Improved effective equation for the Rashba spin-orbit coupling in semiconductor nanowires

Semiconductor Rashba nanowires are quasi-one dimensional materials that have large spin-orbit (SO) coupling arising from a broken crystal potential symmetry due to an external electric field. There exist parametrized multiband models that can describe accurately this effect. However, simplified single band models are highly desirable to study geometries of recent experimental interest, since they may allow to incorporate the effect of the electrostatic environment of the nanowires at a reduced computational cost. We demonstrate here that an effective equation for the linear Rashba SO coupling of the semiconductor conduction band can reproduce the behaviour of more sophisticated eight-band k$\cdot$p model calculations. This is achieved by adjusting a single effective parameter that depends on the nanowire crystal structure and its chemical composition. We further compare our results with the Rashba coupling extracted from magnetoconductance measurements in several experiments on InAs and InSb nanowires, finding excellent agreement. This approach may be relevant in systems where Rashba coupling is known to play a major role, such as in spintronic devices or Majorana nanowires.


I. INTRODUCTION
The spin-orbit (SO) interaction is a relativistic effect that couples the electron's spin and momentum in the presence of an electric field.Among crystalline solids it is particularly strong in some semiconductors [1] and, although it is typically small compared to other characteristic energies, it produces a splitting of otherwise degenerate energy bands around the Fermi level [2].This can have tremendous consequences in the transport of electrons, as is manifested in the field of spintronics [3] and, more recently, in spin-orbitronics [4].In particular, the SO interaction is the driving mechanism behind the existence of topological insulators [5,6] such as the quantum spin Hall effect [7].It is also essential in the search of Majorana zero modes in topological superconductors [8][9][10][11][12][13], such is the case of hybrid superconductorsemiconductor nanowires [14,15].In these wires, the SO term contributes to create non-degenerate bands with spin-momentum locking, a key ingredient behind the topological phase transition.Moreover, in the topological phase of the wire, the minigap which protects the Majorana modes from decoherence is directly proportional to the SO coupling [16].
For their connection to Majorana physics as well as for other spin-related mechanisms, semiconductor nanowires with strong SO coupling have come to the forefront of condensed-matter research [17][18][19].There are several good reasons for their choice.First, they can be grown to a high degree of perfection, almost at the atomic scale [20][21][22].Second, they can be proximitized [23] both by depositing superconductors on top of them as well as by growing them epitaxially on the nanowire, forming heterostructures [24].They can also be easily contacted with metallic leads to an external circuit, and their properties are highly tunable through gate electrodes and external fields.In particular, it is possible to tune them to enhance the SO coupling [25][26][27].
Inside these nanowires, electrons are subject to nonuniform electrostatic potentials.When a charged particle moves in an electric field, it experiences an effective magnetic field that couples to the particle's spin through the Zeeman effect [2].The corresponding Hamiltonian is usually written as where k is the electron's wavevector, σ are the Pauli matrices in spin space and α is the so-called SO coupling.This coefficient determines the strength of the coupling between the spin and the momentum of the electron and is related to the effective electric field felt by the electrons inside the wire [1].Because the phenomena and applications mentioned before are very sensitive to the precise value of this coupling [4], a proper description of this mechanism is crucial to predict the actual properties of these nanowires.There are two ways in which an electric field can arise in semiconductor nanowires [1].On the one hand, the crystal itself creates an intrinsic electric field when there is a lack of an inversion centre (bulk inversion asymmetry).This gives rise to the Dresselhaus SO coupling α D [28].On the other hand, an electric field arises when there is a lack of inversion symmetry due to an external confining potential (structural inversion asymmetry), due to e.g.crystal interfaces or metallic gates.This case corresponds to the Rashba coupling α R [29].arXiv:2001.04375v1[cond-mat.mtrl-sci]13 Jan 2020 Depending on the source of the SO interaction, some theoretical methods may be more advantageous to describe them.Dresselhaus couplings, being an intrinsic interaction depending only on the crystal's unit cell structure and composition, are usually computed using ab initio calculations [30,31].Instead, Rashba couplings that depend on the electrostatic environment and/or interfaces with other materials are less amenable to ab initio methods and tend to be described using effective models.In particular, multiband k•p theory has been successfully used to compute the energy bands of Rashba semiconductors including several conduction and hole bands [1,[32][33][34][35][36][37].
Multiband effective models are specially suited to three-dimensional infinite systems whose bands depend on a single momentum k.However, when treating low-dimensional systems (such as 2DEGs or nanowires), multiband models can be computationally challenging due to the extra degrees of freedom introduced by the transverse momentum subbands [34].This is because, for each transverse subband, one has to take into account several valence and conductions bands.This is further aggravated when treating heterojunctions with other materials [34].In this situation, it is very desirable to have a simplified effective theory that only takes into account the energy band of interest, typically the first conduction band, and that can incorporate the interaction with other less influential bands (such as heavy and light hole bands) through effective parameters.
