How Cooper Pairs Vanish Approaching the Mott Insulator in Bi 2 Sr 2 CaCu 2 O 8+δ

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1 doi: 1.138/nature7243 SUPPLEMENTARY INFORMATION How Cooper Pairs Vanish Approaching the Mott Insulator in Bi 2 Sr 2 CaCu 2 O 8+δ Y. Kohsaka et al. Supplementary Information I Instruments and Samples The spectroscopic imaging STM (SI-STM) instruments are housed in ultra low vibration laboratories, specifically designed to facilitate these projects. Each consists of an underground concrete vault conditioned to be anechoic, inside which is a nested acoustic isolation room. The inner room (total mass kg) is supported on six pneumatic vibration isolators. Inside the inner acoustic isolation chamber is the cryostat itself whose structure has integrated within it kg of lead. It includes another massive vibration isolation stage with three pneumatic isolators. The liquid helium vessel is suspended from this stage. The SI-STM is usually suspended from a home-made ultra low vibration sub-kelvin refrigerator inside the cryogenic vacuum space. Design of our custom built cryogenic SI-STM systems is summarized in Ref. 51. All Bi 2 Sr 2 CaCu 2 O 8+δ crystals studied are grown by floating-zone furnace techniques in the laboratories of Prof. S. Uchida at the University of Tokyo and of Dr. H. Eisaki at AIST, Tsukuba. The samples with superconducting transition temperatures (T c ) of 2 K and 45 K were doped with Dy at the Ca site, Bi 2 Sr 2 Ca.8 Dy Cu 2 O 8+δ while the others were Bi 2 Sr 2 CaCu 2 O 8+δ. The hole density p for each sample is estimated by combining the results from several different probes, including magnetic susceptibility changes at T c, c-axis length from X-ray, zero-resistivity transition at T c, temperature dependence of resistivity, magnitude of Hall coefficient, effective carrier number from optical conductivity, mean gap values (STM), and Fermi surface volume (ARPES). Inductively coupled plasma optical emission spectroscopy was used to measure the cation ratio relative to the amount of Cu to check for sufficient crystal quality. The transition temperatures and corresponding hole dopings are reported in Table S1. Each crystal is segmented in 1 mm-square plates which are mounted on the sample holder studs for insertion into the cryogenic SI-STM system, is introduced in the cryogenic vacuum space, mechanically cleaved after thermalization, and inserted in the SI-STM head. Once the STM scanner has approached the BiO surface of a crystal, a large 1

2 Compound T c (K) p Bi 2 Sr 2 Ca.8 Dy Cu 2 O 8+δ 2.6 Bi 2 Sr 2 Ca.8 Dy Cu 2 O 8+δ 45.8 Bi 2 Sr 2 CaCu 2 O 8+δ Bi 2 Sr 2 CaCu 2 O 8+δ Bi 2 Sr 2 CaCu 2 O 8+δ Table S1: Sample properties. field of view (FOV) of dimension 5 nm square is chosen for study. It is necessary to achieve highly repeatable sub-atomic resolution and register in topographic images of this surface. The same resolution is required in differential conductance di/dv (r, V ) maps to ensure accurate measurement of both high q-vectors in the quasiparticle interference (QPI) patterns and the r-space structure at high energies. The large > 4 nm FOV is required simultaneously in order to attain sufficient Fourier space resolving power to detect any slowly dispersing q-vectors in the QPI patterns. A repeated series of di/dv (r, V ) maps meeting these specifications and without loss of atomic register to the crystal lattice due to drift or distortion for periods of up to a year, are carried out in each FOV. They are designed to achieve a signal-to-noise ratio in every spectrum of 1 : 1 (since many of the modulation to be studied are only a few percent of the average value of di/dv (E) ) and to achieve simultaneously an energy resolution specification δe 1.5 mev. These procedures were repeated for several samples at each of five different hole-densities. To complete this large data set efficiently, we used three SI-STM systems in parallel. II Modulation q-vector Extraction The analysis of all momentum space properties begins with the Fourier transform power spectral density (PSD) of real space tunneling conductance maps di/dv (r, V ). For all dopings except T c = 86 K [16], we use the ratio of tunneling conductance at opposite bias polarities. The Fourier transform of the tunneling conductance has been shown to yield the expected momentum space properties [16], as has the ratio Z of the conductance maps [19] which eliminates the heterogeneous integrated-dos tunnel junction setup error which severly affects tunneling data from underdoped samples [19,2]. Z is defined in the main text di (r, +V ) Z(r, V ) = dv (S1) di (r, V ) dv 2

