Skew Scattering Mechanism by an Ab Initio Approach: extrinsic spin ...

Solid State Phenomena Vols. 168-169 (2011) pp 27-30
Online available since 2010/Dec/30 at www.scientific.net
© (2011) Trans Tech Publications, Switzerland
doi:10.4028/www.scientific.net/SSP.168-169.27
Skew Scattering Mechanism by an Ab Initio Approach:
extrinsic spin Hall effect in noble metals
M. Gradhand 1, a , D.V. Fedorov 2, b , P. Zahn 2, c 2, 1, d
and I. Mertig
1 Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany
2 Institute of Physics, Martin Luther University Halle-Wittenberg, D-06120 Halle, Germany
a martin.gradhand@physik.uni-halle.de, b dmitry.fedorov@physik.uni-halle.de,
c peter.zahn@physik.uni-halle.de, d ingrid.mertig@physik.uni-halle.de
Keywords: skew scattering, spin Hall effect, impurities in metals, KKR method.
Abstract. We present a first-principles study of the extrinsic spin Hall effect due to skew scattering
at substitutional defects in noble metals (Cu, Ag, and Au). The dependence of the spin Hall angle on
the type of impurity atoms in the host materials is discussed. We perform a detailed analysis based
on the consideration of the total angular momentum dependence on scattering phase shifts of the
impurity and host atoms.
Introduction
The spin Hall effect (SHE) attracts a lot of interest due to the possibility of spin current generation
in nonmagnetic materials. For investigation of the SHE, intrinsic and extrinsic contributions to this
effect can be distinguished. The first one is described by the Berry curvature [1,2], while the second
one includes the so-called side jump [3,4] and skew scattering [5-9] mechanisms. The present work
is based on the density functional theory in the framework of a fully relativistic Korringa-Kohn-
Rostoker method [10-12]. It was successfully applied for the description of the spin-flip scattering
[13] and extrinsic SHE [14] caused by impurities. Here, we describe shortly the formalism of the
solution of the linearized Boltzmann equation including the scattering-in term [14] and apply it for
a detailed analysis of the skew scattering mechanism in noble metals with different substitutional
defects.
Method
The main focus of this paper is the calculation of the spin Hall angle α = σ s yx /σxx , where σxx is the
longitudinal component of the charge conductivity tensor
σ = (e 2 /ћ) (2π) -3 ∫∫ vk ◦ λk dS/|vk| , (1)
and σ s yx is the Hall component of the spin conductivity tensor
σ s = (e 2 /ћ) (2π) -3 ∫∫ sz(k) vk ◦ λk dS/|vk| . (2)
The spin polarization sz(k), the group velocity vk , and the mean free path λk given by the linearized
Boltzmann equation
λk = τk [vk + ∑ Pk´k λk´ ] , (3)
as well as the anisotropic relaxation time τk , correspond to k states on the Fermi surface [14]. Here
Pkk´ denotes the microscopic transition probability for the scattering from an initial state k into a
final state k´. In a nonmagnetic crystal with a space inversion symmetry every k state is twofold
degenerate. It is possible to separate the two degenerate states by their spin polarization to ψ + and
ψ - states [12,13]. The skew scattering mechanism belongs to the non-vanishing difference between