For III-V compound semiconductors, which are broadly used in experiments, there exists an effective equation for the Rashba SO coupling based on a (spinful) single conduction band approximation, as we will show in Eq. ( 5).This equation is derived from the so-called eight-band (8B) k•p model, described in detail in Appx.A and B. Specifically, this equation takes into account the dependence of α R with i) the electric field generated by a spatially-dependent electrostatic potential φ( r), ii) the electron energy E and iii) the crystal structure and atomic composition through three effective parameters.One of these is known as the Kane parameter P [1], which represents the effective coupling between valence and conduction bands of the semiconductor.In this approximation, however, the interaction between valence bands is lost.As a result, the SO coupling obtained with this equation substantially differs from 8B model calculations for specific crystallographic structures.In this work we prove that the conduction band effective equation can accurately take into account different crystal structures and atomic compositions for quasi-1D nanowires if the Kane parameter is substituted by an improved one, which we call P fit .We calculate this parameter by fitting the SO coupling to the 8B model results and provide its value in Table I for InAs, InSb, GaAs and GaSb, both for Zincblende a Wurtzite structures.The improved equation for the Rashba SO coupling constitutes the central result of this work and is given in Eq. ( 8), together with Table I.It retains the simple functional form of the conduction band approximation, but accurately incorporates the complexity of the crystal structure contained in more demanding 8B models through one single parameter, P fit .We test its efficacy for different nanowires in different electrostatic environments and for different transverse subbands, with excellent results.We compare it with other popular oversimplified approximations which cannot capture the crystal structure complexity.Finally, we demonstrate the reliability of our approach by contrasting our predictions with several experimental measurements of the Rashba coupling in InAs and InSb nanowires subject to different electrostatic environments, finding excellent agreement.
This paper is organized as follows.In Sec.II we describe the type of materials of interest and the models and methods that we use.These comprise the 8B k•p Kane model (Sec.II A), the conduction band approximation including the improved effective equation for the SO coupling that we derive (Sec.II B), and the description of the electrostatic environment through the Schrödinger-Poisson equation and the Thomas-Fermi approximation (Sec.II C).In Sec.III we obtain the fitting P fit parameter for various semiconductor compounds and crystal structures, collected in Table I, and compare the SO coupling resulting from the different models and levels of approximation introduced before.In Sec.IV we compare our theoretical results with the Rashba coupling extracted from magnetoconductance measurements in five experiments on InAs and InSb nanowires, discussing the experiments in detail.Finally, in Sec.V we present the conclusions of our work.This paper is complemented with several comprehensive appendixes on the the 8B k•p Kane model for Zinc-blende and Wurtzite crystals (A), the derivation of the conduction band approximation (B), numerical methods (C and D), and a comparison between different semiconductor nanowires (E).

II. MODELS AND METHODS
As mentioned in the introduction, in this work we focus on III-V compound semiconductors such as InAs, InSb, GaAs or GaSb, which tipically exhibit a large Rashba SO coupling [38].We consider semiconductor nanowires with crystallographic structures of Zinc-blende (111) or Wurtzite (0001), since they are the most commonly used in experiments due to their frabrication with low impurity concentrations [20,21,39].We assume that the nanowires are infinite in the specific growth direction but with a finite hexagonal cross-section [24,40] as depicted in Fig. 1.

A. Eight-band k•p Kane model
Multiband k•p models are known to successfully reproduce the energy-band structure of III-V compound semiconductors [1,[32][33][34][35][36][37].The SO coupling can then be directly extracted from the shape of the energy spec- trum [37,[41][42][43].We summarize here this procedure.These effective models, broadly explained and used in the literature (see e.g.Refs. 1 and 44), assume that the electron movement through the crystal is well described by a single-particle Hamiltonian.This Hamiltonian includes relativistic SO effects as well as an effective potential, which arises due to electron-nuclei interactions and thus has the same periodicity than the Bravais lattice.This allows to use the Bloch theorem and to expand the periodic (Bloch) part of the wavefunction around a reciprocal-space point of interest.For the case of III-V semiconductors, the natural expansion is around the Γ point where these compounds exhibit a direct gap [38].The resulting Hamiltonian is then projected over a truncated basis set that includes explicitly the main bands of interest, while their couplings to the remaining bands are included perturbatively using Löwdin perturbation theory [45].Some of the transition matrix elements are forbidden by crystal symmetries.Using group theory arguments, the remaining couplings are substituted by effective parameters.These parameters are called Kane or Luttinger parameters and can be extracted from ab initio calculations or experimental data for a particular material with a specific crystallographic structure.In the 8B model that concerns us, only the four (doubly degenerate) energy bands closer to the semiconductor gap are included in the basis set: the lowest-energy conduction band, and the light-hole, heavy-hole and split-off valence bands.These eight bands are typically sufficient to account for the Rashba SO effects of these materials [44,46,47].For a detailed derivation of this multiband k•p Kane theory and the resulting 8B Hamiltonians for Zinc-blende and Wurtzite crystals, we refer the the reader to Appx.A and references therein [1,48,49].We note that these Hamiltonians are only accurate for the reciprocal space range in which they are fitted to ab initio cal-culations.For the particular Kane parameters provided in Appx.A, this implies k ∈ [−1, 1](nm) −1 .Hence, when the Fermi wavelength is outside this range, the assumptions made for the k•p approximation break down and this model is no longer reliable.For infinite nanowires along the z-direction, the band structure is then calculated by diagonalizing the k•p Hamiltonian for different k z values.To do so, we first replace the momentum operator components across the wire's section by their corresponding derivatives, i.e. k x → −i∂ x and k y → −i∂ y , while replacing the other component k z by a quantum number due to the translational invariance of the wire along this direction.Then, the operators are discretized using the finite difference method in a rectangular mesh for the nanowire section.Special treatment is required in this step in order to avoid spurious solutions in the energy spectrum [50,51].For an extended explanation of the numerical methods used in this work, see Appx. C.
