Scanning tunneling spectroscopy of high-temperature superconductors

Tunneling spectroscopy has played a central role in the experimental verification of the microscopic theory of superconductivity in classical superconductors. Initial attempts to apply the same approach to high-temperature superconductors were hampered by various problems related to the complexity of these materials. The use of scanning tunneling microscopy and spectroscopy (STM and STS) on these compounds allowed the main difficulties to be overcome. This success motivated a rapidly growing scientific community to apply this technique to high-temperature superconductors. This paper reviews the experimental highlights obtained over the last decade. The crucial efforts to gain control over the technique and to obtain reproducible results are first recalled. Then a discussion on how the STM and STS techniques have contributed to the study of some of the most unusual and remarkable properties of high-temperature superconductors is presented: the unusually large gap values and the absence of scaling with the critical temperature, the pseudogap and its relation to superconductivity, the unprecedented small size of the vortex cores and its influence on vortex matter, the unexpected electronic properties of the vortex cores, and the combination of atomic resolution and spectroscopy leading to the observation of periodic local density of states modulations in the superconducting and pseudogap states and in the vortex cores.

Figure 1

(Color online) Principle of STM. (a) Tunneling process between the tip and sample across a vacuum barrier of width $d$ and height $\varphi$ (for simplicity, the tip and sample are assumed to have the same work function $\varphi$). The electron wave functions $\Psi$ decay exponentially into vacuum with a small overlap, allowing electrons to tunnel from one electrode to the other. With a positive bias voltage $V$ applied to the sample, electrons tunnel preferentially from the tip into unoccupied sample states. (b) Schematic view of the scanning tunneling microscope.

Figure 2

(Color online) Generic STM operating modes: (a) constant-current and (b) constant-height imaging.

Figure 3

(Color online) Illustration of the vortex-lattice imaging by STM: (a) Local SIN junction with typical BCS $s$-wave characteristics when the tip is between vortices. (b) Local NIN junction with a constant conductance (for a dirty BCS superconductor) when the tip is positioned over a vortex core.

Figure 4

(Color online) Schematics of a tunnel junction. (a) Geometrical view: ${\mathcal{H}}_L$ and ${\mathcal{H}}_R$ are the Hamiltonians of the isolated left and right materials, respectively, and $T_{\lambda \rho }$ is the probability of tunneling from a state $\mid {\phi }_{\lambda }⟩$ on the left side to a state $\mid {\phi }_{\rho }⟩$ on the right side. (b) Energy diagram: a potential difference $V$ is applied to the junction, resulting in a relative shift of the chemical potentials ${\mu }_L$ and ${\mu }_R$ in both materials.

Figure 5

(Color online) Crystal structure of the cuprate materials. (a) Tetragonal unit cell of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_{8+\delta }$ with $a=b=5.4\mathrm{\AA }$ defined along [110] and $\left[\overline{1}10\right]$, respectively. (b) Orthorhombic unit cell of $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+\delta }$ with $a_0=3.82\mathrm{\AA }$ along [100] and $b_0=3.89\mathrm{\AA }$ along [010]. (c) Schematics of the $d_{x^2-y^2}$ superconducting gap in the unit-cell coordinate system. Note that $a_0$ and $b_0$ designate the lattice constants along the Cu-O-Cu bonds ($Pmmm$ space group for Y123), whereas $a$ and $b$ designate the lattice constants in larger unit cells ($Cmmm$ space group for Bi2212).

Figure 6

(Color online) STM images of the BiO surface of Bi-based HTS cuprates. (a) $13.1\times 13.1{\mathrm{nm}}^2$ area of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Cu}{\mathrm{O}}_6$ at $4.6\mathrm{K}$ (358). (b) $15\times 15{\mathrm{nm}}^2$ area of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ at $4.2\mathrm{K}$ (315). (c) Lead-doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ at $4.3\mathrm{K}$ (201). The bright lattice sites correspond to lead dopant atoms. Note the absence of supermodulation and dark-atom rows. (d) $15\times 15{\mathrm{nm}}^2$ area of zinc-doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ at $4.3\mathrm{K}$ (313).

