Meservey-Tedrow-Fulde effect in a quantum dot embedded between metallic and superconducting electrodes

# Meservey-Tedrow-Fulde effect in a quantum dot embedded between metallic and superconducting electrodes

## Abstract

Magnetic field applied to the quantum dot coupled between one metallic and one superconducting electrode can produce a similar effect as has been experimentally observed by Meservey, Tedrow and Fulde [Phys. Rev. Lett. 25, 1270 (1970)] for the planar normal metal – superconductor junctions. We investigate the tunneling current and show that indeed the square root singularities of differential conductance exhibit the Zeeman splitting near the gap edge features . Since magnetic field affects also the in-gap states of quantum dot it furthermore imposes a hyperfine structure on the anomalous (subgap) Andreev current which has a crucial importance for a signature of the Kondo resonance.

## I Introduction

Already in early days of the tunneling spectroscopy it has been shown that magnetic field (which couples to spin of the charge carriers) is in superconductors responsible for splitting the square root singularities of the tunneling conductance (1) by the Zeeman energy , where is the Bohr magneton. This Meservey-Tedrow-Fulde (MTF) effect has been observed experimentally in the thin superconducting aluminum films applying parallel magnetic field so that orbital diamagnetic effects could be avoided. Similar qualitative results have been recently noticed in the measurements of -axis tunneling for the layered high temperature superconducting compounds (2).

We argue that the MTF effect should be also feasible in various nanostructures consisting of a quantum dot (QD) placed between one metallic and one superconducting electrode. Zero-dimensional character of QDs in a natural way eliminates the influence of orbital effects therefore magnetic field would affect the charge transport only through the Zeeman term. This can in turn manifest itself in the differential conductance. Roughly speaking, the charge current flows if an external bias exceeds the energy gap (necessary to break the Cooper pairs into individual electrons) thereof the resulting conductance has a low voltage onset near the gap edges . In presence of a magnetic field these gap edge singularities are going to split (see section III).

More detailed analysis of the charge tunneling (3) involves however also the additional (anomalous) channels due to mixing of the particle and hole excitations in superconductors. In particular, even at subgap voltages the mechanism of Andreev reflections provides a finite contribution to the conductance. Since the Andreev mechanism is very sensitive to location of the in-gap QD states (4); (5); (6); (7); (8) and the on-dot correlations (9); (10); (11); (12); (13); (14); (15); (16); (17); (18) we shall explore the influence of magnetic field on such subgap conductance. In section IV we discuss a hyperfine structure for the Andreev conductance neglecting the correlations. In the next section V we extend our study taking into account a finite value of the on-dot repulsion . We show that appearance of the low temperature Kondo resonance enhances the zero bias conductance and this feature undergoes the Zeeman splitting when magnetic field is applied.

As concerns some practical aspects, there have been considered the proposals for using the magnetic field tuned Andreev scattering as an efficient cooling mechanism in two dimensional electron gas - superconductor nanostructure (19). There is also considered a possibility to use the, so called, Andreev quantum dot as a magnetic flux detector (20).

## Ii The model

For a general description of transport phenomena through a nanoscopic island placed between external leads one should consider a quantized multilevel structure of QD (21). However, in the case when a level spacing is smaller in comparison to QD hybridization with the electrodes one can restrict to a simplified picture of the Anderson model (5); (6); (9); (10); (11)

 ^H = ^HN+^HS+∑σϵd,σ^d†σ^dσ+U^nd↑^nd↓ (1) + ∑k,σ∑β=N,S(Vkβ^d†σ^ckσβ+V∗kβ^c†kσ,β^dσ).

Operators () denote the annihilation (creation) of electron whose energy level is and is the on-dot Coulomb repulsion between opposite spin electrons. The last terms describe hybridization of QD with the normal () and superconducting () electrodes. Magnetic field eventually shifts the QD level by , where the spin-dependent coefficients are defined as and .

Hamiltonian of the normal (metallic) lead is taken as whereas for the superconducting electrode we choose the usual BCS form with an isotropic energy gap . The relative energies are measured from the chemical potentials . We shall focus on the wide band limit (where ) and consider a small external voltage , which detunes the chemical potentials by inducing the charge flow through N-QD-S junction. We assume to be much smaller than level spacings typical for the realistic QDs (21) so that applicability of the model (1) can be justified.