3 Because Bogoliubov excitations are a coherent superposition of states with the same momentum above and below the chemical potential, the QPI wavevectors of empty states at +E must have the same dispersion as those of filled states at E: q i (+E) = q i ( E). In the superconducting state, Z is the ratio of the of the modulus squared of the coherence factors Z(r, V ) = u(r, ev ) 2 v(r, ev ) 2 for the BCS superconducting wave function [52], Ψ = ] [u(r r ) + v(r r )c (r)c (r ) (S3) The Bogoliubov quasiparticle wave function normalization constraint ensures that the spatial modulations in di/dv (r, E) are not canceled in Eq. (S2). For instance, consider u(r, ev ) 2 with a sinusoidal spatial variation: u(r, ev ) 2 = u + sin(k r). The normalization constraint requires that v(r, ev ) 2 = 1 u(r, ev ) 2 = 1 u sin(k r) so that a local maximum in u(r, ev ) 2 occurs at the same location in space as a local minimum in v(r, ev ) 2. This anti-phase relation between the particle and hole components of Bogliubov QPI modulations, along with the requirement q i (+E) = q i ( E), preserve the 16 dispersing octet q-vectors in the measured ratio Z of Eq. (S1). To analyze the data, all real space Z(r, E) data sets are multiplied by a quadratic window before Fourier transformation to minimize edge effects on low intensity signals. Supplementary Fig. S1 shows Z(r, E) and Z(q, E) pairs for T c = 45 K; a-f show pairs in the dispersive low energy range, while g and h show E = 4 mv as representative of the non-dispersive high energy range. The Z(q, E) have been averaged by reduction averaging and symmetric averaging as defined below. First, the two-dimensional PSD of the Z(q, E) were analyzed to locate the biasdependent local maxima occurring at the elastic scattering q-vectors [15-19]. The PSD is processed using three averaging techniques to increase the signal to noise ratio (S/N). The first technique is to break the original PSD up into blocks of four nearest neighbor pixels, and then replace each block with the average value of the four pixels. As a result of such averaging, a PSD that is originally 256 pixels square becomes 128 pixels square, and the S/N of each pixel increases by a factor of 2. This is discussed in Refs. 17 and 53. We refer to this as reduction averaging. In general, n-pixel averages of the PSD increases the S/N by a factor of n [54]. The next averaging step depends on the type of q-vector. The octet model [15-19] for d-wave Bogoliubov quasiparticle elastic scattering has two types of q-vectors, those constrained to lie along high symmetry directions, and those without constraints. Using the q-vector labeling of Ref. 16, q 1, q 3, q 5, and q 7 are symmetry constrained. For these, linecuts of Z(q, E) data along the particular high symmetry q-direction are taken with multi-pixel averages transverse to each linecut. This 3 (S2)

4 averaging has been discussed in Ref. 17. The peaks in the averaged linecuts are fit with a Lorentzian line shape to extract the peak locations q i. For the unconstrained q 2, q 4, and q 6 scattering vectors, a different averaging technique is used. After the nearestneighbor block averaging described above, the PSD is symmetrized along the crystalline a-axis. This is symmetric averaging. These q-vector peaks are fit with two-dimensional Lorentzians of the form a f(q) = ( ) 2 ( ) 2 (S4) qx q xi qy q yi b x b y yielding the peak locations q i. For both the constrained and unconstrained q-vectors, the fit is to the square root of the averaged PSD, which further doubles the S/N. The square roots of the averaged PSD used to extract the unconstrained q-vectors are shown in the Supplementary Data Movies showing Z(q, E) for a sequence of different p. Fig. S2 shows the resulting q-vector locations for each of the samples with hole-density p spanning the range from 6% < p < 19%. We can only track q 4 for the most underdoped sample. III q-vector Inversion The Bogoliubov quasiparticle dispersion E(k) has banana-shaped constant energy contours which exhibit maxima in the joint-density-of-states at the eight tips of the E(k) bananas - the k-space locations of the Bogoliubov band minima k B (E). Elastic scattering between these eight momentum-space regions k j (E); j=1, 2,..., 8 produces real-space interference patterns in the local-density-of-states whose energy-dispersive N(r, E) modulations exhibit 16 ±q pairs of wavevectors. These q i (E) are measured from the Fourier transform of modulations seen in Z(q, E) and the k B (E) = (k x (E), k y (E)) are then determined along with the superconductor s energy gap structure (k). As in Ref. 16, to determine each momentum vector k, 5-8 statistically independent two component q-vectors are algebraically inverted (using Eq. 1 of the main text) giving 2k x (q) = q 1, q 2x q 2y, q 6y q 6x, (q 3 q 7 )/ 2 or q 5 2 q 7, q 4x 2k y (q) = q 5, q 2x + q 2y, q 6y + q 6x, (q 3 + q 7 )/ 2 or (S5) 2 q 7 q 1, q 4y The two orthogonal Cu-O directions yield two statistically independent q 1 and two statistically independent q 5 values. Thus, at each bias voltage these equations yield numerous values for the location of one point of the Bogoliubov-band minima k B (E) = (k x (E = ev ), k y (E = ev )) in one octant of the first Brillouin zone and the corresponding superconducting gap value (θ k ). The k-vectors plotted in Fig. 3a of the main text are the mean of this sample. In Fig. S3 we plot the k-vector mean for each doping and here include 4