Figure 7

STM topography and spectroscopy of a cleaved ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\mathrm{Cu}}_2{\mathrm{O}}_8$ thin film. (a) $10\times 10{\mathrm{nm}}^2$ topographic image of a BiO-terminated surface. (b) $6.4\times 6.4{\mathrm{nm}}^2$ topographic image of a $\mathrm{Cu}{\mathrm{O}}_2$-terminated surface (inset $0.8\times 0.8{\mathrm{nm}}^2$). Tunneling spectra taken at the indicated distances along a straight line (c) on the BiO plane and (d) on the $\mathrm{Cu}{\mathrm{O}}_2$ plane. Spectra are offset for clarity, and all data were taken with $V=200\mathrm{mV}$ and $I=200\mathrm{pA}$. From 274.

Figure 8

$5\times 5{\mathrm{nm}}^2$ STM images of the (a) BaO and (b) CuO chain surfaces of Y123. From 311.

Figure 9

Ideal STM vacuum tunnel junction between a Au tip and Bi2212 at $4.2\mathrm{K}$. (a) Tunneling spectra measured as a function of increasing tip-to-sample distance. The current is changed from $I=1.0$ (short distance, top curve) to $0.2\mathrm{nA}$ (large distance, bottom curve) in $0.1\text{−}\mathrm{nA}$ increments at constant sample voltage $V=0.4\mathrm{V}$. (b) Same data plotted on a normalized conductance scale to emphasize the distance independence. Adapted from 330.

Figure 10

$c$-axis SIN vacuum tunneling conductance spectra of selected superconductors (single crystals) measured by STM. (a) Nb at $335\mathrm{mK}$ (circles), BCS fit with $\Delta =1.0\mathrm{meV}$ (solid line). From 310. (b) Optimally doped Bi2212 $(T_c=92\mathrm{K})$ at $4.8\mathrm{K}$ (solid line), $s$-wave BCS fit with $\Delta =27.5\mathrm{meV}$ and pair-breaking parameter $\Gamma =0.7\mathrm{meV}$ (dash-dotted line), and low-energy V-shaped $d$-wave BCS conductance (dashed line). From 330. (c) Y123 at $4.2\mathrm{K}$. From 244. (d) Overdoped Bi2201 $(T_c=10\mathrm{K})$ at $2.5\mathrm{K}$ (solid line) and $82\mathrm{K}$ (dashed line). Adapted from 223. (e) Optimally doped ${\mathrm{Sr}}_{0.9}{\mathrm{La}}_{0.1}\mathrm{Cu}{\mathrm{O}}_2$ at $4.2\mathrm{K}$. Adapted from 434. (f) Underdoped Bi2223 $(T_c=111\mathrm{K})$ at $4.2\mathrm{K}$. From 221.

Figure 11

Gap amplitude ${\Delta }_p$ as a function of the number $n$ of $\mathrm{Cu}{\mathrm{O}}_2$ planes in the bismuth and mercury series. Optimally doped $\mathrm{Hg}{\mathrm{Sr}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+2+\delta }$ (circles), gap values from 414. Slightly overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{n-1}{\mathrm{Cu}}_n{\mathrm{O}}_{2n+4+\delta }$ (squares), gap values for $n=1$ from 223, for $n=2$ from 334, and for $n=3$ from 221. Each compound’s $T_c$ is indicated next to the symbol. The dashed line is a guide to the eye.

Figure 12

Large-energy-scale SIN tunneling characteristics of Bi2212 ([336]).

Figure 13

(Color) Fit of the differential tunneling conductance measured on optimally doped Bi2212. The fit (red line) to be compared with the experimental data (circles) is the sum of the contributions from antibonding (dotted line) and bonding (green line) bands describing the $\mathrm{Cu}{\mathrm{O}}_2$ bilayer splitting. Adapted from 160.