Let us start by establishing the QD Green’s function in the equilibrium situation, i.e. for . Fourier transform of the retarded Green’s functions can be formally expressed by the Dyson equation

 \boldmath{G}σ(ω)−1 ≡ ⎡⎣⟨⟨^dσ;^d†σ⟩⟩ω⟨⟨^dσ;^dσ⟩⟩ω⟨⟨^d†−σ;^d†σ⟩⟩ω⟨⟨^d†−σ;^d−σ⟩⟩ω⎤⎦−1 (2) = [ω−εd,σ00ω+εd,−σ]−\boldmath{Σ}0d,σ(ω)−\boldmath{Σ}Ud,σ(ω)

where denotes the selfenergy of noninteracting QD () and accounts for the correlation effects. For a simple understanding of the MTF effect it would be helpful to focus first on the uncorrelated QD when the selfenergy is known exactly. Further corrections due to contribute a renormalization of the spectral function (14) whose impact on the charge transport will be discussed separately in section V.

For convenience we introduce the hybridization coupling and define the following spin-dependent energy . Imaginary part of the selfenergy for is given by while at large energies it takes the following form (12); (15)

 Im\boldmath{Σ}0d,σ(ω)= −12⎡⎢ ⎢ ⎢ ⎢ ⎢⎣ΓN+ΓS|~ωσ|√~ω2σ−Δ2ΓSΔsgn(~ωσ)√~ω2σ−Δ2ΓSΔsgn(~ωσ)√~ω2σ−Δ2ΓN+ΓS|~ωσ|√~ω2σ−Δ2⎤⎥ ⎥ ⎥ ⎥ ⎥⎦. (3)

The corresponding real parts can be determined using the Kramers-Krönig relations.

Imaginary part of the selfenergy has thus the square root singularities at energies , so in presence of magnetic field there are altogether 4 such points. They show up as kinks in the spectral function , where

 ρd,σ(ω)=−1πIm⟨⟨^dσ;^d†σ⟩⟩ω+i0+. (4)

We shall see below that appearance of such characteristic points leads to the MTF effect observed in the tunneling conductance.

## Iii Meservey-Tedrow-Fulde effect

To compute the tunneling current we adopt the formalism outlined in the previous studies (9); (8); (12) extending it here on a situation with the spin sensitive transport due to magnetic field. The steady charge current is defined as . We carry out the time derivative and determine the expectation value using the nonequilibrium Keldysh Green’s functions.

In analogy to the standard Blonder-Tinkham-Klawijk theory (3) we express the current as composed of two contributions

 I(V)=I1(V)+IA(V). (5)

The first part stands for a contribution which at low temperatures appears practically outside the energy gap . Its magnitude is expressed by the Landauer-type formula

 I1(V)=eh∑σ∫dωT1,σ(ω)[f(ω+eV)−f(ω)], (6)

where . The transmittance is nonvanishing only outside the energy gap and is given by the following parts of the retarded Green’s functions (8); (12)

 T1,σ(ω) = ΓNΓS|~ωσ|√~ω2σ−Δ2(∣∣⟨⟨^dσ;^d†σ⟩⟩ω∣∣2+∣∣⟨⟨^dσ;^d−σ⟩⟩ω∣∣2) (7) − 2ΓNΓSΔ√~ω2σ−Δ2Re{⟨⟨^dσ;^d†σ⟩⟩ω⟨⟨^dσ;^d−σ⟩⟩∗ω}.

The second part in (5) originates from the mechanism of Andreev reflections (3); (9); (12)

 IA(V)=eh∑σ∫dωTA,σ(ω)[f(ω+eV)−f(ω−eV)]. (8)

Its transmittance is finite even inside the energy gap (8); (12)

 TA,σ(ω)=Γ2N∣∣⟨⟨^dσ;^d−σ⟩⟩ω∣∣2. (9)

Physically such process occurs when an incident electron from electrode (of arbitrary energy) is converted into a pair on QD (with a simultaneous reflection of a hole) and it propagates in electrode as a Cooper pair. This anomalous Andreev current is closely related to the off-diagonal order parameter induced in the QD (proximity effect) (15); (14).

Figure (1) illustrates the influence of magnetic field on the total differential conductance obtained for N-QD-S junction. We clearly notice the Zeeman splitting of the square root singularities resembling the former experimental observation for N-I-S (I-insulator) junction (1). However, in a present case the conductance does not saturate to a finite value far outside the gap because the QD spectrum spreads only nearby (usually in realistic multilevel QDs there would be seen the quantum oscillations of (21)). The in-gap features related to the Andreev current are discussed in the next section.