5 the standard deviations from the sets of k x (E = ev ) and k y (E = ev ) as the error bars. Since the generated k x, k y are not statistically independent, the covariance between their samples is used in all momentum space fits described below. This covariance C(k B (E)) is C(k B (E)) = 1 N 1 N {k xj (E) k x (E) } {k yj (E) k y (E) } (S6) j where j spans all available statistically independent solutions from Eq. (S5). The final step in k-vector inversion is to reflect each k-vector sample across the Brillouin zone (π, π) diagonal to completely populate the first quadrant. The model Bogoliubov arc is then a least-squares one parameter fit of this k-space data to a quarter circle. The model fits are plotted over the k-vectors in Fig. S3 and the fit parameters are listed in Table S2, where we include the χ 2 divided by the degrees of freedom (D.O.F) statistic output by the least-squares fit [55]. Sample T c (K) Radius (π/a ) χ 2 /(D.O.F.) ± ± ± ± ±.5.1 Table S2: Bogoliubov arc fit parameters. To model a large Fermi surface Luttinger count of the hole density, the fitted Bogoliubov arc quarter circle was extended beyond the (π/a, ) (, π/a ) line with horizontal and vertical lines as shown in Fig. S4a. For comparison, Fig. S4b shows the locus of quasiparticle poles for the single particle Green s function with the self-energy proposed in Ref. 11. This self-energy generates a line of zeroes in the Green s function at the chemical potential along the (π/a, ) (, π/a ) line in k-space. IV SC Energy Gap To extract the d-wave gap dispersion, we parameterize each k-vector by its angle θ k about (π, π) and fit the gap to the function (θ k ) = QPI [B cos(2θ k ) + (1 B) cos(6θ k )] (S7) This form was introduced earlier in ARPES studies [56] and has been found to describe both tunneling data [16] and Raman spectroscopy [29]. These fits are independent of the Bogoliubov arc fits. The error on θ k is found from the k x, k y standard deviations as well as the k x, k y covariance using partial derivative error propagation [55]. The 5

6 error is the RMS bias modulation amplitude of the lock-in amplifier used to make the spectroscopic tunneling maps. The θ k points and corresponding fits are plotted for each doping in Fig. 3b of the main text. The fit parameters are reported in Table S3. We find that the doping dependence of the parameter B is not inconsistent with that found by ARPES [56] and Raman spectroscopy [29]. Sample T c (K) QPI (mev) B χ 2 /(D.O.F.) ± 2.82 ± ± 3.87 ± ± 3.81 ± ± 6.77 ± ± ±.9 Table S3: Superconducting gap fit parameters. We emphasize that the octet inversion outlined above is not a direct fit of the model to the spectroscopic data. But with 5-8 q-vectors measuring the same k-vector, the model is heavily over determined, allowing us to replace the statistical sample by its mean and standard deviation in the above k-space fits. To demonstrate the degree of internal consistency of this statistical model we plot as solid lines in Fig. S2 the q-vectors that the model fits generate over the measured q-vectors (symbols). V QPI Termination As the bias voltage moves farther away from the chemical potential, the peak amplitudes of q 2, q 3, q 6, and q 7, decay away until they fall below the noise floor. This is shown in Fig. S5a-d. These four scattering vectors are the d-wave superconducting state s response to perturbing ordinary scalar impurity potentials via the d-wave coherence factors. Simultaneously, the octet k-vectors disperse away from the nodal region. At a certain bias voltage, the k dispersion suddenly stops. The point in k-space where the octet model dispersion terminates is near the line connecting the (π/a, ) (, π/a ) k-points within the octet sample error. This is demonstrated in Table S4 which lists for each doping the deviation k of the terminating k-point from the segregation line, and the octet sample standard deviations δk x, δk y which represent the error. Just as the dispersion stops, the peak amplitudes of q 2, q 6, and q 7 approach the noise floor and disappear a few millivolts later. In contrast, the q 1 and q 5 peaks remain well above the noise floor beyond the dispersive termination. In the non-dispersive regime we label these q 1 and q 5 and plot with filled symbols in Fig. S2. The peak amplitudes for q 1, q 1 and q 5, q 5 are shown in Fig. S5e-h. This behavior of dispersive q-vectors at 6

