# Spin Hall and Edelstein effects in metallic films: from 2D to 3D

## Abstract

A normal metallic film sandwiched between two insulators may have strong spin-orbit coupling near the metal-insulator interfaces, even if spin-orbit coupling is negligible in the bulk of the film. In this paper we study two technologically important and deeply interconnected effects that arise from interfacial spin-orbit coupling in metallic films. The first is the spin Hall effect, whereby a charge current in the plane of the film is partially converted into an orthogonal spin current in the same plane. The second is the Edelstein effect, in which a charge current produces an in-plane, transverse spin polarization. At variance with strictly two-dimensional Rashba systems, we find that the spin Hall conductivity has a finite value even if spin-orbit interaction with impurities is neglected and “vertex corrections” are properly taken into account. Even more remarkably, such finite value becomes “universal” in a certain configuration. This is a direct consequence of the spatial dependence of spin-orbit coupling on the third dimension, perpendicular to the film plane. The non-vanishing spin Hall conductivity has a profound influence on the Edelstein effect, which we show to consist of two terms, the first with the standard form valid in a strictly two-dimensional Rashba system, and a second arising from the presence of the third dimension. Whereas the standard term is proportional to the momentum relaxation time, the new one scales with the spin relaxation time. Our results, although derived in a specific model, should be valid rather generally, whenever a spatially dependent Rashba spin-orbit coupling is present and the electron motion is not strictly two-dimensional.

## I Introduction

Spin-orbit coupling gives rise to several interesting transport phenomena arising from the induced correlation between charge and spin degrees of freedom. In particular, it allows one to manipulate spins without using magnetic electrodes, having as such become one of the most studied topics within the field of spintronics. Hirsch (1999); Zhang (2000); Murakami et al. (2003); Sinova et al. (2004); Engel et al. (2007); Raimondi et al. (2006); Culcer and Winkler (2007a, b); Culcer et al. (2010); Tse et al. (2005); Galitski et al. (2006); Tanaka and Kontani (2009); Hankiewicz and Vignale (2009); Vignale (2010) Among the many interesting effects that arise from spin-orbit coupling, two stand out for their potential technological importance: the spin Hall effectDyakonov and Perel (1971) and the Edelstein effectLyanda-Geller and Aronov (1989); Edelstein (1990). The spin Hall effect consists in the appearance of a -polarized spin current flowing in the -direction produced by an electric field in the -direction.Kato et al. (2004); Sih et al. (2005); Wunderlich et al. (2005); Stern et al. (2006, 2008) The generation of a perpendicular electric field by an injected spin current, i.e. the inverse spin Hall effect, has been observed in numerous settings and presently provides the basis for one of the most effective methods to detect spin currents.Valenzuela and Tinkham (2006); Kimura et al. (2007); Seki et al. (2008) The Edelstein effectLyanda-Geller and Aronov (1989); Edelstein (1990) consists instead in the appearance of a -spin polarization in response to an applied electric field in the -direction. It has been proposed as a promising way of achieving all-electrical control of magnetic properties in electronic circuits.Mihai Miron et al. (2010); Kato et al. (2004); Sih et al. (2005); Inoue et al. (2003); Yang et al. (2006); Chang et al. (2007); Koehl et al. (2009); Kuhlen et al. (2012) The two effects are deeply connected,Gorini et al. (2008); Raimondi and Schwab (2009); Shen et al. (2014) as we will see momentarily.

There are, in principle, several possible mechanisms for the spin Hall effect, and it is useful to divide them in two classes. We call them either extrinsic or intrinsic, depending on whether their origin is the spin-orbit interaction with impurities or with the regular lattice structure. In this work we will focus exclusively on intrinsic effects. This means that the impurities (while, of course, needed to give the system a finite electrical conductivity) do not couple to the electron spin.

Bychkov and Rashba devised an extremely simple and yet powerful modelBychkov and Rashba (1984)
describing the intrinsic spin-orbit coupling of the electrons in a 2-Dimensional Electron Gas (2DEG)
in a quantum well in the presence of an electric field perpendicular to the plane in which the electrons move.
In spite of its apparent simplicity, this analytically solvable model has several subtle features,
which arise from the interplay of spin-orbit coupling and impurity scattering.
The best-known feature is the vanishing of the Spin Hall Conductivity (SHC) for a uniform and constant
in-plane electric field.Mishchenko et al. (2004); Raimondi and Schwab (2005); Khaetskii (2006)
This would leave spin-orbit coupling with impurities (not included in the original Bychkov-Rashba model)
as the only plausible mechanism for the experimentally observed spin Hall effect
in semiconductor-based 2DEGs Kato et al. (2004); Sih et al. (2005).
^{1}

