Orbital contributions to the electron -factor in semiconductor nanowires
Recent experiments on Majorana fermions in semiconductor nanowires [Albrecht et al., Nat. 531, 206 (2016)] revealed a surprisingly large electronic Landé -factor, several times larger than the bulk value — contrary to the expectation that confinement reduces the -factor. Here we assess the role of orbital contributions to the electron -factor in nanowires and quantum dots. We show that an coupling in higher subbands leads to an enhancement of the -factor of an order of magnitude or more for small effective mass semiconductors. We validate our theoretical finding with simulations of InAs and InSb, showing that the effect persists even if cylindrical symmetry is broken. A huge anisotropy of the enhanced -factors under magnetic field rotation allows for a straightforward experimental test of this theory.
Early electron spin resonance experiments in the 2D electron gas (2DEG) formed in AlGaAs/GaAs heterostructures found a reduced Landé -factor of electrons Stein et al. (1983), which was later theoretically explained to arise due to the electronic confinement Lommer et al. (1985); Ivchenko and Kiselev (1992); Kiselev et al. (1998). It is by now well established that confinement in a nanostructure leads to a reduction in the -factor Winkler et al. (2003); exc () – the subband confinement increases the energy gap which is inversely proportional to , where is the effective and the free electron -factor Roth et al. (1959); Winkler et al. (2003). Surprisingly, experiments in InAs Csonka et al. (2008); Schroer et al. (2011) and InSb Nilsson et al. (2009); van Weperen et al. (2013) nanowires found -factors surpassing the corresponding bulk -factors by up to 40%.
Recently, this discrepancy has attracted interest due to the experimental discovery of a zero bias conductance peak in semiconductor nanowires proximity coupled to an -wave superconductor Das et al. (2012); Deng et al. (2012); Mourik et al. (2012); Albrecht et al. (2016); Zhang et al. (2016), which is believed to be a signature of the Majorana bound state Kitaev (2001); Lutchyn et al. (2010); Oreg et al. (2010) having possible applications in topological quantum computation Kitaev (2003); Nayak et al. (2008). The electron -factor of the semiconductor nanowire determines the strength of magnetic field required to trigger the topological phase transition in these systems. It is desirable to keep the magnetic field low since it also suppresses superconductivity, and thus a large -factor semiconductor is desired. Furthermore, Majorana proposals based on magnetic textures Kjaergaard et al. (2012); Fatin et al. (2016); Matos-Abiague et al. (2017) and various spintronic devices Žutić et al. (2004) require large -factors. Small band-gap semiconductors like InAs and InSb are therefore the materials of choice for Majorana nanowires, having large -factors and strong spin-orbit coupling (SOC).
In a recent experiment with InAs nanowires -factors 111We measure the -factors in units of the Bohr-magneton and use the sign convention where the free electron -factor is . more than three times larger than the bulk -factor ( Pidgeon et al. (1967); Winkler et al. (2003)) were measured Albrecht et al. (2016). Moreover, it was found that the -factor depends very strongly on the chemical potential tuned by the gate potential Vaitiekėnas et al. (). For low small -factors where found which can be explained by the bulk -factor of InAs. The anomalously large -factors have been only detected at high chemical potential .
In this work, we present a mechanism that can lead to very large -factors in higher subbands of nanowires and similarly shaped nanostructures. With this we can explain both the large -factors observed in Refs. Csonka et al., 2008; Schroer et al., 2011; Nilsson et al., 2009; Albrecht et al., 2016, and the chemical potential dependence Vaitiekėnas et al. (). In particular, we find that the orbital angular momentum in the confined nanostructure plays a crucial role. The lowest conduction subband/state is characterized by no or only small orbital angular momentum. In this case the usual reasoning applies and confinement does lead to a reduction of the -factor. Higher subbands/states, however, can have nonzero orbital angular momentum in an approximately cylindrical structure. Due to strong SOC in small band-gap semiconductors one finds an -type spin alignment if the orbital angular momentum is nonzero. Kramers pairs of opposite orbital angular momentum form at , and thus the -factor obtains an additional contribution resulting from the coupling of the orbital angular momentum to the magnetic field. A similar orbital enhancement of the g-factor is known from the theory of the hydrogen atom Landau and Lifshitz (1981) and has also been observed in carbon nanotubes Kuemmeth et al. (2008); Laird et al. (2015). However, due to the small effective mass the -factor enhancement can be orders of magnitude larger in the semiconducting structures investigated here.
