Zeeman Spin Splitting Research Articles

Electron temperature and tunnel coupling dependence of zero-bias and almost-zero-bias conductance peaks in Majorana nanowires

A one-dimensional semiconductor nanowire proximitized by a nearby superconductor may become a topological superconductor hosting localized Majorana zero modes at the two wire ends in the presence of spin-orbit coupling and Zeeman spin splitting (arising from an external magnetic field). The hallmark of the presence of such Majorana zero modes is the appearance of a zero-temperature quantized zero-bias conductance peak in the tunneling spectroscopy of the Majorana nanowire. We theoretically study the temperature and the tunnel coupling dependence of the tunneling conductance in such nanowires to understand possible intrinsic deviations from the predicted conductance quantization. We find that the full temperature and the tunneling transmission dependence of the tunnel conductance does not obey any simple scaling relation, and estimating the zero-temperature conductance from finite-temperature and finite-tunnel-broadening tunneling data is difficult in general. A scaling relation, however, does hold at the extreme weak-tunneling low-temperature limit where the conductance depends only on the dimensionless ratio of the temperature and tunnel broadening. We also consider the tunneling contributions from nontopological Andreev bound states which may produce almost-zero-bias conductance peaks, which are not easy to distinguish from the Majorana-induced zero-bias peaks, finding that the nontopological almost-zero-modes associated with Andreev bound states manifest similar temperature and transmission dependence as the topological Majorana modes. We comment on the Zeeman splitting dependence of the zero-bias conductance peak for finite temperature and tunnel coupling.

Conductance of a superconducting Coulomb-blockaded Majorana nanowire

In the presence of an applied magnetic field introducing Zeeman spin splitting, a superconducting (SC) proximitized one-dimensional (1D) nanowire with spin-orbit coupling can pass through a topological quantum phase transition developing zero-energy topological Majorana bound states (MBSs) on the wire ends. One of the promising experimental platforms in this context is a Coulomb blockaded island, where by measuring the two-terminal conductance one can in principle investigate the MBS properties. We theoretically study the tunneling transport of a single electron across the superconducting Coulomb blockaded nanowire at finite temperature to obtain the generic conductance equation. By considering all possible scenarios where only MBSs are present at the ends of the nanowire, we compute the nanowire conductance as a function of the magnetic field, the temperature, and the gate voltage. In the simplest 1D topological SC model, the oscillations of the conductance peak spacings (OCPSs) arising from the MBSs overlap from the two wire ends manifest an increasing oscillation amplitude with increasing magnetic field (in agreement with theories without Coulomb blockade and in disagreement with a recent experimental observation). We develop a generalized finite temperature master equation theory including not only multiple subbands in the nanowire, but also the possibility of ordinary Andreev bound states in the non-topological regime. Inclusion of all four effects (temperature, multiple subbands, Andreev bound states, and MBSs) provides a complete picture of the tunneling transport properties. Based on this complete theory, we indeed obtain OCPSs whose amplitudes decrease with increasing magnetic field in qualitative agreement with recent experimental results, but this happens only for rather high temperatures with multisubband occupancy and the presence of both Andreev bound states and MBSs.

Strong Landau-quantization effects in high-magnetic-field superconductivity of a two-dimensional multiple-band metal near the Lifshitz transition

We investigate the onset of superconductivity in magnetic field for a clean two-dimensional multiple-band superconductor in the vicinity of the Lifshitz transition when one of the bands is very shallow. Due to small number of carriers in this band, the quasiclassical Werthamer-Helfand approximation breaks down and Landau quantization has to be taken into account. We found that the transition temperature TC2(H) has giant oscillations and is resonantly enhanced at the magnetic fields corresponding to full occupancy of the Landau levels in the shallow band. This enhancement is especially pronounced for the lowest Landau level. As a consequence, the reentrant superconducting regions in the temperature-field phase diagram emerge at low temperatures near the magnetic fields at which the chemical potential matches the Landau levels. The specific behavior depends on the relative strength of the intraband and interband pairing interactions and the reentrance is most pronounced in the purely interband coupling scenario. The reentrant behavior is suppressed by the Zeeman spin splitting in the shallow band, the separated regions disappear already for very small spin-splitting factors. On the other hand, the reentrance is restored in the resonance cases when the spin-splitting energy exactly matches the separation between the Landau levels. The predicted behavior may realize in the gate-tuned FeSe monolayer.