Figure 14

(Color) Schematic doping phase diagram of electron- and hole-doped high-$T_c$ superconductors showing, in particular, the superconducting (SC) and antiferromagnetic (AF) phases.

Figure 15

(Color) Pseudogap in Bi2212. (a) Doping dependence of STM tunneling spectra measured on Bi2212 (334). (b) Compilation of the gap of Bi2212 measured by various techniques in a wide doping range. The numerical values are listed in Appendix . The dashed line is a linar fit of the average gap at a given doping. Note the absence of scaling between the gap and $T_c$.

Figure 16

(Color online) $56\times 56{\mathrm{nm}}^2$ maps of the spatial gap distribution in (a) underdoped and (b) as-grown Bi2212 single crystals. Adapted from 227. (c) Range of tunneling conductance spectra measured on inhomogeneous superconducting Bi2212. From 265. (d) Scatter plot of the superconducting gap versus the integrated local DOS in optimally doped Bi2212. From 315.

Figure 17

(Color online) Spatially resolved STM tunneling characteristics measured along different traces on (a) Bi2212 (227), (b) Bi2212 (adapted from 334), (c) Bi2223 (221), and (d) Y123 (243). Note the high degree of homogeneity in (b)–(d).

Figure 18

(Color online) Spectroscopy above a Zn atom in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\left({\mathrm{Cu}}_{1-x}{\mathrm{Zn}}_x\right)}_2{\mathrm{O}}_{8+\delta }$ with $x=0.6\%$ and $T_c=84\mathrm{K}$. (a) Spectra taken at the center and away from the Zn impurity, (b) $60\times 60{\mathrm{\AA }}^2$ topography, and (c) conductance map at $-1.5\mathrm{meV}$ [logarithmic color (gray) scale]. (d) $30\times 30{\mathrm{\AA }}^2$ representation of the square $\mathrm{Cu}{\mathrm{O}}_2$ lattice lying below the BiO surface layer. The Zn atom is at the center and the conductance maxima observed in (c) are indicated by circles. Adapted from 313 and 25.

Figure 19

(Color online) Spectroscopy above a Ni atom in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2\mathrm{Ca}{\left({\mathrm{Cu}}_{1-x}{\mathrm{Ni}}_x\right)}_2{\mathrm{O}}_{8+\delta }$ with $x\approx 0.2\%$ and $T_c=85\mathrm{K}$. (a) Spectra taken at the center and away from the Ni impurity. (b) (c) $35\times 35{\mathrm{\AA }}^2$ conductance maps at positive $(+9\mathrm{mV})$ and negative $(-9\mathrm{mV})$ bias. (d) Position of the Ni atom (center) with respect to the Cu atoms in the invisible $\mathrm{Cu}{\mathrm{O}}_2$ plane, and schematics of the $d_{x^2-y^2}$ order parameter. Adapted from 171 and 25.

Figure 20

STM tunneling spectra measured relative to a $\left[\overline{1}10\right]$ step edge (45° boundary) on Bi2212 at $4.2\mathrm{K}$. (a) Differential conductance spectra measured at increasing distances in the direction perpendicular to the step edge. (b) Differential conductance spectra measured along the step edge. The ZBCP is measured only for tunnel junctions close to the step edge. From 275.

Figure 21

(Color online) Spatially resolved cross-sectional STM tunneling spectroscopy at $4.2\mathrm{K}$. (a) $31\text{−}\mathrm{nm}$ trace (running parallel to [001]) on Y123 showing the presence of a ZBCP on the (110) surface of a hole-doped HTS. Adapted from 433. (b) $35\text{−}\mathrm{nm}$ trace on NCCO showing the systematic absence of the ZBCP on the (110) surface of an electron-doped HTS. Adapted from 196.