7 T c (K) k(π/a ) δk x (π/a ) δk y (π/a ) Table S4: Deviation of QPI termination point from the segregation line. low biases followed by a loss of dispersion at a high biases is entirely consistent with the results of autocorrelation ARPES studies in the superconducting state [37, 38]. VI Doping Dependence of Non-dispersive Wave-vectors at Extinction The non-dispersive wave-vectors q 1 and q 5 at biases above the extinction energy follow the doping dependence of the Bogoliubov-arc termination. This is illustrated by the arrows in the schematic Brillouin zone of Fig. S6a. These non-dispersive features are not harmonics tied to a static 4a modulation: q 1 is not locked at (1/4) (2π/a ) and q 5 is not locked at (3/4) (2π/a ) although their sum adds to 2π/a. Thus we demonstrate that they are determined by the point of intersection of the Bogoliubov Arc and the line between (, π) to (π, ). This is displayed in the Z( q ) line-cut data of Fig. S6c which shows the evolution of the 48 mv q 1 and q 5 peaks with doping. We focus in Fig. S6d on the q 5 peak from these data, overlaying the fits used to extract the peak location as well as the terminating k y point determined from data at biases more than 1 mv lower. VII Local Pseudogap Energy Rescaled Images We display in Fig. S7 images of the conductance ratio Z(r, e(r)) rescaled in energy to the fraction e of the local pseudogap value, e(r) = E/ 1 (r) as discussed in the main text. The energy rescaled Z(r, e(r)) is displayed for six values of e between and 1.4 demonstrating that the bond-centered pattern of Ref. 2 actually exhibit maximum intensity and contrast at e = 1 and thus, this r-space structure is that of pseudogap excitations E = 1 (r). References [51] Pan, S. H., Hudson, E. W. & Davis, J. C. 3 He refrigerator based very low temperature scanning tunneling microscope. Rev. Sci. Instrum. 7, 1459 (1998). 7

8 [52] Fujita, K. et al., Bogoliubov angle and visualization of particle-hole mixture in superconductors. (27). [53] Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. Numerical Recipes in C, 2nd Ed. (Cambridge University Press, 1997). [54] Oppenheim, A. V. & Schafer, R. W. Discrete-Time Signal Processing (Prentice Hall, 1989). [55] Melissinos, A. Experiments in Modern Physics (Academic Press, 1969). [56] Mesot, J. et al., Superconducting Gap Anisotropy and Quasiparticle Interactions: A Doping Dependent Photoemission Study. Phys. Rev. Lett. 83, 84 (1999). 8

9 a b (2π,) Z(r,E=8mV) Z(q,E=8mV) (,2π) c d (2π,) Z(r,E=16mV) Z(q,E=16mV) (,2π) e f (2π,) Z(r,E=24mV) Z(q,E=24mV) (,2π) g h (2π,) Z(r,E=4mV) Z(q,E=4mV) (,2π) Figure S1 Z(r,E), Z(q,E) pairs for =45K. a-f are in the dispersive energy range, while g and h are for E=4mV, representative of non-dispersive energies. The red lines in the Z(q,E) label the Cu-O directions for the pairs.