Magnetic impurities in spin-split superconductors

Hybrid semiconductor-superconductor quantum dot devices are tunable physical realizations of quantum impurity models for a magnetic impurity in a superconducting host. The binding energy of the localized sub-gap Shiba states is set by the gate voltages and external magnetic field. In this work we discuss the effects of the Zeeman spin splitting which is generically present both in the quantum dot and in the (thin-film) superconductor. The unequal $g$-factors in semiconductor and superconductor materials result in respective Zeeman splittings of different magnitude. We consider both classical and quantum impurities. In the first case we analytically study the spectral function and the sub-gap states. The energy of bound states depends on the spin-splitting of the Bogoliubov quasiparticle bands as a simple rigid shift. For the case of collinear magnetization of impurity and host, the Shiba resonance of a given spin polarization remains unperturbed when it overlaps with the branch of the quasiparticle excitations of the opposite spin polarization. In the quantum case, we employ numerical renormalization group calculations to study the effect of the Zeeman field for different values of the $g$-factors of the impurity and of the superconductor. We find that in general the critical magnetic field for the singlet-doublet transition changes non-monotonically as a function of the superconducting gap, demonstrating the existence of two different transition mechanisms: Zeeman splitting of Shiba states or gap closure due to Zeeman splitting of Bogoliubov states. We also study how in the presence of spin-orbit coupling, modeled as an additional non-collinear component of the magnetic field at the impurity site, the Shiba resonance overlapping with the quasiparticle continuum of the opposite spin gradually broadens and then merges with the continuum.

Adsorption-enhanced spin–orbit coupling of buckled honeycomb silicon

We have studied the electronic structures of quasi-two-dimensional buckled honeycomb silicon (BHS) saturated by atomic hydrogen and fluorine by means of first-principles calculations. The graphene-like hexagonal silicon with chair configurations can be stabilized by atomic hydrogen and fluorine adsorption. Together with a magnetic ground state, large spin–orbit coupling (SOC) of BHS saturated by hydrogen on either side (Semi-H-BHS) indicated by the band splitting of σ bond at Γ point in the Brillouin zone is attributed to the intermixing between the density of states of hydrogen atoms and π bonds of unpassivated Si2 around the Fermi level. The Zeeman spin splitting is most likely caused by the internal electric field induced by asymmetric charge transfer.

Electromagnetic coupling of spins and pseudospins in bilayer graphene

We present a detailed theoretical study of bilayer-graphene's electronic properties in the presence of electric and magnetic fields. Using group-theoretical methods, we derive an invariant expansion of the Hamiltonian for electron states near the K point of the Brillouin zone. In contrast to known materials, including single-layer graphene, any possible coupling of physical quantities to components of the external electric (magnetic) field has a counterpart where the analogous component of the magnetic (electric) field couples to exactly the same combination of quantities. For example, a purely electric spin splitting appears as the magneto-electric analogue of the familiar magnetic Zeeman spin splitting. The measurable thermodynamic response induced by magnetic and electric fields is thus completely symmetric. The Pauli magnetization induced by a magnetic field takes exactly the same functional form as the polarization induced by an electric field. Our findings thus reveal unconventional behavior of spin and pseudospin degrees of freedom in their coupling to external fields. We explain how these counterintuitive couplings are consistent with fundamental principles such as time reversal symmetry. For example, only a magnetic field can give rise to a macroscopic spin polarization, whereas only a perpendicular electric field can induce a macroscopic polarization of the sublattice-related pseudospin degree of freedom characterizing the intravalley orbital motion in bilayer graphene. These rules enforced by symmetry for the matter-field interactions clarify the nature of spins versus pseudospins. We also provide numerical values of prefactors for relevant coupling terms. While our theoretical arguments use bilayer graphene as an example, they are generally valid for any material with similar symmetries.