— We start by considering cylindrical nanowires and estimate the maximally achievable -factor for subbands as a function of their orbital angular momentum. Initially, we assume independent SU(2) spin rotation symmetry (no SOC) and time-reversal (TR) invariance without magnetic field. We then introduce magnetic field parallel to the wire, thus preserving the rotational invariance (both in real space and spin) around the axis of the wire ( direction in the following).
As the wire is translationally invariant in the direction, and the conduction band minimum is at , we restrict to in the following and investigate the wavefunction in the plane only. As a consequence of separate real space and spin rotation symmetries, the states can be classified by their orbital angular momentum etc. and spin (for brevity we drop the subscript in the following and use the lower case letters for angular momentum in units of ). The lowest subband is twofold spin degenerate , higher subbands with being fourfold .
In a simple quadratic band with an effective mass , the momentum and electrical current are related as . Using the orbital angular momentum the orbital magnetic moment is expressible as
We see that the orbital magnetic moment is enhanced by the low effective mass of the bands. Because of the fourfold degeneracy, we cannot unambiguously calculate -factors and thus next include spin-orbit coupling.
With SOC the orbital and spin angular momentum is no longer separately conserved, but the total angular momentum is still conserved and takes half-integer values. Without magnetic field the system is TR invariant. As angular momentum is odd under TR, the degenerate Kramers-pairs have opposite . Turning on SOC splits the fourfold degeneracy of the subbands into two degenerate pairs: and stay degenerate () and so do and (), as shown in Fig. 1 (a). Even though the orbital and local angular momenta are no longer separately conserved their expectation values remain similar for realistic SOC strengths.
The magnetic field couples to the total magnetic moment Kiselev et al. (1998). Using Eq. (1), the Zeeman splitting of a Kramer’s pair and for a magnetic field in -direction is given by and the resulting effective -factor can be read off
Below we will see from numerical simulation that this is a good approximation even in a less ideal case.
This result is analogous to the well known Landé -factor of the Hydrogen atom when taking relativistic SOC into account: the splitting induced by weak external magnetic field has contributions from both the orbital and spin angular momentum Landau and Lifshitz (1981). This effect is amplified in semiconductor nanostructures because the small effective mass increases both the orbital magnetic moment and the bulk -factor .
— We next validate our theoretical findings with simulations of nanowires using an eight-band -model for zincblende semiconductors Kane (1957); Foreman (1997); Winkler et al. (2003). At first, we assume perfect cylindrical symmetry of a nanowire, grown in 001 direction, and employ the axial approximation Sercel and Vahala (1990); Vahala and Sercel (1990); Çakmak et al. (2003); sup (). In this case, the wavefunctions can be written as Voon et al. (2005)
where are the basis states of the 8-band Hamiltonian with local angular momentum 222Here takes the role of in the previous argument, as in these materials the -type orbitals have nonzero local orbital angular momentum and are treated as spin- degrees of freedom.. Since the Hamiltonian conserves the total angular momentum one obtains the orbital part of each component as . If we furthermore focus on an infinite wire in the -direction the problem is reduced to a 1D boundary value problem in which we solve using the finite difference method sup ().