Once the band structure is obtained, the SO coupling can be extracted by fitting each band j by the following effective dispersion relation where j and ± are the band and spin indexes, m eff is the effective mass and E (j) is the band energy at k z = 0.The other two parameters, α (j) eff and β (j) eff , take into account possible Rashba and/or Dresselhaus SO effects.While the Rashba contribution to the SO coupling is known to be mainly linear in k z irrespective of the crystal structure, the Dresselhaus one can be both linear and quadratic with k z for Wurtzite crystals, and only linear for Zincblende (111) crystals [37].Hence, it is not possible, in principle, to separate the contributions of the Rashba and Dresselhaus coefficients in the linear term of the SO coupling.In practice, the linear Dresselhaus contribution turns out to be zero for Zinc-blende (111) crystals [31] (because their crystallographic structure is symmetric), and negligible for Wurtzite (0001) ones [44].Thus, we assume that the linear coefficient of the SO coupling α (j) eff is dominated by the Rashba contribution.

B. Conduction band approximation
As argued in the introduction, for low dimensional systems such as the wires considered here, and specially when they form heterostructures with other materials, the 8B Kane model described before is computationally expensive.For this reason, we look for a simplified model that takes into account the band of interest, the conduction band in our case, but still captures the main SO effects as described by the 8B Kane model (see Appx.B for a derivation).
Starting from the 8B Hamiltonian of a specific crystal, the key approximation behind this procedure consist in ignoring the couplings among the valence bands while maintaining the conduction-valence band ones [1,48].This simplified Hamiltonian allows to decouple the conduction and valence bands, making it possible to describe the conduction band with the following analytical reduced Hamiltonian where the effective mass coefficient is given by and the linear Rashba SO coupling by Here P is one of the Kane parameters of the 8B model and is related to the coupling between conduction and valence bands, φ( r) is the electrostatic potential inside the wire, ∆ g and ∆ soff are the semiconductor and splitoff gaps, respectively, and [, ] is a commutator.We note that the previous expressions depend on the electron energy E, which is itself the solution to the eigenvalue problem of Hamiltonian of Eq. (3).To remove this dependence, which would require a self-consistent solution for the Hamiltonian eigenvalues, it is usually assumed in the literature [1,48,52] that ∆ g eφ( r)+E.Expanding the expressions above and truncating them to lowest order, the effective mass gets and the SO coupling This last (oversimplified ) equation has been used in previous works [52][53][54] to describe the Rashba SO coupling in nanowires, in an attempt to go beyond the use of a rough constant coefficient.Nevertheless, we note that this expression has no information about the energy E, the crystal structure information encoded in the valence band couplings has been lost, and its application range, ∆ g eφ( r) + E, is indeed very restrictive (actually, this condition is violated in most of the experimental situations of interest).In the following section we will show how the oversimplified Eq. ( 7) fails to predict both the quantitative and qualitative behaviour of the Rashba coupling.Moreover, the original equation derived directly from the reduced Hamiltonian, Eq. ( 5), also fails in predicting the quantitative behaviour expected for Zincblende and Wurtzite crystals.
The previous inaccuracies arise because the intravalence band couplings have been ignored in the reduced Hamiltonian.In order to overcome this shortcoming without complicating the functional form of Eq. ( 5), we make the assumption that the Kane parameter P can be substituted by an improved one, P fit , chosen so as to reproduce the Rashba SO coupling extracted from the 8B model of Zinc-blende or Wurtzite nanowires.In this way, we are conjecturing that the lost information about the intra-valence band couplings can be recovered, at least partially, by one fitting parameter.Remarkably, as we will show in the rest of the paper, this assumption turns out to be very accurate.On the other hand, the total energy for an infinite nanowire can be written as E = E (j) + E(k z ), where E (j) is the transverse subband energy and E(k z ) the z-dependent part.Since for the small k z -range for which the 8B model applies ∆ g + eφ( r) + E (j)  E(k z ), we can write the effective Rashba SO coupling for each transverse subband j as where P fit is extracted by fitting this equation to the Rashba coupling obtained with the 8B model.We will show in Sec.III that this improved equation produces results that perfectly match those of 8B model calculations at a considerably reduced computational cost.

C. Electrostatic environment
The Rashba SO coupling given in Eq. ( 8) depends on the gradient of the electrostatic potential.This is a direct manifestation that only when there is a structural inversion asymmetry, i.e. an inhomogeneous electrostatic potential, there is a non-zero Rashba SO coupling.Hence, the SO coupling is sensitive to the precise electrostatic environment, as well as to electron-electron interactions.To compute the electrostatic potential corresponding to an arbitrary environment we use the Poisson equation given by Here ( r) is the inhomogeneous dielectric permittivity, which we take as constant inside each material and with abrupt changes at the interfaces (therefore, it encodes the geometry information of the environment), φ( r) is the electrostatic potential in the entire system and ρ T is the total charge of the wire.This source term includes two parts [53], The first one represents the mobile charge inside the wire that can be changed using gates.The second one represents the surface charge that is typically present at the boundaries of these semiconducting wires [55].This surface charge cannot be removed using gates and it thus gives an intrinsic contribution to the electrostatic potential.In this work we choose ρ surf = 5 • 10 −3 e nm 3 , similarly to previous theoretical works [53,54,56], and in agreement with experimental evidence [39,57].This value however does not play any fundamental role but simply results in a small particular contribution to the intrinsic doping of the wire and its SO coupling [54].The numerical methods used to solve Eq. ( 9) are described in Appx.D.
We note that the solution of the coupled Schrödinger-Poisson equation typically possess a rather demanding numerical problem, as explained in Appx.D. However, it can be simplified by relying on the Thomas-Fermi approximation, as shown in previous works for semiconducting nanowires [52,54,58].This approximation allows to decouple both equations by assuming that the charge density of the wire is indeed very similar to that of a 3D free electron gas, where E F is the Fermi energy of the wire and f (E) is the Fermi-Dirac distribution for a given temperature.