## Iv Magnetic field effect on the Andreev current

The mechanism of Andreev reflections transmits the charge current even for the subgap voltages. To focus solely on this anomalous current it is convenient to consider the extreme limit as proposed by Tanaka et al (14). In such case can be completely discarded from our analysis. Using (3) we obtain the selfenergy simplified to (9); (15); (14)

 \boldmath{Σ}0d,σ(ω)=−12[iΓNΓSΓSiΓN] (10)

Upon neglecting the Coulomb correlations one can analytically determine the Green’s function (2), where the spin dependent spectral function (4) acquires the BCS structure (14)

 ρd,σ(ω) = 12[1+εdEd]1πΓN/2(~ωσ−Ed)2+(ΓN/2)2 (11) + 12[1−εdEd]1πΓN/2(~ωσ+Ed)2+(ΓN/2)2

with a quasiparticle energy . The in-gap QD states (often referred as Andreev bound states) form around as illustrated in the upper panel of figure 2. Their line broadening is given by and in absence of magnetic field the particle-hole splitting is controlled by (14); (15) (the dashed line in figure 2). Magnetic field further enforces the Zeeman splitting of these in-gap states.

Above mentioned behavior has an indirect effect on the off-diagonal parts of the Green’s function (2) which in turn determine the Andreev transmittance. In the limit (9) reduces to

 TA,σ(ω)=Γ2N(ΓS/2)2[(~ωσ−Ed)2+(ΓN/2)2][(~ωσ+Ed)2+(ΓN/2)2]

The subgap Andreev conductance is thus characterized by a four peak structure as shown in the bottom panel of figure 2. Obviously the weights of particle and hole peaks of the spectral function (11) as well as their weights in the Andreev transmittance (LABEL:T_A_ingap) depend on the QD level . Variation of the Andreev conductance with respect to (, ) is plotted in figure 3. We can notice that optimal conditions for the subgap current occur when the QD level is located near the energy gap center, otherwise the proximity effect is less efficient.

On top of the particle-hole structure seen in the Andreev states there is an additional Zeeman splitting brought by magnetic field. In figure 4 we sketch the Andreev conductance in (,) plane for , where the dark areas correspond to a maximal value . There appears a characteristic diamond shape marking the positions of such maximal conductance . We believe that this hyperfine structure could be probed experimentally.

To complete the discussion of the subgap Andreev current we briefly comment on a possible influence of an asymmetry between the hybridization couplings , . We explore for this purpose the zero bias conductance . At low temperature we find from equation (LABEL:T_A_ingap) that

 GA(0)= (13) 4e2hΓ2N(ΓS/2)2[(μBB−Ed)2+(ΓN2)2][(μBB+Ed)2+(ΓN2)2].

In figure 5 we show the influence of magnetic field on the zero bias Andreev conductance for several values of the asymmetry rate . If then a line-broadening of the Andreev states diminishes so in consequence the particle and hole peaks become well separated. Under such conditions the subgap conductance has maxima around the quasiparticle states at (where the ideal conductance is reached). Let us recall, that in absence of magnetic field the equation (13) reproduces for the well known result (14). For the symmetric coupling it yields (9).

## V Influence of the Coulomb correlations

In the limit the selfenergy becomes a static quantity (10) therefore the role of superconducting lead can be exactly replaced by the on-dot gap parameter . Instead of (1) we can thus use the following auxiliary Hamiltonian

 ^H = ^HN+∑k,σ(VkN^d†σ^ckσβ+h.c.)+∑σϵd,σ^d†σ^dσ (14) + (Δd^d†↑^d†↓+h.c.)+U^nd↑^nd↓,

which turns out to be very convenient for investigating the correlations. Tanaka and coworkers (13); (14) were able to rigorously prove that the selfenergy must have a diagonal structure due to invariance of term on the Bogoliubov-Valatin transformation.

In the remaining part of this section we shall focus on the subgap Andreev current transmitted through the correlated QD. The matrix Green’s function (2) simplifies in the limit to the following (exact) structure

 \boldmath{G}σ(ω)= (15) (ω−εd,σ−ΣN,σ(ω)12ΓS12ΓSω+εd,−σ+Σ∗N,−σ(−ω))−1.