Figure 1 (b) shows the subband edges of an InSb nanowire of 40 nm diameter. At one generically finds the lowest conduction subband to originate from the state without SOC. At higher energy there are the and states and then another state with a higher radial quantum number (not shown). This order of states is generic as long as the conduction band is approximately quadratic Robinett (2003). Figure 1 (c) zooms in on the subbands. Due to SOC the and states are split at by . If a magnetic field (see Fig. 1 (c)) is turned on a splitting between states of opposite orbital angular momentum is observed and thus, enhanced -factors according to Eq. (2). However, when the magnetic field is large, , states of the same orbital angular momentum bundle together and their relative slope with respect to corresponds to the normal -factor without orbital contributions. Thus a splitting is a crucial ingredient for enhanced -factors.
Figure 2 shows the dependence on the diameter of the nanowire. From the dependence it is evident that the wire cannot be made too thick to experimentally observe the effect with a detectable energy scale, e.g. to distinguish the split energy levels using Coulomb oscillations Tarucha et al. (1996). Figures 2 (c) and (d) show that at large wire diameters Eq. (2) is reproduced perfectly by numerics, but for small diameters the -factor enhancement is reduced by the confinement. Thus, the optimal diameter range where enhancement of the -factor is strong and at the same time and are large enough is in between 10 and 100 nm. We see that the -factors of higher subbands can be very large — enhancements of an order of magnitude compared to the bulk -factor are possible.
The splitting is generic if SOC is present, since in a typical semiconductor wire with SOC there is no symmetry that would protect the degeneracy between states of different total angular momentum. The conduction band of zincblende semiconductors has a purely -orbital character at the -point of the Brillouin zone (BZ), which is insensitive to SOC. Thus, also the conduction subbands of a zincblende nanowire are mostly derived from -orbitals. Any nonzero splitting results from -like hole contributions to the conduction band due to confinement. This explains why the splitting in the conduction band is so small compared to the split-off energy of the valence bands , which is 0.81 eV for InSb and 0.38 eV for InAs Winkler et al. (2003).
Since results from the scattering of states at the surface of the wire the boundary conditions impact the numerical value, and even the sign, of sup (). Abrupt boundaries can be problematic in simulations Rodina et al. (2002), therefore, we use tight-binding (TB) simulations to check the robustness of our results. The effective tight-binding Hamiltonian is generated from the first-principles and -like Wannier functions Mostofi et al. (2014), calculated using the Vienna ab initio simulation package (VASP) Kresse and Hafner (1993, 1994); Kresse and Furthmüller (1996, 1996) with the projector augmented-wave method Blöchl (1994); Kresse and Joubert (1999), a cut-off energy of 300 eV, a 888 Monkhorst-Pack mesh and using the HSE06 hybrid functional Kim et al. (2009); Heyd et al. (2003); Heyd and Scuseria (2004). Furthermore, the TB model includes the Dresselhaus term which was neglected for the zincblende simulations since its effect is found to be very small sup (). In Fig. 3 (a) we show the magnetic field dependence of the subbands in a hexagonal InSb wire. The -factors of -59 and +40 and agree qualitatively with the -results.
While in zincblende wires boundary effects are dominating, in wurtzite wires the situation is different: There, the conduction band has a mixed and -character. Thus, wurtzite wires have an intrinsic splitting independent of confinement Lew Yan Voon et al. (1996). Using a -model for wurtzite semiconductors Faria Junior et al. (2016), we find a nearly size-independent of order 1 meV for  grown wurtzite InAs wires for experimentally used diameters of 40 to 160 nm Krogstrup et al. (2015), see Fig. 3 (b). At very large wire diameters nm the confinement induced subband splitting becomes smaller than , leading to a reduction of , and at very small diameters nm the cubic Dresselhaus term dominates over the linear Rashba term, causing a sign change in sup (); Gmitra and Fabian (2016).
— We now consider the effects of broken cylindrical symmetry and solve the full 2D cross section of hexagonal zincblende wires, grown in the 111 direction, using a 2D discretization of the -model sup (); Nijholt and Akhmerov (2016). We allow for symmetry breaking by electric field and off-axis magnetic field, see Fig. 4 (a) for the definitions of the relevant directions. In experimental situations, the symmetry is generally broken by electric fields, e.g. due to the backgate for tuning the electron density in the wire Csonka et al. (2008); Nilsson et al. (2009); Albrecht et al. (2016); Vaitiekėnas et al. (). We find that, especially in higher subbands, the enhanced -factors are quite robust to an external electric field.
