Spin-resolved Andreev levels and parity crossings in hybrid superconductor-semiconductor nanostructures
Updated version of this manuscript is available at Nature Nanotech. 9 (2014) 79.
The hybrid combination of superconductors and low-dimensional semiconductors offers a versatile ground for novel device concepts Sdfreview (), such as sources of spin-entangled electrons Hofstetter (); Herrmann (); DasCooper (), nano-scale superconducting magnetometers Cleuziou (), or recently proposed qubits based on topologically protected Majorana fermions Sau (); Lutchyn (); vonOppen (). The underlying physics behind such hybrid devices ultimately rely on the magnetic properties of sub-gap excitations, known as Andreev levels. Here we report the Zeeman effect on the Andreev levels of a semiconductor nanowire quantum dot (QD) strongly coupled to a conventional superconductor. The combination of the large QD -factor with the large superconductor critical magnetic field allows spin degeneracy to be lifted without suppressing superconductivity. We show that a Zeeman-split Andreev level crossing the Fermi energy signals a quantum phase transition in the ground state of the superconductivity-induced QD, denoting a change in the fermionic parity of the system. This transition manifests itself as a zero-bias conductance anomaly appearing at a finite magnetic field, with properties that resemble those expected for Majorana fermions in a topological superconductor MajoDelft (); MajoWeizmann (); MajoLund (); Finck (); Churchill (). Although the herein reported zero-bias anomalies do not hold any relation with topological superconductivity, the observed parity transitions can be regarded as precursors of Majorana modes in the long-wire limit Stanescu ().
When a normal-type () conductor is connected to a superconductor (), superconducting order can leak into it giving rise to pairing correlations and to an induced superconducting gap. This phenomenon, known as the superconducting proximity effect, is expected also when the conductor is reduced to a small quantum dot (QD) with a discrete electronic spectrum. In this case, the superconducting proximity effect competes with the Coulomb blockade phenomenon, which follows from the electrostatic repulsion among the electrons of the QD. While superconductivity priviliges the tunneling of electron pairs with opposite spin, thereby favoring QD states with even numbers of electrons and zero total spin (i.e. spin singlets), the local Coulomb repulsion enforces a one-by-one filling of the QD, thereby stabilizing not only even but also odd electron numbers.
In order to analyse this competition, let us consider the elementary case of a QD with a single, spin-degenerate orbital level. When the QD is singly-occupied, two ground states (GSs) are possible: a spin-doublet (), , or a spin singlet (), , whose nature has two limiting cases. In the large superconducting gap limit (), the singlet is superconducting-like, , corresponding to a Bogoliubov-type superposition of the empty, , and doubly-occupied, , states. In the strong coupling limit, where the QD-S tunnel coupling, , is much larger than , quasiparticles in the superconductor screen the local magnetic moment in the QD and the singlet is Kondo-like. Although the precise boundary between these limiting cases is not well-defined Yamada (), it is possible to unambiguously detect changes in the parity of the ground state of the system, i.e., whether the GS is a singlet (even fermionic parity) or a doublet (odd fermionic parity), as we show here. The competition between the singlet and doublet states is determined by different energy scales: , , the charging energy, , and the energy of the QD level relative to the Fermi energy of the electrode (see Fig. 1a) Glazman (); Rozhkov99 (); Vecino03 (); Hewson04 (); Hewson07 (); Choi (); Simon (); Domanski (); Koenig2013 (). Previous works addressing this competition have focused either on the Josephson current in S-QD-S devices sdf2006 (); Lindelof (); Wernsdorfer12 () or on the sub-gap structure in S-QD-S and N-QD-S devices Buitelaar (); Kanai (); Jesp (); Eichler (); Pillet (); Deacon2010 (); Mason (); Lee2012 (); Chang (); Pillet2013 (); Kumar ().
Here we investigate the magnetic properties of the lowest-energy states in a -QD- geometry, where the contact acts as a weakly coupled tunnel probe. In this geometry, a direct spectroscopy of the density of states (DOS) in the QD- system can be performed through a measurement of the differential conductance, , as a function of the voltage difference, , between and . In such a measurement, an electrical current measured for is carried by so-called Andreev reflection processes, each of which involves two single-electron transitions in the QD. For example, an electron entering the QD from induces a single-electron transition from the QD GS, i.e. or , to the first excited state (ES), i.e. or , respectively. The ES relaxes back to the GS through the emission of an electron pair into the superconducting condensate of and a second single-electron transition corresponding to the injection of another electron from (the latter process is usually seen as the retroreflection of a hole into the Fermi sea of ). The just described transport cycle yields a resonance, i.e. an Andreev level, at , where is the energy difference between ES and the GS, i.e. between or , or vice-versa (see Fig. 1a). The reverse cycle, which involves the same excitations, occurs at , yielding a second Andreev level symmetrically positioned below the Fermi level.