10 .8 q (2π /a ).6 T c =86K 2 4 a q (2π /a ).6 T c =88K q (2π /a ).6 T c =74K 2 4 b c q (2π /a ) , 6 5 Tc =45K q (2π /a ).6 T c =2K d e 6 Figure S2. q-space locations of the scattering vectors as a function of bias voltage. The solid lines are the scattering vectors of the octet model. This model is not fit directly to the q- vectors and these plots represent the internal consistency of the model. The scales are identicalineachplot.a. For the = 86K sample, there was no analysis of high biases 18. b- e. For the other dopings the vectors q 1 and q 5 can tracked over a larger bias range than the other q-vectors, and become non-dispersive in the range where the other q-vectors cannot be found.

11 1. 1. =86K =88K k y (π /a ).5 k y (π /a ).5.. a.5 k x (π /a ) 1... b.5 k x (π /a ) =74K =45K k y (π /a ).5 k y (π /a ).5.. c.5 k x (π /a ) 1... d.5 k x (π /a ) =2K k y (π /a ).5.. e.5 k x (π /a ) 1. Figure S3 a-e. The doping dependence of the Bogoliubov arcs generated by the octet model. We emphasize that each k-vector in one octet is generated from a different bias voltage. The octet is then reflected across the line k y = k x. The length of the error bars is one standard deviation of the measurement sample for each point. The solid lines are fits of the data to a quarter-circle.

12 1. =86 =88 =74 =45 =2 k y (π/a ).5.5 k x (π/a ) 1. b Figure S4 a. Plots of the Fermi arc quarter circle fits with exteneded horizontal and vertical lines used to determine the Luttinger count of the hole density. b. For comparison we reproduce from Ref 11 a plot of the locus of poles in k-space of the single particle Green's function at the chemical potential containing a proposed form for the self-energy. This self-energy generates zeroes in the Green's function along the (π/a,) - (,π/a ) line, which we plot as the dashed line.

13 = 88K Peak amplitude (Za /2π) a q 7 q 2,6 q 3 Peak amplitude (Za /2π) e q 1 q 1 * q 5 q 5 * Peak amplitude (Za /2π). = 74K b 2 4 q 7 q 2,6 q 3 Peak amplitude (Za /2π). f q 1 q 1 * q 5 q 5 * Peak amplitude (Za /2π). = 45K c 2 4 q 7 q 2,6 q 3 Peak amplitude (Za /2π). g q 1 q 1 * q 5 q 5 * Figure S5 Plots of the peak amplitude density for the scattering vectors a-d: q 2,q 3,q 6, and q 7.e-h: q 1 and q 5. Comparison of the peak amplitudes for the =86K data set 18 cannot be made because the analysis was performed on conductance maps which suffer from the constant current setup effect 22. The peak amplitude density is the peak amplitude of the discrete Fourier transform divided by the square root the size of the two-dimensional frequency bin. We present the data this way so that the amplitude information is independent of the discretization size. Peak amplitude(za /2π). = 2K. d q 7 q 2,6 q 3 4 Peak amplitude (Za /2π).. h q 1 q 1 * q 5 q 5 * 6

14 X ½q 1 * ½q 5 * (2π,) Γ a M b q 1 * c 3. d Normalized Amplitude 2 =2K =45K q 5 * Normalized Amplitude 1.5 =2K =45K =74K =88K =74K =88K. 5.5 q (2π /a ) q (2π /a ).9 Figure S6. a. Schematic diagram of the Brillouin zone illustrating the relationship of non-dispersive q 1 * and q 5 * to the ends of the Bogoliubov arc. b. Z(q, V=48mV) for =74K. The red line schematically indicates the source of the data in c. and d. The arrow locates the reciprocal lattice vector. c. Doping dependence of linecuts of Z(q, V=48mV) extracted along the Cu-O bond direction. d. The peak is q 5 *, and the lines over the data are fits used to extract its location. The short vertical lines are the terminating 2k y values derived from lower bias data. The veritcal dashed line at.75*2π/a demonstrates that the non-dispersive q-vectors at high biases are not harmonics of a 4a periodic modulation, but are instead directly related to the extinction point of the Fermi arc. The data in c. and d. have been normalized to the peak amplitude of q 5 * and offset vertically for clarity.

15 2nm a. e= b. e=.6 c. e=.8 d. e=1. e. e=1.2 f. e= Figure S7 a-f. A series of images displaying the real space conductance ratio Z as a function of energy rescaled to the local psuedogap value, e = E/Δ 1 (r). Each pixel location was rescaled independently of the others. The common color scale illustrates that the bond centered pattern appears strongest in Z exactly at E = Δ 1 (r).