In the next section we compare the results provided by both approaches in some specific cases, showing that the Thomas-Fermi approximation predicts roughly the same Rashba coupling as Schrödinger-Poisson.

III. RESULTS
We first discuss how to obtain the improved parameter P fit of Eq. ( 8) for Zinc-blende (111) and Wurtzite (0001) III-V compound semiconductors.Specifically, we focus on InAs, GaAs, InSb and GaSb.To do so, as already mentioned, we fit Eq. ( 8) to calculations based on the 8B model.We consider the geometry depicted in Fig. 1, where an infinite semiconductor nanowire with hexagonal cross section is placed on top of a dielectric substrate,   2).Solid lines correspond to αR obtained from the effective conduction band (CB) approximation discussed in this work.In red (blue) the result for a Zinc-blende (Wurtzite) crystal using the improved Eq. ( 8) with the corresponding parameter P fit displayed in Table I.In green the result using the original P Kane parameter.
In black the Rashba coupling obtained in the approximation ∆g eφ( r)+E, i.e. using the oversimplified Eq. ( 7).Parameters are: Wwire = 80nm, W substrate = 20nm, W layer = 10nm, wire = InAs = 15.15,substrate = HfO 2 = 25, V layer = 0V and ρ surf = 5 • 10 −3 e nm 3 .The charge density of the wire ρ mobile has been neglected for simplicity.The 8B model parameters are given in Table A 2. particularly Hf 2 O. Below the substrate, there is a bottom gate, which allows to tune the chemical potential inside the wire.To increase the gradient of the electrostatic potential, and therefore enhance the SO coupling, the upper facet of the wire is covered by a grounded metallic layer.Despite the arbitrary choice of this electrostatic environment, the fitting parameter P fit proves to be insensitive to it.This is essential for the applicability of Eq. ( 8) in arbitrary conditions.The motivation for considering this particular environment is simply that this kind of geometries are typically used in experimental setups for spintronics [25,26,59] and Majorana nanowire devices [41,42].Since our sole concern in this subsection is to obtain P fit , we ignore for the moment the electronelectron interactions by fixing ρ mobile = 0.
The modulus [60] of the Rashba coupling obtained from the 8B model through Eq. ( 2) is plotted with dots in Fig. 2 for the case of InAs nanowires.The first four transverse subbands are considered in (a-d), represented versus the back gate voltage.Red (blue) dots correspond to a Zinc-blende (Wurtzite) crystallographic structure.
The corresponding 8B Hamiltonians used for the calculation are provided in Appx.A, together with the values of their parameters, displayed in Tables A 2 and A 3. We can see that the SO coupling exhibits a minimum around −0.04V, independently of the transverse mode.This is because, at this gate voltage value, the electric field inside the wire is basically zero.It occurs at V gate = 0 due to the surface charge present at the nanowire facets, which introduces a small pinned electrostatic field [57].As was noticed in previous works [44], the SO coupling is larger for the Zinc-blende crystal than for the Wurtzite one.
Now we proceed to fit the previous 8B model results with those obtained with the improved Eq. ( 8) through , where is the transverse eigenstate for subband j.We use the values for ∆ g and ∆ soff provided in Ref. 38 and shown in Appx. A. The resulting P fit values for Zincblende and Wurtzite InAs crystals are collected in Table I.We represent | α R | with solid red and blue lines in Fig. 2. We note that we can use the same fitting parameter, calculated for the first mode, for all the different transverse modes since P fit depends only very slightly on the subband energy.This is not a priori obvious since the intra-valence band couplings ignored in the conduction band approximation could, in principle, depend on the specific subband.Fortunately, as shown in Fig. 2, the fit is very good also for higher transverse modes.We further note that the SO coupling decreases with the number of the transverse mode, i.e. it is larger for the lowest energy subband.This can be directly deduced from Eq. ( 8) since it is inversely proportional to the subband energy E (j) .
We have performed equivalent calculations for Zincblende (111) InSb, GaAs and GaSb nanowires.These can be found in Appx.E and the corresponding values of P fit in Table I.
We now compare the previous results for specific crystallographic structures with the ones that one would obtain directly from the reduced Hamiltonian.In Fig. 2 we show with a solid green line the SO coupling obtained with the original Eq. ( 5) (where E is replaced by E (j) ) or, equivalently, with Eq. ( 8) but with the original P Kane parameter (instead of P fit ).We note that this approximation underestimates the SO coupling for Zinc-blende crystals and overestimates it for Wurtzite ones.
Finally, the black solid lines in Fig. 2 correspond to simulations using the oversimplified Rashba coupling given by Eq. ( 7), frequently used in the literature.We observe that this equation provides not only different quantitative results than those predicted for Zinc-blende or Wurtzite crystals, but also qualitative ones.In particular, the oversimplified equation predicts a linear behaviour with the electric field ∇φ, while the original one goes like ∼ φ −2 ∇φ.In addition, it produces larger SO coupling values for higher-energy modes, in contrast to 8B model results.Hence, we conclude that this oversim- plified equation does not describe accurately the Rashba SO coupling in type III-V semiconductors.

A. Schrödinger-Poisson vs Thomas-Fermi
The previous simulations were performed ignoring the charge density of the wire, ρ mobile .Including it requires a self-consistent solution of the coupled Schrödinger-Poisson equations.Specially for the 8B model, this is a difficult task due to the large number of bands involved [34].As explained in Sec.II C, the Thomas-Fermi approximation helps to reduce the computational cost of the simulations.