In Fig. 4 (b-d) we simulate a hexagonal InSb wire, of 40 nm diameter, in a perpendicular external electric field . The point group of the wire at is and crossings between states of different angular momentum are protected, as illustrated in Figure 4 (c). At nonzero the different angular momentum eigenstates hybridize, which reduces their orbital angular momentum expectation value. However, as shown in Figs. 4 (b) and (d), the orbital contribution to the -factor remains very significant until very large fields are applied. Bands with larger values of have larger splitting and, therefore, the orbital contribution to their -factors is more robust and can remain significantly larger than the bulk -factor until large electric fields, e.g. see the cyan and magenta lines corresponding to in Fig. 4 (b).
The electron -factor anisotropy in the magnetic field of 2DEGs is well established Ivchenko and Kiselev (1992); van Kesteren et al. (1990); Peyla et al. (1993); Winkler et al. (2003). In our case of orbitally enhanced -factors in nanowires we expect an even stronger anisotropy. Indeed, the electron spins in subbands with feel a very strong orbital magnetic field that aligns them (anti-) parallel to the wire axis. Therefore, a perpendicular magnetic field first needs to overcome this orbital effect to create a Zeeman splitting of the states Kuemmeth et al. (2008); Laird et al. (2015).
This is illustrated in Fig. 4 (e), where we simulate a hexagonal InSb wire of 40 nm diameter in a magnetic field of 0.2 Tesla. We show there the g-factor as a function of the angle between the magnetic field and the nanowire axis. While the -factor of the lowest subband is unaffected by the direction of , the -factor for bands with almost vanishes for perpendicular magnetic field. This strong anisotropy of the electron -factor can be used in experiments to prove the important role of orbital angular momentum in nanowires.
In a Majorana wire circular symmetry breaking by gate potentials and band bending is mandatory to create a Rashba effect in the wire Lutchyn et al. (2010); Oreg et al. (2010); Chang et al. (2015). The results shown above suggest that even in such an environment orbital effects still dominate the -factors of certain subbands in wires. This is illustrated in Fig. 5 (a) and (b), where we simulate an InAs wire proximity coupled to an Al superconductor (see the Supplemental Material sup () for the details of the simulation). When the chemical potential is tuned to the and subbands, the -factors, extracted from the slope of the Majorana state forming Andreev bound state, are 23 and 43 333Our simulations do not include the renormalization effects of the superconductor Stanescu et al. (2011); Sticlet et al. (2017), which could lead to a reduction of the resulting -factor., respectively. These -factors are significantly larger than the bulk -factor of InAs, thus reproducing the experimental result of Ref. Albrecht et al., 2016.
Conclusions and Outlook —
In summary, we have provided a theory for the previously unexplained large -factors observed in nanowires. Our findings help to better understand and optimize Majorana experiments. Similar results apply to quantum dots. For cylindrical quantum dots we find that orbital -factor enhancements are still significant if the length of the dot is much shorter than its diameter, see the Supplemental Material sup, for more details. Due to the observed robustness of the effect, it also applies in irregularly shaped quantum dots and can explain -factor fluctuations there.
Acknowledgements.Acknowledgments. We would like to thank L Kouwenhoven, S Vaitiekėnas, MT Deng, CM Marcus, K Ennslin, TD Stanescu, AE Antipov, E Rossi, and RM Lutchyn for useful discussions and QS Wu for providing first-principles derived tight-binding models. This work was supported by Microsoft Research, the Netherlands Organization for Scientific Research (NWO), the Foundation for Fundamental Research on Matter (FOM), the European Research Council through ERC Advanced Grant SIMCOFE, the Swiss National Science Foundation and through the National Competence Centers in Research MARVEL and QSIT.
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