Linear magnetoresistance caused by mobility fluctuations in n-doped Cd(3)As(2).

Cd(3)As(2) is a candidate three-dimensional Dirac semimetal which has exceedingly high mobility and nonsaturating linear magnetoresistance that may be relevant for future practical applications. We report magnetotransport and tunnel diode oscillation measurements on Cd(3)As(2), in magnetic fields up to 65 T and temperatures between 1.5 and 300 K. We find that the nonsaturating linear magnetoresistance persists up to 65 T and it is likely caused by disorder effects, as it scales with the high mobility rather than directly linked to Fermi surface changes even when approaching the quantum limit. From the observed quantum oscillations, we determine the bulk three-dimensional Fermi surface having signatures of Dirac behavior with a nontrivial Berry phase shift, very light effective quasiparticle masses, and clear deviations from the band-structure predictions. In very high fields we also detect signatures of large Zeeman spin splitting (g∼16).

Intersubband optical absorption in InSb stepped quantum wells: Effect of spin sublevels crossing

We study linear and non-linear coefficients of the intersubband absorption in InSb-based stepped quantum wells subjected to an in-plane magnetic field. We consider also a transverse electric field to achieve near resonance conditions. Taking into account the two deepest conduction levels and their corresponding Zeeman spin splitting sublevels, we calculate dispersion relations by means of an improved version of Kane model. Besides the known anti-crossing between down and up spin split sublevels, we obtain an extra spin level crossing for some determined parameters. This crossing clearly modifies the absorption spectrum for transitions among the four sublevels considered. We study a low electron density case, when only the first deepest sublevel is occupied, and a high density case with only the highest sublevel empty. We find a similar behavior of the absorption spectrum in both cases.

A modified transfer matrix approach to include Zeeman spin splitting in dispersion relations for coupled quantum wells

In our work, we calculate the dispersion relations for InGaAs – InAlAs based double quantum wells (narrow gap structures). We have developed an improved 4 × 4 version of the Transfer Matrix Approach, considering contributions from external fields when tunneling through central barrier exists. The transverse electric field is necessary to reach the resonance of electronic levels in asymmetric structures. The in-plane magnetic field induces the Zeeman effect and the spin splitting of the resonant levels. We have also included abrupt barrier effects due to the nature of the interfaces between the above materials.

Hole-spin dynamics and holeg-factor anisotropy in coupled quantum well systems

Due to its $p$-like character, the valence band in GaAs-based heterostructures offers rich and complex spin-dependent phenomena. One manifestation is the large anisotropy of Zeeman spin splitting. Using undoped, coupled quantum wells (QWs), we examine this anisotropy by comparing the hole-spin dynamics for high- and low-symmetry crystallographic orientations of the QWs. We directly measure the hole $g$ factor via time-resolved Kerr rotation, and for the low-symmetry crystallographic orientations (110) and ($113a$), we observe a large in-plane anisotropy of the hole $g$ factor, in good agreement with our theoretical calculations. Using resonant spin amplification, we also observe an anisotropy of the hole-spin dephasing in the (110)-grown structure, indicating that crystal symmetry may be used to control hole-spin dynamics.