In order to test the validity of the Thomas-Fermi approximation for the determination of of the SO coupling and whether the electron-electron interaction plays a relevant role, we have performed the same simulations as before in the conduction band approximation but including ρ mobile [61].In Fig. 3 we show the SO coupling as a function of gate voltage in the absence of mobile charge (dashed line), including it in the Thomas-Fermi approximation (solid line) and using the full Schrödinger-Poisson approach (dots).We find that Thomas-Fermi provides an excellent approximation that matches the Schrödinger-Poisson results for Zinc-blende, Wurtzite, as well as simplified structures.Moreover, we see that taking into account the charge density of the wire is essential to describe the behaviour of α R for large positive V gate values (large dopings).This is due to the inhomogeneous electrostatic potential profile created by the charge, which contributes to the structural inversion asymmetry that in turn leads to the Rashba coupling.Once again, we note that the quantitative results predicted by the oversimplified equation (black curves) deviate considerably from the ones predicted for specific crystal structures (red or blue curves).

IV. COMPARISON TO EXPERIMENTS
In order to test the validity of our approach to predict the SO coupling in realistic situations, we compare the results that it provides with the ones obtained in several recent experiments.During the last decade, there has been an increasing interest in measuring the SO coupling in semiconductor nanowires due to their potential applications as spin-FETs [25-27, 62, 63] or Majorana qubit devices [41,42].In most cases, these nanowires were made of InAs due to its large semiconductor band gap and Rashba coupling, although some of them used InSb nanowires [59,64] and other mixed heterostructures [65][66][67][68][69][70][71][72] involving type III-V compound semiconductors.For the comparison, we focus on the works done by Dhara et al. [63], Liang et al. [25], Takase et al. [27], and Scherübl et al. [26], carried out on Zinc-blende InAs nanowires; and by Takase et al. [64] on Zinc-blende InSb nanowires.We choose these experiments because they measure a representative number of SO coupling points versus a wide range of gate potentials.
In all of these experiments, the SO coupling is determined in an indirect way from magnetotransport measurements [73][74][75], which make it possible to access relevant length scales that affect the electron coherence.In particular, the SO coupling can be extracted from the spin-relaxation length l SO as α eff = 2 2m eff lSO .Since this kind of measurements involves the collective transport of electrons around the Fermi energy, the SO coupling extracted from l SO does not correspond to one particular subband but it is instead a weighted sum of all the subbands that contribute to the current.To compute numerically this averaged Rashba coupling in the conduction band approximation, we take the expected value of the SO coefficient where n(E (j) ) is the 1D electron density corresponding to the transverse subband j (see Appx.C for further details).In Fig. 4 we compare the experimental data (dots) with the numerical results obtained with the improved Eq. ( 8) and the P fit values of Table I (red curves).The electrostatic potential is calculated using the Thomas-Fermi approximation.For completeness, we also show the results provided by the oversimplified Eq. ( 7) (blue curves).FIG. 4. Electrostatic environment modelling of some experimental setups (left), and corresponding effective Rashba couplings (right) obtained with magnetoconductance measurements (dots) and with conduction band (CB) numerical simulations (solid lines).In red we present results using the improved Eq. ( 8) for Zinc-blende crystals, while in blue using the oversimplified Eq. ( 7).The shown experimental data corresponds to (a) Dhara et al.   [63].This is one of the first works that used magnetotransport measurements to determine the Rashba coupling in InAs nanowires.The device is quite simple (see sketch on the left): an 80nm wide InAs nanowire is placed on top of a SiO 2 substrate and 300nm below the substrate there is a bottom gate that is used to tune the electrostatic potential inside the wire.The large V gate range explored in this experiment with a relatively small variation of the SO coupling, see right panel, is due to the rather large thickness of the substrate.Both theoretical curves predict the same qualitative behaviour but, although they are quantitatively similar, our approach gives a somewhat better agreement with the experimental values.For this particular setup, the oversimplified equation works reasonably well because the electrostatic potential created by the gate is small.Thus, the condition ∆ g eφ( r) + E is almost fulfilled.Figure 4(b) refers to the experiment carried out by Liang et al. [25].They prove that it is possible to enhance the Rashba coupling by using an appropriate electrostatic environment.To do so, a 40nm wide InAs nanowire is suspended inside the (ionic) dielectric PEO+LiClO 4 .At 50nm below the wire there is a SiO 2 substrate of thickness 250nm sitting on a grounded gate.On top of the device, 500nm above the wire, there is another gate in contact with the dielectric.The PEO+LiClO 4 dielectric is characterized by a large permittivity, which allows to subject the wire boundaries to almost the same potential as it is applied to the top gate.It is thus possible to significantly increase the wire's doping with a small gate voltage.The origin of the SO coupling in this setup is the inhomogeneous distribution of the charge along the radius of the wire.In the right panel we can see that the red curve is in good agreement with the experimental data, whereas the blue one deviates, specially at large V gate , due to the large electrostatic potentials involved.
Figure 4(c) deals with the experiment done by Takase et al. [27].This work follows the same spirit than the previous one, but they look for a long-lived device that could be used as a spin-FET for spintronic applications.To this end, they fabricate a gated-all-around (GAA) device in which a 100nm wide nanowire is covered with a 2nm thick Al 2 O 3 dielectric, a 4nm thick HfO 2 dielectric and a potential gate (see sketch on the left).Due to the small distance between the gate and the wire, the potential applied to the gate and at the boundary of the wire is almost the same.As in the previous example, the theoretical prediction resulting from our improved equation exhibits an excellent agreement with the experimental data, while the oversimplified one largely deviates from it.In this case, and motivated by their transport measurements, we have chosen the Fermi level at E F = 100meV, as if the wire was initially doped with holes.