Influence of the correlations have been so far analyzed for the Hamiltonian (1) using various techniques (9); (10); (11); (12); (13); (14); (15). Here we estimate the diagonal selfenergy within (14) by the equation of motion method (22); (25)

 ω−εd,σ−ΣN,σ(ω)=[ω−εd,σ−Σ0d,σ(ω)][ω−εd,σ−U−Σ0d,σ(ω)−Σ3d,σ(ω)]+UΣ1d,σ(ω)ω−εd,σ−Σ0d,σ(ω)−Σ3d,σ(ω)−U[1−⟨^nd,−σ⟩] (16)

where are given by (22)

 Σνd,σ(ω) = ∑k|VkN|2(1ω+ξkN−εd,−σ−εd,σ−U+1ω−ξkN+εd,−σ−εd,σ)[f(ω,T)]3−ν2. (17)

Approximation (16,17) qualitatively reproduces the following properties caused by on-dot correlations: (i) the charging effect and (ii) a possible appearance of the Kondo resonance for temperatures smaller than . The latter one is related to screening of the quantum dot spin by itinerant electrons of the metallic lead. In the case when energy level is located slightly below the hybridization induces effectively antiferromagnetic interaction between the QD and metallic lead. In consequence the bound singlet state can be formed giving rise to the resonance at for temperatures . Magnetic field eventually splits this resonance as illustrated in figure 6.

Any features present in the QD spectrum are further showing up in the measurable differential conductance. This is also valid for the Kondo resonance. Since it forms near the chemical potential therefore its signatures appear predominantly in the low voltage current. In fact, it has been shown that Kondo resonance enhances at low temperatures the zero bias Andreev conductance (9); (15), however its magnitude remains much smaller than the unitary limit value typical for N-QD-N systems in the Kondo regime. In the present context we emphasize that magnetic field enforces the Zeeman splitting of the zero bias Andreev anomaly in much the same way as it affects the zero bias anomaly for the QD coupled to both metallic leads (23); (24).

The zero bias enhancement of the Andreev conductance is a feature whose presence might be difficult to notice (9); (10); (15) unless some stringent requirements are fulfilled (25). It turns out that optimal conditions for the low temperature enhancement of take place when is comparable to (see figure 8) and is located slightly below the energy gap center. For an increasing asymmetry between the hybridizations , the magnitude of low voltage Andreev conductance diminishes (similarly as we have been shown in section IV upon neglecting the correlations). On the other hand, for moving far aside from the superconductor’s gap center the proximity effect becomes weakened and the overall Andreev conductance is again suppressed.

In general it seems that an interplay between the on-dot pairing (absorbed from the superconducting electrode) and the Kondo state (due to screening of QD spin by the metallic lead electrons) has the same character as a competition of superconductivity versus magnetism in the solid state physics. Since this is outside the main scope of the present topic we shall discuss it separately (25). A combination of the Kondo physics, superconductivity and the Zeeman polarization is a complex problem and to our knowledge only few papers have so far attempted to address this challenging issue (26).

## Vi Summary

We have explored the effect of magnetic field on charge transport through the quantum dot attached to one normal and one superconducting electrode. For a bias voltage we find the Zeeman splitting of the square root singularities in the differential conductance. This resembles the experimental result of Meservey, Tedrow and Fulde observed in the N-I-S junction (1) which for the N-QD-S structures it seems rather easy to achieve.

We have extended our study also on the in-gap Andreev current. Due to the proximity effect the particles and holes of the quantum dot get mixed and effectively the spectrum acquires the BCS-like structure (11). Differential conductance of the in-gap current indirectly probes such structure of the bound Andreev states. We have shown that magnetic field leads to appearance of four peaks via the combined particle-hole and Zeeman splittings. We hope that this result might stimulate a search for the experimental detection of above mentioned structures.

Moreover, we have explored influence of the on-dot Coulomb interactions on the subgap Andreev current assuming the extreme limit . In general, the on-dot correlations contribute to the QD spectrum: (i) appearance of the Coulomb satellite near (charging effect), and (ii) at sufficiently low temperatures can produce the narrow Kondo resonance at the chemical potential . Magnetic field imposes the hyperfine splitting onto such spectrum in a similar way as has been observed in N-QD-N junctions (24). The Kondo effect alone is exemplified in the zero bias Andreev conductance where under appropriate conditions (25) a low temperature enhancement can be seen if and the gate voltage tunes nearly to the energy gap center.

It would be of interest to use some more sophisticated methods for treating the on-dot interaction in order to check whether there exist a minimal magnetic field necessary for splitting the Kondo peak (as theoretically predicted for N-QD-N junctions (23)) observable in the Andreev conductance. One can also study QD coupled with -wave superconductor, where the square root singularities are replaced by weaker kinks. We think that the Meservey-Tedrow-Fulde effect would be observable there too (but in a less pronounced manner) whereas the subgap conductance might qualitatively change.

Acknowledgment We thank A.M. Gabovich for discussion leading to this study and P. Fulde for interest and encouragement. This work was partly supported by the Ministry of Science and Education under the grants NN202187833 and NN202373333.

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