However it has been recently pointed out that the vanishing of the SHC need not occur
in systems which are not strictly two-dimensional, as explicitly shown
in a model schematically describing the interface of the two insulating oxides LaAlO and
SrTiO (LAO/STO)Hayden et al. (2013).
Even more recently Wang et al. (2013), it has been suggested that a large SHC could be realized
in a thin metal (Cu) film that is sandwiched between two different insulators,
such as oxides or the vacuum.^{2}

The inversion symmetry breaking across the interfaces produces interfacial Rashba-like spin-orbit couplings, thus allowing metals without substantial intrinsic bulk spin-orbit to host a non-vanishing SHC. The spin-orbit coupling asymmetry – or, more generally, the fact that the spin-orbit interaction is not homogeneous across the thickness of the film – is the core issue in this novel approach. In this paper we will study the influence of the interfacial spin-orbit couplings on the Edelstein and spin Hall effects in this class of heterostructures.

Before proceeding to a detailed study of the model depicted in Fig. 1, it is useful to recall the deep connectionGorini et al. (2008); Raimondi and Schwab (2009); Shen et al. (2014) that exists between the spin Hall and Edelstein effects in the Bychkov-Rashba model, described by the Hamiltonian

(1) |

where is the effective electron mass and is the Bychkov-Rashba spin-orbit coupling constant given by , with the materials’ effective Compton wavelength, the electric field perpendicular to the electron layer, and the absolute value of the electron charge. It is convenient to describe spin-orbit coupling in terms of a non-Abelian gauge field , with and . Tokatly (2008); Gorini et al. (2010); Takeuchi and Nagaosa (2013) If not otherwise specified, superscripts indicate spin components, while subscripts stand for spatial components. The first consequence of resorting to this language is the appearance of an magnetic field , which arises from the non-commuting components of the Bychkov-Rashba vector potential. Such a spin-magnetic field couples the charge current driven by an electric field, say along , to the -polarized spin current flowing along . This is very much similar to the standard Hall effect, where two charge currents flowing perpendicular to each other are coupled by a magnetic field. The drift component of the spin current can thus be described by a Hall-like term

(2) |

It is however important to appreciate that this is not yet the full spin Hall current, i.e. is not the full SHC. In the diffusive regime is given by the classic formula , where is the “cyclotron frequency” associated with the magnetic field, is the elastic momentum scattering time, and is the Drude conductivity. For a more general formula see Eq. (6) below.

In addition to the drift current, there is also a “diffusion current” due to spin precession around the Bychkov-Rashba effective spin-orbit field. Within the formalism this current arises from the replacement of the ordinary derivative with the covariant derivative in the expression for the diffusion current. The covariant derivative, due to the gauge field, is

(3) |

with a given quantity being acted upon. The normal derivative, , along a given axis is shifted by the commutator with the gauge field component along that same axis. As a result of the replacement diffusion-like terms, normally proportional to spin density gradients, arise even in uniform conditions and the diffusion contribution to the spin current turns out to be

(4) |

where is the diffusion coefficient, being the Fermi velocity. In the diffusive regime the full spin current can thus be expressed as

(5) |

For a detailed justification of Eq. (5) we refer the reader to Refs. Gorini et al., 2010; Raimondi et al., 2012. The factor in front of the spin density in the first term of Eq.(5) can also be written as an effective velocity . Here is the typical spin length due to the different Fermi momenta in the two spin-orbit split bands, whereas is the Dyakonov-Perel spin relaxation time. In terms of and one has

(6) |

which is indeed equivalent to the classical surmise given after Eq. (2). If we introduce the total SHC and the Edelstein Conductivity (EC) defined by

(7) |

we may rewrite Eq.(5) as

(8) |

In the standard Bychkov-Rashba model a general constraint from the equation of motion dictates that under steady and uniform conditions . Therefore the EC reads

(9) |

which is easily obtained by using the expressions given above and the single particle density of states in two dimensions, . The remarkable thing is that this expression remains unchanged for arbitrary ratios between the spin splitting energy and the disorder broadening of the levels. However, in a more general situation with a non-zero SHC the EC would consist of the two terms appearing in Eq. (8). The latter equation is the “deep connection” mentioned earlier between the Edelstein and the spin Hall effect. The first term on the r.h.s. is the “regular” contribution to the EC, the only surviving one in the Bychkov-Rashba model where the full SHC vanishes. The second term is “anomalous” in the sense that it does not appear in the standard Bychkov-Rashba model, but it does appear in more general models such as the one we discuss in this paper. Notice that the “regular” term is proportional to (see Eq. (6)), while the “anomalous” term, being proportional to the Dyakonov-Perel relaxation time and, in the diffusive regime, is inversely proportional to the momentum relaxation time.

