Non-local detection of the Shiba state

Presence of a non-local signal

The differential conductance of the tunnel probe, G_T is shown on Fig. 5.3a-c as the function of V_B and V_P for three different values of the magnetic fields, 0 mT on panel a, 150 mT on panel b and 250 mT on panel c. The conductance of the right arm, G_QD is quenched in the whole gate range. In the absence of magnetic field (panel a) two pairs of parallel resonance lines are present on top of a smooth background of about 0.1 G₀ conductance (G₀ = 2e²/his the conductance quantum). As I will discuss in the following, the enhanced conductance lines are the non-local transport signatures of the Shiba state formed on the quantum dot. In zero magnetic field the conductance enhancement is about 0.005 G₀.

Upon applying an in-plane magnetic field smaller than critical field of the supercon-ductor the signal is significantly increased (see Fig. 5.3b measured in 150 mT). The largest peak is close to 0.1 G₀, corresponding to a 20 times enhancement compared to the zero-field value.

In 250 mT magnetic field, above the critical field of the superconductor,B_c= 230 mT the conductance enhancement vanishes (see Fig. 5.3c). This indicates that the origin of the additional resonance lines is indeed superconductivity-related.

2The sample was fabricated at the University of Basel by Gerg˝o F¨ul¨op.

5.1. Magnetic field enhancement of non-local Shiba signal

-0.5

-0.6 VP (V)

-1.45 -1.4

V_B (V) -1.45 -1.4

V_B (V) -1.45 -1.4

V_B (V) 0.09 0.193

GT (G0) B = 0 mT B = 150 mT B = 250 mT

a) b) c)

Figure 5.3: Conductance of the tunnel probe as the function of V_B and V_P gate voltages measured in c)B = 0 mT, d)B = 150 mT, e) B= 250 mT. The two pair of parallel resonance lines are non-local transport signatures of the Shiba state. The magnitude of the conductance enhancement significantly increases with magnetic field (panel b), and vanishes above the critical field,Bc= 230 mT of the superconductor (panel c). Note that the same color scale is used for the three panel for better comparison. On panel c the conductance barely changes in the used gate voltage range, it follows the homogenous, yellow color. The line cuts at V_B = −1.497 V, shown in panels a-c), further illustrate the strong dependence of the conductance enhancement on the magnetic field.

Line cuts made at V_B=−1.497 V shown on each panels of Fig. 5.3 further illustrates the strong dependence of the conductance enhancement on the magnetic field. From now on I will focus on the resonance pair marked by a circle and a triangle.

In the following I will prove that the conductance enhancement in G_T is indeed the signature of the Shiba state, when it is tuned to zero energy, by carrying out a simultaneous measurement of the current in leads N and N_QD after opening up the barrier between the dot and N_QD.

Direct characterization of the Shiba state

Increasing V_B to more positive values opens up the barrier to N_QD and allows for the direct transport characterization of the quantum dot. Fig. 5.4a&b shows the conductance of the tunnel probe G_T and the dot, G_QD, respectively, in a larger gate voltage window for B = 0 mT. Let us follow the marked resonance lines to more positive V_B values. In G_T the resonance are preserved, and above above V_B ≈ −1.1 V G_QD also sets in. The similarities of the conductance enhancement in G_T and the resonances of G_QD implies that the enhancement of GT is related to the level structure of the quantum dot.

Note 1: the magnitude of the non-local signal is increased when G_QD is increased, which I will discuss in the following section.

Note 2: Analyzing the fine structure of GQD reveals that right arm of the nanowire hosts not one but two quantum dots. The slope of the second dot’s resonances – being almost vertical, indicated by the gray line on Fig. 5.4b – implies that this dot is situated further

5.1. Magnetic field enhancement of non-local Shiba signal

-1.5 -1

-0.5

V_B (V) VP (V)

-0.6

-0.7

1.15

0 G_QD (G₀)

-0.5

VP (V) -0.6

-0.7-1.5 -1

V_B (V)

0.21

0.094 G_T (G₀)

a) b)

c)

0.2

0

-0.2 Vbias (mV)

-0.02 0.21

-0.66 -0.62

G_QD (G₀) c)

V_P (V) 4.4b)

QD N SC

InAs NW QD

SC N_QD

Figure 5.4: Conductance of the tunnel probe, G_T (on panel a) and the quantum dot, G_QD (on panel b) in a larger gate voltage range, in zero magnetic field. The white dashed rectangle marks the gate voltage region already presented on Fig. 5.3a. For more positiveVB values GQD

is restored, while the conductance enhancement persists in G_T. The similarities between the conductance lines on panel a and b indicate that the tunnel probe measures the level structure of the quantum dot non-locally. c) Finite-bias direct characterization of the quantum dot along the dashed line on panel b. The eye-shaped crossing is the usual signature of the Shiba state.