Figure 4 (d) shows the last case with InAs.It was performed by Scherübl et al. [26] to prove that it is possible to tune the Rashba coupling without changing the elec-tron occupation inside the nanowire.To that end, they use a bottom gate together with two side ones.As shown in the sketch, a 77nm wide nanowire is placed over a SiO 2 substrate, a bottom gate is 1µm below the substrate and the two side gates are placed at 70nm from the corners of the wire.The main origin of the strong SO coupling in this device is the structural inversion symmetry created by the difference between the gate potentials applied to the side gates.In the right panel, the SO coupling is plotted versus one side gate, while the other side gate is changed accordingly to keep the total charge inside the wire constant (see Ref. 26 for further information).We find here that both theoretical methods give a good match, although our improved equation produces a better agreement with the data.Actually, in order to explain the discrepancy between the prediction of the oversimplified equation and the experimental data, in Ref. 26 the authors add ad hoc a built-in intrinsic SO coupling of ∼ 5(meV•nm), which is not necessary using our effective equation.
Finally, Fig. 4(e) refers to the experiment realised by Takase et al. [64] in InSb nanowires.This work proves that the SO coupling of InSb nanowires is much larger than that of InAs ones.Their device consists of a 182nm wide wire covered by a 6nm thick Al 2 O 3 dielectric, placed directly over a metallic gate.The small thickness of the dielectric allows to tune (almost) perfectly the wire.In this last case we also find a good agreement between the red curve and the experimental measurements, while the oversimplified equation fails to predict the proper behaviour.Deviations at small V gate may arise due to the small length of the wire in this experiment (500nm).This may cause that the leads used for the transport measurements have an impact on the electrostatic potential profile at the wire edges [79], changing unintentionally the precise SO coupling value.

V. SUMMARY AND CONCLUSIONS
Multiband k•p effective models are successfully used to indirectly extract the SO interaction of semiconductors from their band structure, but they can be computationally demanding in low-dimensional heterostructures of current interest.In this work, we perform a single band approximation and introduce an analytical expression that accurately describes the Rashba SO coupling for the lowest-energy conduction band of III-V compound semiconductors, Eq. ( 8).This equation takes into account not only the dependence of α R with the spatiallydependent electrostatic potential, which accounts for possible structural inversion asymmetries, but also with the transverse subband energy in low-dimensional systems.Equation ( 8) depends on two parameters, the semiconductor gap and the split-off gap between valence bands.Additionally, it approximately takes into account the crystallographic structure of the compound semiconductor, lost, at least partially, in the conduction band ap-proximation, through one improved effective parameter that we call P fit .We compute this parameter by fitting the SO coupling given by Eq. ( 8) to results provided by realistic 8B k•p calculations.The results for Zinc-blende (111) InAs, InSb, GaAs and GaSb nanowires, and for Wurtzite (0001) InAs nanowires are collected in Table I.
We compare the results provided by Eq. ( 8) with other approximations and with exact 8B model calculations.The improved equation works well regardless of the particular electrostatic environment surrounding the wire, the parallel mode and the chemical composition of the wire.As a final proof of the validity of our approach, we simulate the experimental conditions of five magnetoconductance experiments on InAs and InSb nanowires realized in recent years.We find that the Rashba SO coupling resulting from Eq. ( 8) is in excellent agreement with the experimental data over wide ranges of external gate potentials.
We believe that our work may be useful for reducing the computational cost of accurate Rashba SO coupling computations in realistic low-dimensional semiconductors of current interest.It may also help to understand and design better devices where the Rashba SO coupling is key.
matical Software 37, 1 (2010).We provide here an introduction to the multiband k•p theory extracted from several references [1,80,81].This theory can be derived starting from the single-electron Hamiltonian where p = −i ∇ is the momentum operator, m 0 the (bare) electron mass, c the speed of light and σ the vector of Pauli matrices for the spin degree of freedom.The first term of this Hamiltonian corresponds to the kinetic energy of the electron, the second one takes into account relativistic SO effects, and the last one corresponds to the effective electrostatic potential energy experienced by the electron inside some material.If the crystal is translational invariant, this potential can be described using a periodic function.As a consequence, the electron wave function satisfies the Bloch's theorem, i.e., where k is the wavevector (usually restricted to the first Brillouin zone), n the quantum number that labels the different possible energy bands, and u n,k ( r) is the so-called envelope function that encodes the periodic part of the wavefunction.Hence, the electron wave function satisfies the condition pΨ n,k = e i k• r ( k + p)u n,k ( r), what allows to rewrite the Hamiltonian as This Hamiltonian is known as the k•p Hamiltonian (because it includes a k • p term).It describes the motion of an electron inside a periodic crystal.
In general, the k•p Hamiltonian has no analytical solution, so it is usually solved perturbatively.The most common way to do so is to expand the Hamiltonian around a point k 0 in reciprocal space, whose solution is known, and then use Löwdin perturbation theory [45] to perform the expansion over a reduced basis set.Within this technique, the states are separated into two classes A and B. Class A includes the truncated basis set elements that describe the main aspects of the crystal.In principle, this basis set could include all the orbitals on each atom of the unit cell, but this would not help to decrease the complexity of the problem.Because of that, the less influential states not included in A are collected in class B. The couplings between class B and A states are the ones treated perturbatively.