At variance with the Bychkov-Rashba model, the one we choose for our system is not strictly two-dimensional, and we take into account several states of quantized motion in the direction perpendicular to the interface (). Another crucial feature of this model is the occurrence of two different spin-orbit couplings at the two interfaces. The difference arises because (i) the interfacial potential barriers and are generally different, and (ii) the effective Compton wavelengths and , characterizing the spin-orbit coupling strength at the two interfaces, are different.

Our central results for the generic asymmetric model are

(10) |

and

(11) |

the sums running over the filled -subbands of the thin film. To each subband there correspond a Fermi wavevector (without spin-orbit) , an intraband spin-orbit energy splitting with a linear- and a cubic-in- part

(12) | |||||

(13) |

and a Dyakonov-Perel spin relaxation time

(14) |

In the above formulas is the film thickness and .
Two particularly interesting regimes are apparent. First, a “quasi-symmetric” configuration,
defined by equal spin-orbit strengths, , but different barrier heights, .
In this case (due to Ehrenfest’s theorem^{3}

(15) |

At the same time the “anomalous” EC is at its largest. A second very interesting configuration is a strongly asymmetric insulator-metal-vacuum junction, and . In this case the SHC becomes directly proportional to the gap

(16) |

Notice however that the SHC cannot be made arbitrarily large simply by engineering a large , since the above result holds provided .

The paper is organized as follows. In Sec. II we introduce and discuss the model. In Secs. III and IV we calculate the SHC and the EC, respectively. Both Sections are technically heavy and can be skipped at a first reading, leading straight to Sec. V where the physical consequences of our results are discussed and special regimes are analyzed. Sec. VI presents our summary and conclusions.

## Ii The model and its solution

Following Ref. Wang et al., 2013, we model the normal metallic thin film via the following Hamiltonian

(17) |

where the first term represents the kinetic energy associated to the unconstrained motion in the plane and is the standard two-dimensional momentum operator. The finite thickness of the metallic film is taken into account by a confining potential

(18) |

where is the height of the potential barrier at and is the Heaviside function. The third term in Eq.(17) describes the Rashba interfacial spin-orbit interaction in the plane located at

(19) |

where are the effective Compton wavelengths for the two interfaces, are the Pauli matrices. The last term in Eq.(17) represents the scattering from impurities affecting the motion in the plane and is the coordinate operator. The impurity potential is taken in a standard way as a white-noise disorder with variance , where is the two-dimensional density of states previously introduced. We will assume throughout that the Fermi energy in each subband is much larger than the level broadening and use the self-consistent Born approximation.

The eigenfunctions of the Hamiltonian (17) have the form

(20) |

where is the area of the interface, is the in-plane wave vector, is the position in the interfacial plane and is the coordinate perpendicular to the plane. is the angle between and the axis. These states are classified by a subband index , which plays the role of a principal quantum number, an in-plane wave vector , and an helicity index, or which determines the form of the spin-dependent part of the wave function.

By inserting the wave function (20) into the Schrödinger equation for the Hamiltonian (17) we find the following equation for the functions describing the motion along the -axis

(21) |

where the full energy eigenvalues are

(22) |

By taking into account the continuity of the wave function at and the discontinuities of its derivatives we obtain for the eigenvalue the following transcendental equation

(23) |

where the energy is measured in units of set by the thickness of the film. In the absence of spin-orbit coupling () and for infinite heights of the potential (), the solution reduces to the well-known energy levels . In the general case with both and finite we use perturbation theory by assuming large. There are two natural length scales associated with the confining potential so that we expand in the small parameters . Since all the energy scales are set by , we find useful to describe the spin-orbit coupling in terms of the parameters in such a way that the product has the dimensions of a velocity, just as the typical Rashba coupling parameter. In the following we make an expansion to first order in and up to third order in .

For the eigenvalues of (21) we find

(24) |

and the eigenfunctions

(25) |

where

(26) |

Notice that the sign of the coefficients and depends on the relative strength of the spin-orbit coupling and barrier heights . To avoid troubles with minus signs in the following calculations, we assume that the couplings are labeled in such a way that , and so that .

