Temperature and field dependence of the intrinsic tunnelling structure in overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$

We report intrinsic tunneling data for mesa structures fabricated on three over- and optimally-doped ${\mathrm{Bi}}_{2.15}{\mathrm{Sr}}_{1.85}{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ crystals with transition temperatures of 86–78 K and 0.16–0.19 holes per ${\mathrm{CuO}}_2$ unit, for a wide range of temperature ($T$) and applied magnetic field ($H$), primarily focusing on one overdoped crystal (OD80). The differential conductance above the gap edge shows a clear dip structure which is highly suggestive of strong coupling to a narrow boson mode. Data below the gap edge suggest that tunnelling is weaker near the nodes of the $d$-wave gap and give clear evidence for strong $T$-dependent pair breaking. These findings could help theorists make a detailed Eliashberg analysis and thereby contribute towards understanding the pairing mechanism. We show that for our OD80 crystal the gap above $T_c$, although large, is reasonably consistent with the theory of superconducting fluctuations.

Figure 1

Typical $I-V$ curves for the three mesas at low bias voltages. Red points show data taken at 10 K while increasing $I$ to an appropriate maximum value and then decreasing it. This generates a series of curves in which, from left to right, the Josephson currents of an increasing number of junctions are suppressed because there is a finite voltage across them. The blue lines show fits of the form $I=m_{1}V+m_2{V}^3+m_3{V}^5$ to the ($N-1$)th curve. The coefficients $m_1, m_2$, and $m_3$ are then scaled by ${[(N-1)/n]}^i$where $i=1$, 3, and 5, respectively, and $n$ is an integer between 1 and $N$. Differences between the red data points and the blue lines give an indication of possible nonuniformity in junction areas, or more likely, their resistances.

Figure 2

(a) Log-log plots of $dI/dV$ curves measured for the three mesas at 1.4 K in zero magnetic field when sweeping $I$ down from its maximum value. The green curve agrees with an earlier calculation [25] and shows that purely incoherent tunneling gives completely different behavior. (b) $dI/dV$ for mesa OD80 at 1.4 K in 0 and 13 T fields after normalizing to the normal state conductance. The solid black curve corresponds to a normalized coherent part calculated from Eqs. (1) and (2) with $M^2\propto {(cos2\theta )}^4$, and multiplied by 0.8 plus a normalized incoherent part multiplied by 0.2. It gives a good description of the data below $V={\mathrm{\Delta }}_0$ after adding a small residual term, 0.012, to $G\left(V\right)/G_N\left(V\right)$. The longer and shorter dashed curves show the calculations for purely coherent tunneling with $M^2$ constant and $M^2\propto {(cos2\theta )}^4$, respectively.

Figure 3

Calculation showing sensitivity of structure above the gap edge to the resistance of individual junctions. The black curve corresponds to the case where all 10 junctions have the same resistance and $dI/dV$ as measured for the best ITJ OD80. The purple curve shows the effect of having three out of 10 junctions with $10\%$ higher resistance, i.e., $10\%$ larger values of $V$ for the same $I$.

Figure 4

Zero bias conductance for the three mesas at 1.4 K obtained from straight line fits to $I/V$ vs $V^2$ curves, for $V$ typically between 0.009 and 0.012 $V$, at various fields, $H$ applied along the $c$ axis. The normal state conductance at zero bias $G_N\left(0\right)$ is obtained from the polynomials that give states-conserving fits, see Ref. [38]. The dashed lines show fits for $20\%$ incoherent tunneling of unpaired quasiparticles near the nodes in one layer, generated by the Volovik effect, to zero energy non-nodal states in the neighboring layer (see text).

Figure 5

(a) $dI/dV$ curves for OD80 at 1.4 K in various magnetic fields applied perpendicular to the ${\mathrm{CuO}}_2$ planes plotted vs $V/N$, the bias voltage per junction. Numerical data is available [44]. (b) and (c) show details of the field dependence of the lower and upper dips. Here $dI/dV$ curves have been normalized by dividing through by the polynomial given in Ref. [38]. Values of $({\mathrm{\Delta }}_0+\mathrm{\Omega })/e, (2{\mathrm{\Delta }}_0+\mathrm{\Omega })/e$, and $2({\mathrm{\Delta }}_0+\mathrm{\Omega })/e$ are shown by arrows (see text).

Figure 6

(a) $dI/dV$ curves for OD80 at selected values of $T$. Numerical data for 31 values of $T$ between 1.4 and 300 K is available [44]. (b) $T$ dependence of the $d$-wave gap $2{\mathrm{\Delta }}_0\left(T\right)$ up to $T_c$ from the main peaks in $dI/dV$ for OD80 (black circles), OD78 (red squares), and OP86 (blue triangles). For OD80, green squares above $T_c=80$ K give voltages of broad maxima in $dI/dV$. Green dashed lines show their increased breadth by marking regions where $dI/dV\ge 0.95{(dI/dV)}_{\mathrm{MAX}}$.