We used devices based on single InAs/InP core/shell nanowires (NWs), where vanadium (gold) was used for the () contact Giazotto (). A device schematic and a representative image are shown in Figs. 1c and 1d, respectively. The fabricated vanadium electrodes showed meV and an in-plane critical magnetic field T ( NW axis). The QD is naturally formed in the NW section between the and contacts. We find typical values of a few meV (i.e., ). The QD properties are controlled by means of two bottom electrodes crossing the NW, labeled as plunger gate and -barrier gate, and a back gate provided by the conducting Si substrate. To achieve the asymmetry condition , the -barrier gate was positively biased at V. We used the plunger gate voltage to vary the charge on the QD, and the back-gate voltage to finely tune the tunnel coupling.
Figure 2a shows a series of measurements for three different . The top row refers to the weakest . In this case, the spanned range corresponds to a horizontal path in the phase diagram that goes through the doublet GS region (see right diagram). Let us first consider the leftmost plot taken at magnetic field . On the left and right sides of this plot, the QD lies deep inside the singlet GS regime. Here the doublet ES approaches the superconducting gap edge, yielding an Andreev-level energy . By moving towards the central region, the two sub-gap resonances approach each other and cross at the singlet-doublet phase boundaries, where . In the doublet GS regime between the two crossings, the sub-gap resonances form a loop structure with maximal at the electron-hole symmetry point. Increasing corresponds to an upward shift in the phase diagram. The middle row in Fig. 2a refers to the case where is just large enough to stabilize the singlet GS over the full range (see right diagram). At , the Andreev levels approach the Fermi level without crossing it. A further increase in leads to a robust stabilization of the singlet GS (bottom row). At zero-field, the sub-gap resonances remain distant from each other coming to a minimal separation at the electron-hole symmetry point (). Similarly to the case of superconducting single-electron transistors Devoret (), the QD occupation increases with in units of two without going through an intermediate odd state.
We now turn to the effect of on the Andreev levels (middle and right columns in Fig. 2a). Starting from the weak coupling case (top row), a field-induced splitting of the sub-gap resonances appears, yet only in correspondence of a singlet GS. This is due to the fact that these resonances involve excitations between states of different parity. For a singlet GS, the spin degeneracy of the doublet ES is lifted by the Zeeman effect resulting in two distinct excitations (see Fig. 1b). By contrast, for a doublet GS, no sub-gap resonance stems from the excitation, because these two states have the same (odd) number of electrons. The energy of the only visible Andreev level, associated with the transition, increases with . The formation of a loop structure in the rightmost panel of the middle row shows that a QPT from a singlet to a spin-polarized GS can be induced by when the starting is sufficiently small. In the bottom row, Zeeman-split Andreev levels can be seen all over the spanned range. At T, the inner levels overlap at the Fermi level, indicating a degeneracy between the and states. In Fig. 2b we show theoretical plots of a -QD- Anderson model calculated by means of self-consistent Hartree-Fock theory Rodero12 () (see Suppl. Information). The full phenomenology explained above is recovered, supporting our interpretation in terms of spin-resolved Andreev levels and a QPT.
Interestingly, the splitting of Andreev levels appears to be gate dependent. It tends to vanish when the system is pushed deep into the singlet GS, and it is maximal near the phase boundaries. To further investigate this behaviour, we have measured for fixed values of . These measurements were carried out on a second similar device (see Suppl. Information). The mid-panel of Fig. 3b displays the dependence of the sub-gap resonances measured at position 1 in Fig. 3a. Initially, the energy of the Andreev levels increases, as expected for a doublet GS (see left panel). From a linear fit of the low-field regime, i.e. , where is the Zeeman energy and is the Bohr magneton, we obtain a -factor . For 0.7 T, the field-induced closing of the gap bends the Andreev levels down to zero-energy. Finally, above the critical field, a split Kondo resonance is observed, from which 5.5 is estimated, consistent with the value extracted from the Andreev level splitting. The mid-panel of Fig. 3c displays a similar measurement taken at position 2 in Fig. 3a, where the GS is a singlet. The splitting of the Andreev levels is clearly asymmetric. The lower level decreases to zero according to a linear dependence , with 6.1, which is close to the value measured from the split Kondo resonance in the normal state. The higher energy level, however, exhibits a much weaker field dependence. Both the non-linear field dependence for T in Fig. 3b and the asymmetric splitting in Fig. 3c can be explained in terms of a level-repulsion effect between the Andreev levels and the continuum of quasiparticle states. This interpretation is confirmed by numerical calculations shown in the right panels of Figs. 3b and 3c, which are in good agreement with the respective experimental data. In the mid-panel of Fig. 3c, the inner sub-gap resonances cross around 1.5 T, denoting a field-induced QPT. Above this field, however, they remain pinned as a zero-bias peak (ZBP) up to T. This peculiar behavior, reproduced by the numerical results, can be attributed to the level-repulsion effect discussed above, in combination with the rapid shrinking of with .