Dipolar excitons in a potential trap in a magnetic field

The conditions for observing the Zeeman spin splitting compensation in an exciton Bose gas have been investigated. The magnetoluminescence of spatially indirect, dipolar excitons in a 25-nm-wide GaAs/AlGaAs quantum well upon their accumulation in a lateral electrostatic trap has been studied in the Faraday geometry. The critical magnetic field Bcr below which the spin (paramagnetic) splitting of the luminescence line for a heavy-hole exciton at the trap center is almost completely compensated due to the exchange interaction in a dense Bose gas has been found to increase linearly with exciton concentration in qualitative agreement with the theory. Using a potential trap is fundamentally important. Incomplete compensation is observed in a homogeneous photoexcitation spot for dipolar excitons: the splitting is considerably smaller than that for a spatially direct exciton but differs noticeably from zero. The spin splitting compensation effect is observed only under neutral charge balance conditions—there is no Zeeman splitting suppression in a charged quantum well.

Magnetic quantum coherence effect in Ni4 molecular transistors

We present a theoretical study of electron transport in Ni4 molecular transistors in the presence of Zeeman spin splitting and magnetic quantum coherence (MQC). The Zeeman interaction is extended along the leads which produces gaps in the energy spectrum which allow electron transport with spin polarized along a certain direction. We show that the coherent states in resonance with the spin up or down states in the leads induces an effective coupling between localized spin states and continuum spin states in the single molecule magnet and leads, respectively. We investigate the conductance at zero temperature as a function of the applied bias and magnetic field by means of the Landauer formula, and show that the MQC is responsible for the appearence of resonances. Accordingly, we name them MQC resonances.

Hall conductivity as bulk signature of topological transitions in superconductors

Topological superconductors (TSCs) may undergo transitions between phases with different topological numbers which are related to the presence of gapless (Majorana) edge states. In a TSC the charge Hall conductivity, , is not quantized and evolves continuously between different topological phases. Here we show that in two-dimensional -TSC the derivatives of display sharp features signaling the topological transitions. We consider in detail the case of a triplet superconductor with p-wave symmetry in the presence of Rashba spin-orbit coupling and externally applied Zeeman spin splitting. We generalize to the cases where the normal system is already topologically non-trivial.

Strong coupling expansion of the extended Hubbard model with spin-orbit coupling

We study the strong coupling limit of the extended Hubbard model in two dimensions. The model consists of hopping, on-site interaction, nearest-neighbor interaction, spin-orbit coupling and Zeeman spin splitting. While the study of this model is motivated by a search for topological phases and in particular a topological superconductor, the methodology we develop may be useful for a variety of systems. We begin our treatment with a canonical transformation of the Hamiltonian to an effective model which is appropriate when the (quartic) interaction terms are larger than the (quadratic) kinetic and spin-orbit coupling terms. We proceed by analyzing the strong coupling model variationally. Since we are mostly interested in a superconducting phase we use a Gutzwiller projected BCS wavefunction as our variational state. To continue analytically we employ the Gutzwiller approximation and compare the calculated energy with Monte-Carlo calculations. Finally we determine the topology of the ground state and map out the topology phase diagram.

Density of states of disordered topological superconductor-semiconductor hybrid nanowires

Using Bogoliubov-de Gennes equations we numerically calculate the disorder averaged density of states of disordered semiconductor nanowires driven into a putative topological $p$-wave superconducting phase by spin-orbit coupling, Zeeman spin splitting, and the $s$-wave superconducting proximity effect induced by a nearby superconductor. Comparing with the corresponding theoretical self-consistent Born approximation (SCBA) results treating disorder effects, we comment on the topological phase diagram of the system in the presence of increasing disorder. Although disorder strongly suppresses the zero-bias peak (ZBP) associated with the Majorana zero mode, we find some clear remnant of a ZBP even when the topological gap has essentially vanished in the SCBA theory because of disorder. We explicitly compare effects of disorder on the numerical density of states in the topological and trivial phases.

Observation of acoustic edge magnetoplasmons near the filling factor ν = 1

Investigation of acoustic edge magnetoplasma excitations in micrometer two-dimensional electron disks has been carried out. It has been shown that additional acoustic edge magnetoplasma modes associated with the existence of incompressible strips in the two-dimensional electron system caused by Zeeman spin splitting can appear in the system at temperatures below the Zeeman energy. The magnetic dispersion of the first “spin” branch of acoustic edge magnetoplasmons has been studied. It has been shown that this mode vanishes as the filling factor ν = 1 is approached. The dependence of the relative amplitude of acoustic edge magnetoplasmons on the filling factor has also been investigated.