Figure 22

(Color online) $T$ dependences of the DOS measured by STM. (a) Junction between a gold sample and a niobium tip with $T_c\approx 9\mathrm{K}$, ${\Delta }_p=1.5\mathrm{meV}$. Adapted from 310. (b) Junction between an iridium tip and UD Bi2212 with $T_c=83\mathrm{K}$, ${\Delta }_p=44\mathrm{meV}$, and $T^{*}$ near room temperature. Adapted from 334.

Figure 23

(Color online) $T$ dependence of the superconducting gap for $T<T_c$. The $4.2\mathrm{K}$ experimental data [334; see also Fig. 22b] are fitted by a $d$-wave function (4 K, dashed line), which is then used to recalculate the spectra at each temperature corresponding to the experimental data, assuming two different gap $T$ dependences: (a) a BCS $d$-wave $T$ dependence and (b) a constant gap magnitude.

Figure 24

(Color online) Tunneling spectra vs temperature in overdoped (OD) Bi2212. (a) Strongly OD sample with $T_c=74.3\mathrm{K}$ and ${\Delta }_p=34\mathrm{meV}$. Adapted from 334. (b) Slightly OD sample with $T_c=81\mathrm{K}$ and ${\Delta }_p=42\mathrm{meV}$. Adapted from 258. The tunnel resistances are of the order of $1\mathrm{G}\Omega$.

Figure 25

(Color online) $T$ dependence of the DOS of Bi2212 for two different junction types and tunneling techniques. (a) SIN vacuum tunneling using STM, $T_c=83\mathrm{K}$, ${\Delta }_p=44\mathrm{meV}$, $T^{*}\approx 300\mathrm{K}$, and $R_t=0.1–1\mathrm{G}\Omega$. From 334. (b) SIS junction using PC-BJT, $T_c=82\mathrm{K}$, ${\Delta }_p\approx 27\mathrm{meV}$, $T^{*}\approx 180\mathrm{K}$, and $R_t=10–100\mathrm{k}\Omega$; From 85.

Figure 26

(Color online) Psendogap in Bi2201. (a) $T$ dependence of the DOS in Bi2201. (b) Top-view representation of (a). (c) Doping phase diagram showing the values of $T_c$, ${\Delta }_p$, and $T^{*}$ from Table . The doping level $p$ was determined as described by 18. From 223.

Figure 27

(Color online) Temperature evolution in Y123. (a) $T$ dependence of optimally doped Y123 ($T_c=92\mathrm{K}$, ${\Delta }_p=20\mathrm{meV}$). The spectra (solid lines) are normalized to the background conductance taken at $V>100\mathrm{mV}$ and extrapolated to $V=0$. Dotted spectra correspond to the $4.2\text{−}\mathrm{K}$ spectrum thermally smeared to the different temperatures. From 246. (b) $T$ dependence of Nd123 ($T_c=95\mathrm{K}$, ${\Delta }_p=30\mathrm{meV}$). Adapted from 300.

Figure 28

Temperature evolution in electron-doped cuprates. (a) $T$ dependence by STS of a ${\mathrm{Nd}}_{1.85}{\mathrm{Ce}}_{0.15}\mathrm{Cu}{\mathrm{O}}_4$ single crystal (optimally doped, $T_c=20.5\mathrm{K}$, ${\Delta }_p\approx 4.5\mathrm{meV}$). The tip is parallel to the [110] direction. From [143]. (b) $T$ dependence by grain-boundary junction tunneling (GBJT) on a ${\mathrm{Pr}}_{1.85}{\mathrm{Ce}}_{0.15}\mathrm{Cu}{\mathrm{O}}_4$ thin film (optimally doped, $T_c\approx 24\mathrm{K}$, ${\Delta }_p=3.1\mathrm{meV}$). Note that these junctions are of the SIS type and in plane. From 208.

