Experimental Josephson Current-Voltage Characteristics

In the following I will make predictions for theI(VB) andI(VJ J) we expect to measure

for Josephson tunnelling between a Bi2Sr2CaCu2O8+δsample and the Bi2Sr2CaCu2O8+δ

nano-flake tips characterised in the previous section. The physics of ultra-small Joseph- son junctions such as these was discussed in section 5.3.1. The strong non-linearity of theI(VJ J) characteristics expected for these Josephson junctions means that the form of

circuit. Here I will include details of the measurement circuit used in our SJTM exper- iments and predict I(VB) and I(VJ J) that can be compared directly with experiment.

Figure 5.15 (a) shows a diagram of the hybrid STM/SJTM measurement circuit used in our experiments. A load resistor RB = 10 MΩ is biased, in series with the Josephson

junction formed between the STM tip and the sample, by a voltageVB. The tip-sample

junction is represented by the parallel combination of an ideal Josephson element with intrinsic critical current,IJ, a junction capacitance,CJ, and non-linear resistor,R(VJ J),

that represents the SIS single-particle tunnelling channel. The voltage dropped across the junction is denoted asVJ J. The ideal ammeter in this diagram is a low-noise current-

to-voltage pre-amplifier in actual experiments, as detailed in figure 2.9.

To model theI(VB) characteristics of the circuit I will take the current due to incoherent

Cooper-pair tunnelling, IP(VJ J), to be,

IP(VJ J) =aIJ2

VJ J

V_{J J}2 +V_C2 , (5.47)

which is the form predicted by Ivanchenko and Zi’lberman for classical phase diffusion in the limit CJ → 0 (see section 5.3.1) [152]. Here I use this function only as an

approximate analytic form for the pair current because it exhibits a pair current peak at voltageVC 6= 0 which scales in magnitude withIJ2. As detailed in section 5.3.4 these

are generic features of ultra-small Josephson junctions in a low impedance environment and so this approach is justified. Equation 5.47 is not intended to be used to extract the absolute value of IJ.

Although the Josephson junction could be regarded as current biased due to the use of a large series resistance, it can be effectively treated as voltage biased, as discussed in detail in section 5.3.2 [155, 157]. This is because the parasitic capacitance between the leads on either side of the junction, CP, is typically much larger than the capacitance

of the junction itself. As a result, the voltage across CP can be well approximated as

independent of time even though the current flowing through the Josephson junction consists of current spikes corresponding to tunnelling events. This renders the junction effectively voltage biased.

A

+ _

Figure 5.15: (a) Circuit diagram of the hybrid STM/ SJTM setup used in these experiments. The voltage bias VB is from the usual STM bias controller; the load

resistorRB = 10MΩ; the single-particle tunnelling resistance of the Josephson junction

formed between the tip and sample is R(VJ J), and the voltage actually developed

across the junction isVJ J. (b)The dynamics of this circuit produces two predominant

effects. First there is a discontinuous change in the I(VB) characteristic when the

current reaches a valueIc∝IJ2measured with the external ammeter shown. The second

predicted effect is strong hysteresis depending on which direction the external voltage is swept; this is shown as the difference between the solid-red and dashed-blue lines. Both effects are seen very clearly and universally in the measuredI(VB) throughout our

studies. (c)The current as in (b) but plotted against the voltage across the Josephson junctionVJ J.

The DCI(VB) for the circuit shown in figure 5.15 (a) is then given by the simultaneous

solution of I = VR RB =aI_J2 VJ J V2 J J+VC2 (5.48a) and VB =VR+VJ J . (5.48b)

We set R(VJ J) = ∞ because the shunt provided by the SIS single-particle tunnelling

Josephson junction for small VJ J.

Figure 5.15 (b) shows the numerical solutions to equations 5.48. The red and blue lines show the trajectories that the forwards and backwards sweeps take through the

I(VB) plane. The non-linearity of the circuit introduces discontinuities in the measured

I(VB). A discontinuity occurs at the point where the current reaches the maximum pair

current that the junction can pass,I =Ic, and thus makes this current experimentally

identifiable by the sharp I(VB) feature associated with it. Figure 5.15 (c) shows the

same trajectories but as a function of the voltage across the junction VJ J.

This circuit’s non-linearity also generates hysteresis in the I(VB) characteristic even if

the dynamics of the Josephson junction itself are non-hysteretic. This hysteresis is indeed observed throughout our SJTM studies of Bi2Sr2CaCu2O8+δ and systematic errors due

to the hysteresis are avoided during our I = Ic(~r) imaging experiments by sweeping

the applied voltage always in the same direction once the Josephson junction has been formed at each location~r.

We are now in a position to compare the above predictions with the Josephson tunnelling signatures detected in our experiment. Figure 5.16 shows the I(VJ J) characteristic

between our Bi2Sr2CaCu2O8+δ nano-flake tip and a Bi2Sr2CaCu2O8+δ sample, forT =

45 mK and fixed~r, as a function of decreasing junction resistance,RJ =Vs/Is, and thus

decreasing tip-sample distance.

First notice that, as predicted in the previous section and with comparison to figure 5.15 (c), theseI(VJ J) curves have a sharp discontinuity at their maximum current and exhibit

pronounced hysteresis. Further, as the Josephson critical current IJ increases with di-

minishing distance, the magnitude of the pair current peak increases as expected. These features are in accord with our predictions for I(VJ J) arising from incoherent Cooper

pair tunnelling between a superconducting tip and sample. Taken together they further demonstrate that our Bi2Sr2CaCu2O8+δ nano-flake tip is indeed superconducting.

As discussed in section 5.3, the maximum Cooper pair current arising from incoherent Josephson tunnelling, Ic, is proportional to IJ2 and IJ is directly proportional to the

amplitude of the sample superconducting order parameter |Ψ|. Thus, Ic as measured

from one of ourI(VJ J) curves is related to the sample’s superconducting order parameter

Figure 5.16: Measured evolution of the SJTM I(VJ J) with diminishing tip-sample

distance (diminishingRN) atT = 45 mK (junction formation conditions given in blue

text). The maximum currentIc for a typical I(VJ J) used in anIc(~r) measurement is

indicated by a dashed green arrow.

by measuring I(VJ J) at a series of points across the surface of a sample it is possible

to create a spectroscopic map, Ic(~r), that directly measures spatial variations in the

superconducting condensate throughIc(~r)∝ |Ψ(r~|2∝ρs(~r).

In general, we functionally define the quantity Ic(~r) to be the maximum magnitude of

current achieved on sweeping away fromVB = 0 with the tip at position~ron the surface.

An example of an Ic(~r) map is shown in figure 5.17. The most immediately apparent

features are a strong quasi-periodic modulation along the vertical axis and dark regions whereIc is almost completely suppressed. These are due to the Bi2Sr2CaCu2O8+δ bulk

crystal super-modulation and individual Zn impurity atoms respectively. In the following section I will use these features to validate SJTM as a probe of the superconducting order parameter using their known effect on the superconductivity in Bi2Sr2CaCu2O8+δ.

5nm

Figure 5.17: Typical SJTM Ic(~r) measurement of Bi2Sr2CaCu2O8+δ sample with

p=17% using a Bi2Sr2CaCu2O8+δ nanoflake tip. Strong quasi-periodic modulations

along the vertical axis are due to the Bi2Sr2CaCu2O8+δ bulk crystal super-modulation.

Red dots numbered 1-9 pass through the location of zinc impurity atom and are the locations for a series ofI(VB) curves shown in figure 5.19 that demonstrate the almost

complete suppression ofIc at the impurity site.