Spin-polarized electric currents in diluted magnetic semiconductor heterostructures induced by terahertz and microwave radiation

We report on the study of spin-polarized electric currents in diluted magnetic semiconductor (DMS) quantum wells subjected to an in-plane external magnetic field and illuminated by microwave or terahertz radiation. The effect is studied in (Cd,Mn)Te/(Cd,Mg)Te quantum-wells (QWs) and (In,Ga)As/InAlAs:Mn QWs belonging to the well-known II-VI and III-V DMS material systems, as well as in heterovalent AlSb/InAs/(Zn,Mn)Te QWs, which represent a promising combination of II-VI and III-V semiconductors. Experimental data and developed theory demonstrate that the photocurrent originates from a spin-dependent scattering of free carriers by static defects or phonons in the Drude absorption of radiation and subsequent relaxation of carriers. We show that in DMS structures, the efficiency of the current generation is drastically enhanced compared to nonmagnetic semiconductors. The enhancement is caused by the exchange interaction of carrier spins with localized spins of magnetic ions resulting, on the one hand, in the giant Zeeman spin splitting, and, on the other hand, in the spin-dependent carrier scattering by localized Mn${}^{2+}$ ions polarized by an external magnetic field.

High-Field Fermi Surface Properties in the Low-Carrier Heavy-Fermion Compound URu2Si2

We performed the Shubnikov–de Haas (SdH) experiments of the low-carrier heavy-fermion compound URu 2 Si 2 at high fields up to 34 T and at low temperatures down to 30 mK. All main SdH branches named α, β, and γ were observed for all the measured field-directions ( H ∥[001]→[100], [100]→[110] and [001]→[110]), indicating that they are attributed to the closed Fermi surfaces with nearly spherical shapes. An anomalous split of branch α was detected for the field along the basal plane; the split immediately disappeared by tilting the field to [001] direction, implying a fingerprint of the hidden order state. High-field experiments reveal the complicated field-dependence of the SdH frequencies and the cyclotron masses due to the Zeeman spin-splitting associated with the Fermi surface reconstruction in the hidden order state with small carrier numbers. A new SdH branch named ω with a large cyclotron mass of 25 m 0 was detected at high fields above 23 T close to hidden order instabilities.

Highly nonlinear excitonic Zeeman spin splitting in composition-engineered artificial atoms

Non-linear Zeeman splitting of neutral excitons is observed in composition engineered In(x)Ga(1-x)As self-assembled quantum dots and its microscopic origin is explained. Eight-band k.p simulations, performed using realistic dot parameters extracted from cross-sectional scanning tunneling microscopy, reveal that a quadratic contribution to the Zeeman energy originates from a spin dependent mixing of heavy and light hole orbital states in the dot. The dilute In-composition (x<0.35) and large lateral size (40-50 nm) of the quantum dots investigated is shown to strongly enhance the non-linear excitonic Zeeman gap, providing a blueprint to enhance such magnetic non-linearities via growth engineering.

Spin-polarized current generated by magneto-electrical gating

We theoretically study spin-polarized current through a single electron tunneling transistor (SETT), in which a quantum dot (QD) is coupled to non-magnetic source and drain electrodes via tunnel junctions, and gated by a ferromagnetic (FM) electrode. The I–V characteristics of the device are investigated for both spin and charge currents, based on the non-equilibrium Green's function formalism. The FM electrode generates a magnetic field, which causes a Zeeman spin-splitting of the energy levels in the QD. By tuning the size of the Zeeman splitting and the source–drain bias, a fully spin-polarized current is generated. Additionally, by modulating the electrical gate bias, one can effect a complete switch of the polarization of the tunneling current from spin-up to spin-down current, or vice versa.