Figure 29

(Color) Scaling relations. (a) Comparison of the pseudogap and superconducting gap for Bi2212 and Bi2201. (b) Same data with energy rescaled by ${\Delta }_p$. (c) $T^{*}∕T_c$ as a function of $2{\Delta }_p∕k_B{T}_c$ for various cuprates investigated by tunneling spectroscopy. Dashed line corresponds to the mean-field $d$-wave relation $2{\Delta }_p∕k_B{T}^{*}=4.3$, where $T_c$ is replaced by $T^{*}$. See Table for detailed superconducting parameters. Adapted from 223.

Figure 30

(Color) Comparison of the pseudogap measured above $T_c$ and the LDOS inside the vortex core of Bi2212. (a) Spatial evolution of the LDOS when moving the tip from the center $(Y=0)$ to the periphery of the vortex core $(Y=4.2\mathrm{nm})$ at $4.2\mathrm{K}$ and $6\mathrm{T}$ (Bi2212 OD; 335). (b) $T$ dependence of the DOS. The pseudogap measured at $T\approx T_c=83\mathrm{K}$ is highlighted in white (Bi2212 UD). Adapted from 334. (c) Top: superconducting gap (blue) measured between vortices at $4.2\mathrm{K}$ and $6\mathrm{T}$ compared to a vortex-core spectrum (red). Bottom: pseudogap measured on the same sample in zero field at $T=88.7\mathrm{K}>T_c$ (solid line), compared to the vortex-core spectrum numerically smeared to $88.7\mathrm{K}$ (dashed red) (Bi2212 UD). Adapted from 335.

Figure 31

(Color online) Pseudogap on disordered Bi2212 surfaces. (a) LDOS as a function of tip position on a Bi2212 $300\text{−}\mathrm{\AA }$-thick film at $4.2\mathrm{K}$. Spectra were acquired every $5\mathrm{\AA }$. From 70. (b) LDOS evolution through a locally damaged Bi2212 region. The values on the right indicate the distance of each spectrum relative to spectrum $A$. Adapted from 168.

Figure 32

(Color online) Gap inhomogeneities in Bi2212. (a) Gap map at $4.2\mathrm{K}$ and zero field acquired on a Bi2212 sample which on average is underdoped. The arrow indicates the location of the $80\text{−}\mathrm{\AA }$ trace shown in (b). Adapted from 264. (c) Hypothesis on the local doping variation along the spectral trace.

Figure 33

(Color online) Low-energy coherence. (a) Representative spectra from nanoscale Bi2212 regions exhibiting coherence peaks (labeled 2) and from regions exhibiting the ZTPG (labeled 1) acquired at $4.2\mathrm{K}$. From 265. (b) Comparison of the positive-bias LDOS of a ZTPG and a superconducting region in Bi2212. From 168. (c) Comparison between spatially averaged tunneling conductance spectra on heavily UD ${\mathrm{Ca}}_{2-x}{\mathrm{Na}}_x\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ (bottom curve) and equivalently UD Bi2212 (top curve). From 134.

Figure 34

(Color online) Two scenarios for the hole-doped HTS phase diagram. (a) $T^{*}$ merges with $T_c$ on the overdoped side. (b) $T^{*}$ crosses the superconducting dome (SC) and falls to zero at a quantum critical point (QCP). $T_N$ is the Néel temperature for the antiferromagnetic (AF) state.

Figure 35

(Color online) Doping phase diagrams of various hole-doped HTS cuprates. From 290.

Figure 36

STS images of ordered vortex lattices in conventional superconductors. (a) $400\times 400{\mathrm{nm}}^2$ hexagonal lattice in $\mathrm{Nb}{\mathrm{Se}}_2$ ($T=1.3\mathrm{K}$, $H=0.3\mathrm{T}$). From 331. (b) $250\times 250{\mathrm{nm}}^2$ hexagonal lattice in $\mathrm{Mg}{\mathrm{B}}_2$ ($T=2\mathrm{K}$, $H=0.2\mathrm{T}$). From 102. (c) $170\times 170{\mathrm{nm}}^2$ square lattice in $\mathrm{Lu}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ ($T=4.2\mathrm{K}$, $H=1.5\mathrm{T}$). From 82. The mapping energy was ${\Delta }_p$ for (a) and (c) and 0 for (b), explaining the inverted contrast.