In this representation, the envelope function (in Dirac notation) is written as a superposition of both kind of states where |α and |β are the states in class A and B, respectively.The projection of the Hamiltonian onto the class A basis gives rise to the matrix elements Unfortunately, these matrix elements cannot be evaluated analytically in general because one would need to know all the dipole terms α| k • p |β of all the transitions among the different states, as well as their corresponding transition energies E α,β .For this reason, one firstly invokes symmetry arguments (related to the crystal symmetries) to know which matrix elements are forbidden and, secondly, one substitutes the remaining expressions by parameters, whose and the off-diagonal ones are Here, E F , E h = E F − ∆ g and E soff = E F − ∆ g − ∆ soff are the band edges of the conduction, hole and split-off bands, respectively; ∆ g and ∆ soff are the gaps between the conduction/hole and hole/split-off bands at the Γ point; φ( r) is the electrostatic potential; k ± ≡ k x ± ik y and {γ i } and P are Kane parameters.In this work, we choose the conduction band edge as the reference energy, E F = 0.The Hamiltonian elements whose functional form has been substituted by phenomenological parameters are In order to avoid the spurious solutions coming from the loss of ellipticity of the Schrödinger equation [82][83][84], the Kane parameters {γ i } are renormalized from the Luttinger ones γ where E p = 2m0 2 P 2 is the Kane energy.The specific values that we have used for the Kane parameters are extracted from Ref. 38   The 8-band k•p Kane Hamiltonian for Wurtzite crystals has been derived and described in previous works [44].The basis for Wurtzite type semiconductor is given by Its corresponding Hamiltonian in the Ψ = (Ψ hh,↑ , Ψ lh,↑ , Ψ soff,↑ , Ψ hh,↓ , Ψ lh,↓ , Ψ soff,↓ , Ψ c,↑ , Ψ c,↓ ) basis is where A i , B i , ∆ i , α i , β i , P i , e i and γ 1 are the band and Kane paremeters.As before, E h and E F are the hole and conduction band edges.The explicit form of these parameters is provided in Ref. [44].The specific values we use are extracted from Ref. 44 and shown in Table A 3. We choose the conduction band edge as reference energy, E F = 0.
By performing the following folding-down procedure [52], one can effectively reduce the previous 8×8 Hamiltonian to a 2×2 one, which only involves the conduction band.Performing the matrix multiplications, one obtains where σ i are the spin Pauli matrices (and σ 0 the identity), and We assume now that this Hamiltonian describes a nanowire that is infinite along the growth direction (z direction), and therefore k z can be considered as a good quantum number.Conversely,we assume that in the other directions the nanowire is finite and thus their momentum operators transform as k x → −i∂ x and k y → −i∂ y .Hence, the Hamiltonian can be written as The first term of this simplified Hamiltonian corresponds to the kinetic energy, but the electron has now an effective mass given by The last term of the Hamiltonian corresponds to the Rashba SO interaction, whose Rashba coefficients α x and α y are This equation is the one we call original in Eq. ( 5) of the main text.Equation (B7) describes the Hamiltonian of an electron quasiparticle in the conduction band approximation.We have thus reached a Hamiltonian that involves a smaller number of bands (i.e.only the conduction band), and which allows us to directly introduce additional terms whose functional form we know, as for instance the Zeeman field.We note that this simplified Hamiltonian is equal to that of a bare electron [Eq.(A1)], but with an effective mass m eff ( r) and an effective Rashba coupling α x,y ( r) that depend not only on the band and Kane parameters, ∆ g , ∆ soff and P , but also on the spatial-dependent electrostatic potential φ( r) and the particular transverse subband E that it occupies.We note that due to the energy dependence of these effective quantities, the simplified Hamiltonian has to be solved self-consistently.
To avoid this complication, a further simplification is commonly performed in the literature [1,48,52].If ∆ g and ∆ soff are the largest energies in the system, then it is possible to expand in Taylor series assuming −eφ( r) Truncating the Taylor series up to the zeroth order for the effective mass (up to n = 0 in the above equation), it is possible to obtain a renormalized effective mass which is energy and spatial independent, Similarly, if one truncates the Taylor series up to the first order (n = 1) in the Rashba coupling, it is possible to get an energy-independent Rashba coupling, (B12) These two last equations are the ones frequently used in the literature.In particular, Eq. (B11) is used to estimate the effective masses of typical semiconductors, leading to the widely used values [38] of m eff 0.023m 0 for InAs and m eff 0.014m 0 for InSb semiconductors.Additionally, Eq. (B12) has been used in numerous works [52][53][54] over the last decade to describe or estimate the Rashba coupling in semiconductor nanowires and is the one that we have called oversimplified in the main text, see Eq. (7).However, as we show in this work, in order to accurately predict the behaviour of the Rashba coupling in recent experiments [25-27, 63, 64], it is necessary to go beyond this last approximation and use directly Eq. (B9).This is because most of the experiments, specially those that try to enhance the Rashba coupling through external gates, are outside the regime −eφ( r) − E E h .
to suppress the spurious states.However, if the discretization is not homogeneous, the FDM generates non-Hermitian matrices.For instance, for two consecutive points x i and x i+1 , since, in general, for an arbitrary mesh (x i+1 − x i−1 ) = (x i+2 − x i ).To correct this problem, we follow the ideas of Ref. 87 and we symmetrize the discretization operator, define here as ξ ij ≡ h/|x i − x j | where h ≡ x i+1 − x i , since now the mesh spacing |x i+1 − x i | is not just a constant but a position-dependent operator.With this symmetrization, the derivatives can be written as which are symmetric operators now.