In the next Section we evaluate the SHC assuming that is the topmost occupied subband. In the following we use units such that .

## Iii Spin Hall conductivity

The SHC is defined as the non-equilibrium spin density response to an applied electric field. By using a vector gauge with the electric field given by , the Kubo formula, corresponding to the bubble diagram of Fig.2, reads

(27) |

where we have introduced the spin current operator and the charge current operator . The number current operator, besides the standard velocity component, includes a spin-orbit induced anomalous contribution . Without vertex corrections, the anomalous contribution reads

(28) |

This expression can be written in terms of the exact Green functions and vertices as

(29) |

where is the unit charge, and is the Green function averaged over disorder in the self-consistent Born approximation with self energy

(30) |

After performing the integral over the frequency we obtain

(31) |

where we have introduced the retarded and advanced zero-energy Green functions at the Fermi level

(32) |

and exploited the fact that plane waves at different momentum are orthogonal.

To proceed further we need the expression for the vertices. It is easy to recognize that the standard part of the velocity operator does not contribute since it requires , whereas the matrix elements of differ from zero only for . Explicitly we have

(33) | |||||

(34) | |||||

(35) |

where is half the spin-splitting energy in the -th band. Eq.(34) is straightforwardly obtained from the eigenvalue equation (21) for the functions .

Let us now discuss the overlaps between the wave functions . If we have

(36) |

which is unity plus corrections of order when . If is at least of order . Before continuing our calculation we observe that it is important to distinguish between the intra-band () and the inter-band () contributions. The inter-band contributions are of second order in , because they are proportional to . Since we limit our expansion to the first order in we will from now on neglect these contributions. Notice, however, that this approximation is no longer valid when the intra-band splitting controlled by and vanishes. In this case one cannot avoid taking into account the inter-band contributions. In the same spirit, we also approximate the intra-band overlap , because all of our results are at least linear in and we neglect higher order terms.

The anomalous contribution to the velocity vertex, , can be computed following the procedure described in Ref. Raimondi and Schwab, 2005 according to the equations (see Fig.3)

(37) |

To extend the treatment to the present case, the projection must be made over the states . Assuming that the impurity potential does not depend on , the matrix elements of the effective vertex are:

(38) |

and is given by Eq.(34). The matrix elements and are those of the impurity potential:

(39) | |||||

(40) |

By observing that , one can perform the integration over the direction of in the expression of

(41) |

to get

(42) |

Approximating , summing over , and integrating over with the technique shown in the Appendix yields

(43) |

where we have introduced the spin-averaged Fermi momentum in the -th subband

(44) |

On the other hand is given by

(45) |

where has been replaced by at the required level of accuracy. Combining and as mandated by Eq. (III) we finally obtain

(46) |

Next we project the equation for the vertex corrections in the basis of the eigenstates and get the following integral equation:

(47) |

which, by confining to intra-band processes only, can be solved with the ansatz yielding

(48) |

By performing the integral over momentum and summing over the spin indices in Eq.(31), one obtains the SHC as

(49) |

where is the number of occupied bands.

If vertex corrections are ignored, i.e., if we approximate (cf. Eq.(34)), Eq.(49) gives us

(50) |

which, in the weak disorder limit (), reproduces the result of Ref. Wang et al., 2013, i.e. .

If instead the renormalized vertex (48) is properly taken into account, we obtain

(51) |

Notice that, being proportional to (, ), this result is consistent with the result obtained in Ref. Hayden et al., 2013 for a different but related model. Making use of the explicit expressions for and we finally get the previously reported result of Eq.(10).

## Iv Edelstein conductivity

In the d.c. limit, i.e., for , the Edelstein conductivity (EC) is defined by

(52) |

That can be written as:

(53) |

After performing the integral over frequency we get

(54) |

where we have used again the orthogonality of the eigenvectors with different momentum. As shown in Fig.2, we consider the bare vertex for the spin density and the two vertices for the number current density ,Raimondi and Schwab (2005) – being the renormalized spin-dependent part of the vertex. Clearly, the two parts of the number current vertex yield two separate contributions to the EC and we are now going to evaluate them separately. We then evaluate the (a) diagram in Fig.2 as:

(55) |

where the matrix elements of the spin vertex is

(56) |

Setting and using Eq.(24) for the energy eigenvalues, we can perform the integration over the momentum in Eq.(55) obtaining for the expression

(57) |

Next we evaluate the (b) diagram in Fig.2 as:

(58) |