Figure 37

$100\times 100{\mathrm{nm}}^2$ STS images of (a) a slightly disordered oblique lattice in Y123 (from 244) and (b) a disordered vortex distribution in Bi2212 (from 162). Both images were acquired at $T=4.2\mathrm{K}$ and $H=6\mathrm{T}$. The contrast is defined by the conductance measured at the gap energy ($20\mathrm{meV}$ for Y123 and $30\mathrm{meV}$ for Bi2212) normalized by the zero-bias conductance.

Figure 38

Oblique vortex lattice in Y123. (a) $50\times 50{\mathrm{nm}}^2$ STS image of vortices in Y123. The primitive vectors of the oblique lattice form an angle of 77°, and the cores present an elliptic distortion of about 1.5 ratio. (b) Schematic diagram showing the formation of an oblique lattice, starting from a square lattice distorted by an anisotropy factor of 1.3. Adapted from 332.

Figure 39

(Color online) STS observation of a twin boundary in Y123. (a) $170\times 170{\mathrm{nm}}^2$ image at $H=3\mathrm{T}$; the flux lines are distributed on both sides of the boundary with an equal density. (b) After reducing the field to $1.5\mathrm{T}$, the flux lines pile up on the left domain and reach a density corresponding to about $2.5\mathrm{T}$. From 245.

Figure 40

(Color online) STS images of vortices in Bi2212. (a) $40\times 40{\mathrm{nm}}^2$ image of vortex cores in overdoped Bi2212 at $H=6\mathrm{T}$. From 335. (b) Ordered phase at $H=8\mathrm{T}$, revealing a nearly square symmetry. From 255.

Figure 41

Conductance spectra acquired at $T=0.3\mathrm{K}$, $H=0.05\mathrm{T}$ along a $50\text{−}\mathrm{nm}$ path leading from the center $(r=0)$ to the outside $(r=1200)$ of a vortex core in $\mathrm{Nb}{\mathrm{Se}}_2$ (152). The splitting of the zero-bias peak into one electron and one hole branch shows the presence of many bound states in the core (see text). From 131.

Figure 42

Tunneling conductance spectra acquired at the vortex-core center of ${\mathrm{Nb}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Se}}_2$ at $T=1.3\mathrm{K}$ and $H=0.3\mathrm{T}$ for different impurity dopings $x$. For $x=0$ (clean limit), bound states are observed as a broad ZBCP, which progressively vanishes for larger amounts of impurities (331).

Figure 43

(Color online) Examples of missing zero-bias peak in the vortex cores of conventional BCS superconductors. (a) Spectra acquired at $47\mathrm{nm}$ (dashed line) to the center of the core (dotted line) in $\mathrm{Lu}{\mathrm{Ni}}_2{\mathrm{B}}_2\mathrm{C}$ at $T=4.2\mathrm{K}$ and $H=1.5\mathrm{T}$. From 81. (b) $250\text{−}\mathrm{nm}$-long spectroscopic trace crossing a single vortex core in $\mathrm{Mg}{\mathrm{B}}_2$ at $T=2\mathrm{K}$ and $H=0.2\mathrm{T}$. From 102.

Figure 44

(Color) Core states in Y123. (a) $20\times 20{\mathrm{nm}}^2$ conductance map acquired at $+20\mathrm{mV}$, showing a detailed view of a vortex core ($T=4.2\mathrm{K}$, $H=6\mathrm{T}$). (b) Conductance spectra acquired at the center (red) and at about $20\mathrm{nm}$ from the center of the vortex core (dashed blue). The low-energy core states are located at about $\pm 5.5\mathrm{meV}$. Adapted from 244.