In order to observe a clear -induced QPT from a singlet to a spin-polarized GS, we reduced by tuning closer to the singlet-doublet crossing in Fig. 3a. The corresponding data are shown in Fig. 4a. Contrary to the case of Fig. 3c, the Andreev level splitting is rather symmetric, owing to the reduced importance of the level repulsion effect at energies far from . The inner sub-gap resonances split again after the QPT, which occurs now at T. As expected, the outer sub-gap resonances get simultaneously suppressed (left panel of Fig. 3c). The suppression is not complete though, suggesting a partial population of the ES, possibly favored by thermal activation.
We note that the ZBP at the QPT appears to extend on a sizable field range 150 mT. This range is consistent with the -dominated lifetime broadening of the Andreev levels, i.e. peak width eV. In Fig. 4b we show how the ZBP depends on the in-plane angle, , relative to the NW axis. As varies from 0 to , the ZBP splits into two peaks with decreasing height. This angle dependence is an effect of the -factor anisotropy. For , we find a -factor , i.e. a factor of 2 smaller than for (see Suppl. Information). As a result, the QPT only occurs at a higher field (see Suppl. Information, T), and the split peaks correspond to transitions on the singlet-GS side. Figure 4b shows also a pair of small outer peaks associated with the transitions. Their oscillatory position is as well due to -factor anisotropy.
Noteworthy, the dependences discussed above mimic some of the signatures expected for Majorana fermions in hybrid devices Lutchyn (); vonOppen (); Aguado12b (); Loss12b (); DasSarma12 (); Potter (); MajoDelft (); MajoWeizmann (); MajoLund (); Finck (); Churchill (). A ZBP extending over a sizable range is observed for , and it is suppressed for , i.e. when B is presumably aligned to the Rashba spin-orbit field, MajoDelft (); MajoWeizmann (). While in Fig. 4 the field extension of the ZBP is limited by the ratio between the Andreev-level linewidth and the -factor, Fig. 3b shows a ZBP extending over a much larger range. This stretching effect can be attributed to the field-induced suppression of and the consequently enhanced level repulsion with the continuum of quasiparticle states. In larger QDs or extended nanowires, a similar level-repulsion effect may as well arise from other Andreev levels present inside the gap Aguado12b (); Potter (); Loss12b (); Stanescu ().
A more detailed discussion of the relation between the results presented here and existing experiments concerning Majorana fermions in hybrid devices is given in the Supplementary Information. Interestingly, a recent theoretical work has shown that zero-energy crossings of Andreev levels associated with a change in the ground state parity, similar to those presented here, adiabatically evolve towards zero-energy Majorana modes with increasing nanowire length Stanescu (). This evolution might be experimentally investigated by studying the -evolution of Andreev levels in nanowires of increasing length. Along similar lines, recent proposals have discussed the possibility of exploring the short-to-long wire evolution in chains of magnetic impurities deposited on superconducting surfaces Nadj-Perge (); KlinovajaRKKY (); Franz2013 (); Braunecker (); Pientka (). In such proposals, the Yu-Shiba-Rusinov bound states induced by the individual magnetic impurities (similar to the Andreev levels discussed here) may ultimately evolve towards Majorana modes localized at the edges of the atom chain, upon increasing the chain length.
Device fabrication. The -QD- devices used in this study were based on individual InAs/InP core/shell nanowires grown by thermal evaporation Xiaocheng () (diameter 30 nm, shell thickness 2 nm). The NWs were deposited onto Si/SiO substrates on which arrays of thin metallic striplines [Ti(2.5 nm)/Au(15 nm), width 50 nm] covered by a 8 nm-thick atomic layer deposition (ALD) HfO film had been previously processed. During the measurements, the degenerately-doped Si substrate was used as a global backgate, whereas the striplines were used as local gates. Normal metal [Ti(2.5 nm)/Au(50 nm)] and superconductor [Ti(2.5 nm)/V(45 nm)/Al(5 nm)] leads were incorporated to the devices by means of standard e-beam lithography techniques (lateral separation 200 nm).
This work was supported by the European Starting Grant program and by the Agence Nationale de la Recherche. R. A. acknowledges support from the Spanish Ministry of Economy and Innovation through grants FIS2009-08744 and FIS2012-33521. The authors thank J.-D. Pillet for useful discussions.
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