Once the Hamiltonian is discretized in the transverse section (taking h = 1nm in this work), it is diagonalized for each k z value in order to obtain the band structure E(k z ) and their corresponding wavefunctions Ψ(x, y, k z ).For this we use the standard ARPACK tools provided by the Python package Scipy.For the conduction band model, we assume that the SO length is larger or comparable to the wire diameter, i.e. l SO W wire , as is the case of every experiment analyze in this work.This assumption allows to split the Hamiltonian of Eq. (B7) into transverse and longitudinal parts [88][89][90], Hence, instead of performing one (numerical) diagonalization for each k z value, the conduction band Hamiltonian H CB can be written in the transverse basis and diagonalized analytically for any k z value, E ± (k z ) =

FIG. 3 .
FIG.3.Rashba coupling modulus versus gate voltage for the lowest energy subband within the conduction band approximation.(a) In green the results obtained using the improved Eq. (8) but with the original P Kane parameter.(b) In black, using the oversimplified Eq. (7).(c,d) In red and blue the results of the improved equation using the corresponding P fit parameter displayed in TableIfor a Zinc-blende and a Wurtzite crystal, respectively.Dots are used for full Schrödinger-Poisson simulations, solid lines for the Thomas-Fermi approximation and dotted lines for ρ mobile = 0, i.e. ignoring the mobile charge density.Same parameters as the ones of Fig.2(except ρ mobile ).Temperature T = 10mK.

Figure 4 (
Figure4(a) refers to the experiment performed by Dhara et al.[63].This is one of the first works that used magnetotransport measurements to determine the Rashba coupling in InAs nanowires.The device is quite simple (see sketch on the left): an 80nm wide InAs nanowire is placed on top of a SiO 2 substrate and 300nm below the substrate there is a bottom gate that is used to tune the electrostatic potential inside the wire.The large V gate range explored in this experiment with a relatively small variation of the SO coupling, see right panel, is due to the rather large thickness of the substrate.Both theoretical curves predict the same qualitative behaviour but, although they are quantitatively similar, our approach gives a somewhat better agreement with the experimental values.For this particular setup, the oversimplified equation works reasonably well because the electrostatic potential created by the gate is small.Thus, the condition ∆ g eφ( r) + E is almost fulfilled.Figure4(b) refers to the experiment carried out by Liang et al.[25].They prove that it is possible to enhance the Rashba coupling by using an appropriate electrostatic environment.To do so, a 40nm wide InAs nanowire is suspended inside the (ionic) dielectric PEO+LiClO 4 .At 50nm below the wire there is a SiO 2 substrate of thickness 250nm sitting on a grounded gate.On top of the device, 500nm above the wire, there is another gate in contact with the dielectric.The PEO+LiClO 4 dielectric is characterized by a large permittivity, which allows to subject the wire boundaries to almost the same potential as it is applied to the top gate.It is thus possible to significantly increase the wire's doping with a small gate voltage.The origin of the SO coupling in this setup is the inhomogeneous distribution of the charge along the radius of the wire.In the right panel we can see that the red curve is in good agreement with the experimental data, whereas the blue one deviates, specially at large V gate , due to the large electrostatic potentials involved.Figure4(c) deals with the experiment done by Takase et al.[27].This work follows the same spirit than the previous one, but they look for a long-lived device that could be used as a spin-FET for spintronic applications.To this end, they fabricate a gated-all-around (GAA) device in which a 100nm wide nanowire is covered with a 2nm thick Al 2 O 3 dielectric, a 4nm thick HfO 2 dielectric and a potential gate (see sketch on the left).Due to the small distance between the gate and the wire, the potential applied to the gate and at the boundary of the wire is almost the same.As in the previous example, the theoretical prediction resulting from our improved equation exhibits an excellent agreement with the experimental data, while the oversimplified one largely deviates from it.In this case, and motivated by their transport measurements, we have chosen the Fermi level at E F = 100meV, as if the wire was initially doped with holes.Figure4(d) shows the last case with InAs.It was performed by Scherübl et al.[26] to prove that it is possible to tune the Rashba coupling without changing the elec- [92] A. Logg, K.-A.Mardal, G. N. Wells, et al., Automated Solution of Differential Equations by the Finite Element Method (Springer, Berlin, Heidelberg, 2012).Appendix A: 8-band k•p Kane model 1.Derivation of the model y φ( r) − φ( r)∂ x,y } .
where E(j) are the eigenvalues of the transverse subband Hamiltonian H T (x, y) (found numerically by diagonalizing it), andα ( r)σ y − α y ( r)σ x ) Ψ (j) T (C8)is the projection of the Rashba coupling onto the transverse basis spanned by the eigenvectors Ψ (j) T (x, y) of the transverse Hamiltonian H T (x, y).The charge density for the conduction band approximation is simply given byρ(x, y) = e j |Ψ (j) T (x, y)| 2 • n(E (j) ) = e f (E)  is the Fermi-Dirac distribution and n(E (j) ) is the 1D electron density.This is found by integrating up to the Fermi level the 1D density of states

TABLE I .
Parameter P fit (in meV•nm units) to be used in the improved equation for the Rashba SO coupling, Eq. (8), within the conduction band approximation.This parameter is extracted by fitting to numerical eight-band model calculations.
and shown in Table A 2.

Table A 3
: Band and Kane parameters for the 8-band InAs Wurtzite Hamiltonian, extracted from Ref. 44.