Figure 45

(Color) Conductance spectra taken along a $16.7\text{−}\mathrm{nm}$ path into a vortex core in Y123 at $T=4.2\mathrm{K}$ and $H=6\mathrm{T}$. Together with the vanishing of the $20\text{−}\mathrm{mV}$ coherence peaks, a pair of low-energy peaks emerges when the tip approaches the center of the vortex core $(r=0)$. From 335.

Figure 46

(Color) Conductance spectra taken along a $12\text{−}\mathrm{nm}$ path across a vortex core in overdoped Bi2212 $(T_c=77\mathrm{K})$ at $T=2.6\mathrm{K}$ and $H=6\mathrm{T}$. The spectra acquired at the center of the core reveal a pseudogaplike shape similar to the one observed above $T_c$. From 219.

Figure 47

(Color) Core states in Bi2212. (a) $8.7\times 8.7{\mathrm{nm}}^2$ conductance map of a vortex core in Bi2212, acquired at $-25\mathrm{mV}$ and $6\mathrm{T}$. Inset: The simultaneously acquired topography at the same scale. (b) Spectra taken close to the core center with two peaks at $\pm 6\mathrm{meV}$ corresponding to a pair of core states (red) and outside the vortex core (blue). Adapted from 236.

Figure 48

STS measurements of vortex-core states in Bi2212. (a) Tunneling spectra acquired in various sites of Zn-doped Bi2212 samples, reflecting strong (1) and weak (2) impurity scattering resonances, defect-free superconducting (3), and vortex-core (4) spectra. In the latter case, arrows indicate the position of low-energy core states. Adapted from 312. (b) Tunneling spectra taken in a vortex core showing localized states at $\pm 9\mathrm{meV}$ (solid circles) and outside a vortex core (open circles). From 255.

Figure 49

(Color) $10\text{−}\mathrm{nm}$-long spectroscopic trace across a vortex in overdoped Bi2212, at $T=1.8\mathrm{K}$ and $H=6\mathrm{T}$. Core-state peaks lie at $\pm 6\mathrm{meV}$, and their position does not change across the vortex ([235]).

Figure 50

Energy of vortex-core states $E_{\text{core}}$ as a function of the superconducting gap ${\Delta }_p$. Shown are Bi2212 single crystals of various dopings and optimally doped Y123 at different magnetic fields. The solid line, with a slope of 0.3, is a linear fit to all Bi2212 data points. Inset: Data for Bi2212 samples with different $T_c$, at a fixed field of $6\mathrm{T}$ (164).

Figure 51

(Color online) Vortex-core shapes. (a) $150\times 150{\mathrm{nm}}^2$ zero-bias conductance map taken at $H=500\mathrm{G}$ and $T=0.3\mathrm{K}$, revealing the sixfold shape of a vortex core in $\mathrm{Nb}{\mathrm{Se}}_2$. From 152. (b) $50\times 50{\mathrm{nm}}^2$ spectroscopic image of flux lines in Y123, at $H=6\mathrm{T}$ and $T=4.2\mathrm{K}$. From 243.

Figure 52

Split vortex in Bi2212. (a) $10\times 10{\mathrm{nm}}^2$ zero-bias conductance map acquired in Bi2212 ($T=4.2\mathrm{K}$, $H=6\mathrm{T}$). The whole core pattern is formed by split subcomponents belonging to one single flux line. (b) $8\text{−}\mathrm{nm}$ trace through such a core, illustrating how the spectra recover the superconducting characteristics between two elements, whose positions are marked by arrows. From 163.

Figure 53

Conductance spectra acquired between vortices in Y123 at $T=4.2\mathrm{K}$ under fields ranging from $0\text{to}3\mathrm{T}$. Inset: The zero-bias tunneling conductance as a function of the applied field (244).

