the applied magnetic field (green). If the conducting planes are not oriented parallel, and an external magnetic field is applied to our superconducting sample, then “pancake” vortices form in the conducting planes. These “pancake” vortices consist of trapped magnetic flux that forms a core of normal material. As the applied magnetic field is increased, these vortices eventually merge and drive the sample normal. Experimental Technique The data included in this poster was taken at two different locations. The first location is Clark University's Pulsed Field Magnet Lab (seen on the right). In this lab low temperatures are achieved using a combination 3 He/ 4 He cryostat, which allows us to cool the sample to between 4.2 K and 400 mK. The applied magnetic field is provided by our 43 T pulsed field magnet. The second location where data was taken is the National High Magnetic Field Laboratory in Tallahassee, FL. At this lab low temperatures were achieved using a portable dilution refrigerator. A 33 T DC magnet allowed us to apply an external magnetic field that was swept from 0 T to 33 T and back again. All of these measurements were taken using a method knows as the tunnel diode oscillator (TDO) technique. This technique allows us to study very small samples without having to attach wires to the sample. The sample is placed in a coil that is connected in a self resonant circuit. This resonant circuit is driven by the TDO. As the external magnet field is changed, the frequency of the resonant circuit changes. The change in frequency is proportional to the change in rf penetration depth of the sample. Table of H p Values References [1] C. P. Poole Jr., H. A. Farach and R. J. Creswick, Superconductivity (Academic Press, 1995), ISBN [2] C. Martin et al., Journal of Low Temperature Physics 138, 1025 (2005). [3] F. Zuo et al., Physical Review B 61, 750 (2000). [4] H. Padamsee, J. E. Neighbor and C. A. Shiffman, Journal of Low Temperature Physics 12, 387 (1973). [5] O. J. Taylor, A. Carrington and J. A. Schlueter, Physical Review Letters 99, (2007). [6] S. Wanka et al., Physical Review B 57, 3084 (1998). [7] H. Elsinger et al., Physical Review Letters 84, 6098 (2000). [8] Y. Ishizaki et al., Synthetic Metals , 219 (2003). [9] W. K. Park et al., Physical Review B 72, (2005). [10] R. H. McKenzie, cond-mat/ v2 (1999). [11] J. Müller et al., Physical Review B 61, (2000). [12] F. Zuo et al., Journal of Low Temperature Physics 117, 1171 (1999). [13] M. A. Tanatar et al., Physical Review B 66, (2002). Pauli Paramagnetic Limit Cooper pairs form the basis for superconductivity. Cooper pairs are formed when two electrons interact with each other through an intermediary phonon and form a bound state [1]. Electron subjected to an applied magnetic field: Zeeman Energy Energy Level Splitting From BCS Theory: Energy Gap Increasing applied magnetic field: Cooper pairs are broken and superconductivity is destroyed. Pauli Limiting Field [1]: Organic and heavy fermion superconductors do not follow BCS theory. Therefore, we need a better method to calculate H p. Cooper Pair Zeeman Splitting Proposed Method to calculate the Pauli Paramagnetic Limit (H p ) Energy Gap Ratio found using Padamsee's  -model [4]. Inclusion of many-body effects suggested by Zuo et al. (2000) [3]. Standard form for H p used in Martin et al. (2005) [2].  -model specific heat fits from Taylor et al. [5]. Material  g/g* [10]TcTc HpHp  -(ET) 2 Cu(NCS) 2 d-wave [5] K20.67 T  "-(ET) 2 SF 5 CH 2 CF 2 SO 3 [6] K9.9 T  -(ET) 2 NH 4 Hg(SCN) 4 [7] K2.0 T -(BETS) 2 GaCl 4 [8] K12.38 T CeCoIn 5 [9] K8.78 T Possible Pauli Limit Signature: Organic Superconductor  -(ET) 2 Cu(NCS) 2 In  -(ET) 2 Cu(NCS) 2 we see an interesting feature left over after accounting for all of the vortex effects. In order to be able to visualize this feature better, a straight line was fit to the initial part of the trace (below left picture: black) and then subtracted from the original data ("Corrected Data” in red). This eliminates the background frequency change due to the increasing magnetic field. We then take the derivative of this subtracted data (below left insert: subtracted data in green and the derivative in red). This derivative is then smoothed (below left insert: blue). What remains is a feature that corresponds to our calculated H p (below left insert: black arrow). Clark's Pulsed Field Magnet Lab.  -(ET) 2 Cu(NCS) 2 sample in a TDO coil. Other Organic Superconductors Several other materials also show interesting behavior near our calculated H p. When we plot temperature versus upper critical field (below left: red), we see that the material  "-(ET) 2 SF 5 CH 2 CF 2 SO 3 shows a change is slope at H p. This change in slope is also suggested by data from Müller et al. [11] and Zuo et al. [12] (magenta and green respectively). Interestingly, when we repeated this experiment (blue) we no longer saw this change in slope. We suspect this may be due to the speed our sample was cooled. The red data was rapidly cooled while the blue data was slowly cooled. Cooling speed can affect the ordering of the molecules within the sample. In addition, the red and blue data came from different samples, although they were both from the same batch. The material -(BETS) 2 GaCl 4 also shows differing behaviour around H p for different samples. Sample 2 from Tanatar et al. [13] (below middle: green triangles) shows a change in slope just below the calculated H p. Tanatar’s Sample 1 (green circles) and our sample (red circles) do not show a clear change in slope. More data is needed in order to determine -(BETS) 2 GaCl 4 behaviour at H p. The slope for the material  -(ET) 2 NH 4 Hg(SCN) 4 (below right) flattens out at the calculated H p, which indicates that its upper critical field is limited due to the breaking of Cooper pairs. Elimination of Vortex Effects (cont.) In order to avoid vortex effects, we measure all our samples with their conducting planes parallel to the applied magnetic field. The elimination of vortex effects can be demonstrated by looking at the data for  -(ET) 2 Cu(NCS) 2. In the top right picture is a plot of magnetic field upsweeps for several different orientations about parallel (the traces are vertically shifted to aid visualization). The melting transition (H m ), where magnetic field becomes strong enough that the quasi-2D vortex lattice is able to depin from the conducting layer, is marked with the dashed black line. When we are exactly parallel (green) the melting transition is no longer seen. In the bottom right picture we have plotted upsweeps and downsweeps for the sample off-parallel and parallel. The irreversibility line (H irr ), where the magnetic field becomes strong enough that the vortices are no longer coupled to each other and become a vortex liquid, is no longer clearly defined for the parallel data (green/black). In addition, flux jumps (seen in the low field red/blue data) are also eliminated. Thus, we see that all vortex effects are eliminated when the sample is oriented parallel. This possible signature of H p is seen at several different temperatures (below middle and right). It is seen in both the upsweeps (below right: red) and downsweeps (green) of the magnetic field. The dashed black line is the calculated H p. The upper critical fields for  -(ET) 2 Cu(NCS) 2,  "-(ET) 2 SF 5 CH 2 CF 2 SO 3 and CeCoIn 5 (plotted above right) extend beyond their calculated H p. Since these are Pauli limited materials, it is possible that some type of non-conventional superconductivity is responsible. Future Work Recalculate  -model fits using d-wave fitting where appropriate. Recalculate g/g* using more recent data. Extend calculations to include more materials. Take more data for  "-(ET) 2 SF 5 CH 2 CF 2 SO 3 to determine why one sample/method of cooling shows a change in slope, but the other does not. Take more data for -(BETS) 2 GaCl 4 at lower temperatures.  -(ET) 2 Cu(NCS) 2 parallel to an applied magnetic field The above left equation is the proposed quasi-empirical method to calculate H p. In blue is the standard way to calculate H p used by Martin et al. [3]. Many-body effects, seen in red, are included as suggested by Zuo et al. [2]. The energy gap at zero temperature (  0 ) is seen above right. The energy gap depends on the energy gap ratio (  ), seen in green, and the critical temperature (T C ). For conventional BCS s-wave superconductors  s BCS =1.76 and for d-wave superconductors  d BCS =2.14. Padamsee et al. proposed that  for different superconducting materials could be experimentally found by fitting specific heat data. An example of these  -model fits, using specific heat data, can be seen on the right. These plots are from Taylor et al. [5]. Note that the bottom plot shows that  -(ET) 2 Cu(NCS) 2 is clearly a d-wave superconductor. Conclusions We have proposed a quasi-empirical method to calculate the Pauli limiting field. Method relies on calculating the energy gap by fitting specific heat data using the  -model. Calculated H p corresponds to possible Pauli Limit signature seen in  -(ET) 2 Cu(NCS) 2. Calculated H p for  -(ET) 2 NH 4 Hg(SCN) 4 corresponds to upper critical field limiting due to the breaking of Cooper pairs. The upper critical fields for  -(ET) 2 Cu(NCS) 2 and one of the  "-(ET) 2 SF 5 CH 2 CF 2 SO 3 data sets show a change in slope near their calculated H p. The -(BETS) 2 GaCl 4. data is not clear. Determination of the Pauli Paramagnetic Limit in Quasi 2D Superconductors