5.1. Magnetic field enhancement of non-local Shiba signal

from the superconducting lead. Accordingly, these resonances are not observable inG_T. Note 3: the sudden jumps of the conductance values of G_T is attributed to an instability of the tunnel probe. Apart from the magnitude of the background conductance it does not influence the Shiba state-related non-local signal.

To have further insight into the level structure of the quantum dot the stability di-agram, i.e. finite-bias transport spectrum was measured along the white dashed line on panel b of Fig. 5.4 and is shown on Fig. 5.4c. The eye-shaped crossing of the conductance lines are the usual fingerprint of the Shiba state formed in a SC–QD hybrid (see e.g.

Fig. 2.22). The presence of the Shiba states indicates the strong coupling between the superconductor and the dot, which hybridizes the states of the two.

The previously shown resonance lines of G_QD on Fig. 5.4b correspond to the Shiba state, when its energy is tuned to zero byV_Pplunger gate voltage. As I already mentioned above, comparing panel a and b of Fig. 5.4 reveals that the conductance enhancement of G_T coincides with the resonances of G_QD, when the latter is unquenched. Furthermore, the conductance enhancement persists even for low V_B region, where G_QD is quenched.

From these similarities one can conclude that the conductance enhancement of G_T sets is when the Shiba state of the dot is tuned to zero energy.

Note that in the presented device the tunnel probe is 50 −250 nm away from the quantum, much further, than the typical extension of a Shiba state in STM geometries, hence the presence of the strong non-local signal is surprising.

Effect of magnetic field

A detailed analysis of the finite magnetic field behavior is shown Fig. 5.5. Panel a shows the reduction of superconducting gap with the magnetic field measured on the quantum dot at V_P = −0.7 V and V_B = −0.6 V. Although according to the Ginzburg-Landau theory the gap is reduced as ∆(B) = ∆₀

1− ^BB_c²²

in magnetic field, I have found a somewhat different functional form,

∆(B) = ∆₀ 1−B²/B_c²1/2

, (5.1)

in agreement with observations in thin layer superconductors [193,197]. The white dashed line is a fit of the superconducting gap with zero-field gap, ∆₀ = 250µeV and the critical field, B_c = 230 mT. Note that the here measured induced gap is smaller than the 0.4− 1 meV in Refs. [47, 194] and than the bulk gap of 1.1 meV in Ref. [198].

Fig. 5.5b shows the detailed evolution of the non-local signal with the magnetic field measured along the dashed line on Fig. 5.4a at V_B = −1.25 V and at zero bias. While close to B = 0 the resonance peaks are barely visible, they are strongly enhanced with increasing magnetic field, particularly between 100 and 200 mT. The dotted lines are guide to the eye. Note that the increasing separation of the conductance lines with magnetic field is in line with the expectations of the Zeeman splitting of the Shiba state. The singlet is unaffected by the magnetic field, but the doublet states split. Due to the decreasing energy of the doublet ground state in magnetic field, the degeneracy points move apart.

For details see e.g. Ref. [44].

The contribution of the Shiba state to the tunnel current is quantified by integrating the conductance enhancement peak shown on Fig. 5.5b. In detail, first I subtracted theV_P -independent background conductance for each magnetic field value, then I have integrated

5.1. Magnetic field enhancement of non-local Shiba signal

0.000 0.05 0.10 0.15 0.20 1

2 3 4

0.00 0.05 0.10 0.15 0.20 0.0

0.2 0.4 0.6 0.8 1.0

a)

GQD (10 -2 G0) 1.2

0 0.4

0 -0.4 Vbias (mV)

0 0.1 0.2

B (T)

VP (V) b)-0.52

-0.54

-0.56

0 0.1 0.2

B (T) 0.08 0.2

GT (G0)

B (T) c)

0 1

0.2 0.4 0.6 0.8 d)

Excess current (nA)

0 0.1 0.2

0 1 2 4

I QM Shiba

/ I

QM Shiba

,max

2 2.5 3 3.5 x_T/πξ₀

B (T)

0 0.1 0.2

3

Figure 5.5: a) Reduction of the superconducting gap with magnetic field. The dahsed line is a fit with Eq. (5.1) b) Evolution of the non-local signal with magnetic field along the dashed line on Fig. 5.4a. The gray dotted lines are guide to the eye. c) The magnetic field evolution excess current attributed to the Shiba state, evaluated from the data on panel b (for the procedure see the text). d) NRG simulation of the excess current for different values ofxT/ξ0.

5.1. Magnetic field enhancement of non-local Shiba signal

the peaks along the V_P axis and converted the gate voltage value to energy by using the lever arm of 57.5 estimated from the stability diagram shown on Fig. 5.4c. In the following I will call the obtained quantity excess current, which is shown on Fig. 5.5c to further illustrate the strong dependence of the non-local signal on the magnetic field. Here it is more clear that for low fields, below 150 mT the excess current has a strong upturn, is maximal around 180 mT, above that it gets strongly suppressed and finally vanishes at B_c = 230 mT.

In document Spin-orbit interaction and superconductivity in InAs nanowire-based quantum dots devices (Pldal 99-104)