Figure 54

(Color online) Quasiparticle interference in Bi2212. (a) Atomic-resolution image of a $64\times 64{\mathrm{nm}}^2$ region on near-optimally doped Bi2212. Inset: $16\times 16{\mathrm{nm}}^2$ magnification. (b) Conductance map of the same field of view acquired at $V=-10\mathrm{mV}$. (c) and (d) Fourier-space images of conductance maps at $-8$ and $-10\mathrm{mV}$, respectively. The square of intense points near the corners of each panel corresponds to the atomic lattice, and the arrows in (c) indicate the $q$ vectors of the supermodulation. These two sets of spots do not disperse with energy. The maps clearly show an additional pattern with fourfold symmetry. The corresponding wave vectors disperse in energy. From 267.

Figure 55

(Color online) Representation of the quasiparticle energy along the Fermi surface. ${\mathit{q}}_A$ and ${\mathit{q}}_B$ are two possible vectors connecting quasiparticle states with identical energies, giving rise to interference patterns. From 159.

Figure 56

(Color online) Analysis of the data shown in Fig. 54. (a) Fermi surface; the solid line is a fit to the data, assuming the Fermi surface is the combination of a circular arc joined with two straight lines. The grey band represents the Fermi surface determined by ARPES 84). (b) Superconducting energy gap; the solid line is a $d$-wave fit to the data, and solid circles represent ARPES results (84). From 267.

Figure 57

(Color online) Energy and spatial dependence of the DOS at $100\mathrm{K}$ on Bi2212. (a) A typical spectrum showing the pseudogap. (b) A typical $45\times 19.5{\mathrm{nm}}^2$ topograph showing atomic corrugation and the incommensurate supermodulation along the $b$ axis. (c) and (d) Real-space conductance maps recorded simultaneously at 41 and $6\mathrm{mV}$, respectively. From 398.

Figure 58

(Color online) Fourier analysis of DOS modulations observed at $100\mathrm{K}$ (see also Fig. 57). (a) Fourier-space image of a conductance map $(38\times 38{\mathrm{nm}}^2)$ acquired at $15\mathrm{mV}$. (b) The Fourier-space image shows peaks corresponding to atomic sites (labeled $A$), primary (at $2\pi ∕6.8a_0$) and secondary peaks corresponding to the $b$-axis supermodulation (labeled $S$), and peaks at $\sim 2\pi ∕4.7a_0$ along the $(\pm \pi ,0)$ and $(0,\pm \pi )$ directions (labeled $Q$). From 398.

Figure 59

(Color online) Dispersion of $q$ vectors observed in regions with SC coherence-peaked spectra (bottom curve) and in regions with pseudogaplike spectra (top curve). The vector ${\mathit{q}}^{*}$ has a length $\pi ∕4.5a_0\pm 15\%$ (265).

Figure 60

(Color online) Conductance map on ${\mathrm{Ca}}_{1-x}{\mathrm{Na}}_x\mathrm{Cu}{\mathrm{O}}_2{\mathrm{Cl}}_2$ at $8\mathrm{mV}$ and the corresponding Fourier transform. From 134.

Figure 61

(Color online) Local ordering in the vortex cores of Bi2212 seen in the difference map using the differential conductivity integrated from $0\text{to}12\mathrm{meV}$. The difference map was obtained by subtracting the conductivity map obtained in zero field from the one obtained in the presence of a field of $3\mathrm{T}$. From 158.

Figure 62

(Color) Square modulation in the vortex core. (a) Conductance map at $+6\mathrm{mV}$ of the area shown in Fig. 47a, revealing a square pattern around the vortex center. (b) Fourier transform at $+9.6\mathrm{mV}$ with $a^{*}=2\pi ∕a_0$, $q_1\approx \frac{1}{4}{a}^{*}$, and $q_2\approx \frac{3}{4}{a}^{*}$. (c) Filtered inverse FT image. Inset: The filter applied to the image acquired at $+6\mathrm{mV}$ and which selects the region around the four $q_1$ peaks. (d) Spectra averaged in the seven circles along the arrow in (c). The curves are offset for clarity. Adapted from 236.