Figure 7

Zoom of the structure above the gap edge for OD80 at selected $T$. The data have been normalized for clarity, see Ref. [38]. Values of $({\mathrm{\Delta }}_0\left(0\right)+\mathrm{\Omega })/e, (2{\mathrm{\Delta }}_0\left(0\right)+\mathrm{\Omega })/e$, and $2({\mathrm{\Delta }}_0\left(0\right)+\mathrm{\Omega })/e$ are shown by arrows (see text). Data at 1.4 K for a magnetic field of 13 T $\parallel c$ are also shown.

Figure 8

(a)–(e) Comparison of normalized $dI/dV$ curves with the Dynes formula Eq. (2), with a 4:1 coherence-incoherence ratio, and the same damping factor, $\mathrm{\Gamma }=0.009+0.07x^4/(1+x^2)$ where $x=E/{\mathrm{\Delta }}_0\left(T\right)$, at the selected temperatures shown. (f) Effect of various extra dampings, $\mathrm{\Gamma }=0.09+0.07x^4/(1+x^2)$ etc. at 75 K.

Figure 9

Values of $G_S\left(0\right)/G_N\left(0\right)$ vs temperature for the three mesas. (a) shows the data together with fits to $m_1+m_1{(T/m_3)}^4$ with $m_1=0.0129$, 0.0115, and 0.0122 and $m_3=33$, 36, and 31 K for mesas OD78, OD80, and OP86, respectively. (b) shows the same data and fits to the Arrhenius law shown with $m_4=0.0139$, 0.0119, and 0.0131, $m_5=2.60$, 0.65, and 1.67, $m_6=182$, 158, and 147 K for mesas OD78, OD80, and OP86, respectively. All fits have been made from 1.4 to 40 K. For clarity the data for OD78 and OP86 have been displaced along the logarithmic $y$ axis by the multiplying factors shown.

Figure 10

Values of $\mathrm{\Gamma }\left(T\right)\equiv \sqrt{G_S\left(0\right)/G_N\left(0\right)}\times \left[{\mathrm{\Delta }}_0\left(T\right)\right]$ vs temperature for the three mesas. (a) shows the data together with fits to the formula shown that is expected for electron-electron scattering with $m_1=3.07$, 2.89, and 3.07 meV, $m_2=7.1$, 8.5, and 6.2 meV, $m_3=50$, 55, and 42 K for mesas OD78, OD80, and OP86, respectively. (b) shows the same data and fits to the Arrhenius law shown with $m_4=3.13$, 2.91, and 3.16 meV, $m_5=56$, 49, and 64 meV, $m_6=136$, 139, and 128 K for mesas OD78, OD80, and OP86, respectively. All fits have been made from 1.4 to 50 K. For clarity the data for OD78 and OP86 have been displaced along the logarithmic $y$ axis by the multiplying factors shown.

Figure 11

$T$ dependence of the difference between the maximum of $dI/dV$ at the hump and the minimum at the dip for OD80. The points have been obtained from normalized $dI/dV$ curves such as those shown in Fig. 7. The data show a clear $T^2$ dependence that goes to zero at 50 K.

Figure 12

(a) $dI/dV$ curves measured for OD80 above $T_c$ plus one curve at 78 K for comparison. Numerical data for 31 values of $T$ between 1.4 and 300 K is available [44]. (b) Zoom of data measured for OD80 in the range 110–300 K, after normalizing by a polynomial that gives a states-conserving curve at 1.4 K and 13 T, see Ref. [38]. At higher $T$ the sweep voltage range had to be restricted because of hysteresis associated with voltage-induced changes in the mesa resistance [59], as shown for example by the data at 300 K.

Figure 13

Voltages of maxima in normalized $dI/dV$ curves obtained from the data in Fig. 12, by fitting to $a+b{(V-V_{\max })}^2$ near the maxima. The linear $T$ dependence expected [23] from superconducting fluctuation theory has unit slope on the logarithmic scales used. There is indeed a linear region with approximately the expected slope of $2.0\pi k_B$ between 87 and 110 K. Results are shown for two different normalizing polynomials [38].

Figure 14

Measured values of the zero-bias conductance vs $T$ at all temperatures above $T_c$. The dashed line shows the estimated normal state conductance background $G_N(0,T)$ obtained by assuming that $1/G_N(0,T)=a+bT$ and finding $a$ and $b$ from the points at 250 and 300 K. The superconducting fluctuation part $G_{\mathrm{FL}}$ is also shown.

Figure 15

Normalized values of the fluctuation contribution to the zero-bias conductance, $\delta G_{\mathrm{MEAS}}=[G_N(V=0,T)-G_{\mathrm{MEAS}}(V=0,T)]/G_N(V=0,T)$, obtained from the data in Fig. 14, are plotted vs $ln[1/ln(T/T_{\mathrm{MF}})]$ for the three values of $T_{\mathrm{MF}}$ shown. The slopes of the lines shown give values of the Ginzburg parameter ${\tau }_G$ (see text).