28 Trends in Magnetism
the scattering probabilities of the two states, (P ++ kk´ - P -- kk´). Using time and space inversion
symmetry, one can write (P ++ kk´ - P -- kk´) = (P ++ kk´ - P ++ -k´-k) = (P ++ kk´ - P ++ k´k). Thus, as it was
mentioned already in Ref. [15], the anti-symmetric part of the scattering rate Pkk´ is the reason for
the skew scattering. The formalism described above and used in this work is valid for dilute alloys,
assuming the presence of non-interacting impurities in the host materials [14,16]. In the dilute limit
both conductivities, σxx and σ s yx , are inversely proportional to the number of impurities [14], while
the spin Hall angle is concentration independent by its definition.
Results
As an illustration of skew scattering, the difference between the microscopic transition probabilities
of ψ + and ψ - states in a Au host with Li and
Zn impurities is presented in Fig. 1. Here the
quantization axis is chosen to be in the z
direction. The distribution of (P ++ kk´ - P -- kk´) is
shown for the incoming state k propagating
along the x direction and final states k´
available at the Fermi surface. The clear
left/right asymmetry between the two states
with the opposite spin polarization is the
origin of the spatial spin separation, i.e. of the
Fig.1. The distribution of (P ++ kk´ - P -- kk´) with k in
[100] direction for states k´ on the Fermi surface of
Au (for Li and Zn impurities).
SHE. In addition, as it is demonstrated by
both Fig. 1 and Table 1, the SHE can have
opposite sign depending on the chemical
nature of impurity atoms. Moreover, the sign
as well as the magnitude of the SHE strongly
depend on the considered host material (see Table 1).
TABLE 1. The spin Hall angle α, the spin Hall conductivity σ s yx , and the longitudinal charge
conductivity σxx for different impurity atoms in Cu, Ag, and Au hosts. The conductivities are
shown at an impurity concentration of 1 at.%.
Cu Ag Au
α
10 -3
σ s yx
(10 3 µΩcm) -1
σxx
(µΩcm) -1
α
10 -3
σ s yx
(10 3 µΩcm) -1
σxx
(µΩcm) -1
α
10 -3
σ s yx
(10 3 µΩcm) -1
σxx
(µΩcm) -1
Li 2.3 2.8 1.22 8.66 11.0 1.27 7.2 4.3 0.60
Be 0.37 0.75 2.04 -1.01 -1.7 1.68 -13.4 -12.5 0.93
B 2.0 0.89 0.45 10.4 2.8 0.27 76.4 16.8 0.22
C 6.6 1.0 0.16 11.7 1.4 0.12 96.0 12.0 0.12
N 7.0 0.75 0.11 11.0 1.1 0.10 64.0 5.3 0.08
Mg -1.5 -2.3 1.57 -4.78 -10.0 2.09 -8.2 -5.5 0.67
Ni 5.0 5.92 1.19 2.45 2.7 1.10 2.46 4.79 1.95
Cu − − − 8.92 420 47.1 -0.44 -1.3 2.96
Ag 0.26 7.9 30.2 − − − 4.8 17.0 3.47
Pt 27.0 13.6 0.51 17.2 11.0 0.64 10.0 9.0 0.93
Au 7.8 18.5 2.37 -14.9 -43.0 2.89 − − −
Bi 81.0 18.1 0.22 95.3 14.3 0.15 14.0 1.9 0.13

Solid State Phenomena Vols. 168-169 29
FIG.2. The differences between the scattering
phase shifts of an impurity and the host atom
for the levels j=l±1/2 with l=1 for a) Cu, b)
Ag, and c) Au hosts.
Strong spin-orbit coupling at the scattering
centres is a well-known origin for pronounced
skew scattering [5,6]. In our case this condition is
fulfilled for the Cu(Bi) and Ag(Bi) alloys (Table
1). However, other relatively heavy impurities as
Au and Pt do not provide a large spin Hall angle
comparable to that for Bi impurities. In addition,
the effect is weak in Au(Bi) as well. Surprisingly,
light impurities like B, C, and N in the Au host
provide large α, though their own spin-orbit
interaction is small.
To understand these results, we exploit the fact
that the Fermi surfaces of the noble metals have a
simple topology [12]. It allows us to use a
spherical band approximation with an analysis of
the scattering phase shifts δj at the Fermi level EF
for the total angular momentum j=l±1/2. We
restrict our consideration to p electrons (l=1) only,
since their scattering dominates over the effect in
the noble metals. In Fig. 2 (a), (b), and (c) the
values of ∆δ1/2(EF) = δ1/2 imp (EF) - δ1/2 host (EF) and
∆δ3/2(EF) = δ3/2 imp (EF) - δ3/2 host (EF) are shown for
the Cu, Ag, and Au hosts, respectively. The
absolute values of such effective phase shifts
indicate the scattering strength of the
corresponding partial waves at the impurity site.
Strong scattering leads to a decrease of the
relaxation time. It is well known that for σxx the
first term on the r.h.s. of Eq. 3 is the dominating
one. On the other hand, it was shown in Ref. [14]
that this term does not contribute to σ s yx , and the
scattering-in term (the second one on the r.h.s of
Eq. 3) is mandatory. A general trend for ∆δ1/2(EF)
and ∆δ3/2(EF) as functions of the impurity atom is
almost host independent, since Cu, Ag, and Au are
isovalent elements. The difference |∆δ1/2(EF) -
∆δ3/2(EF)| shows the strength of the left/right
asymmetry of scattering for electrons with
opposite orientations of the spin polarization.
Thus, for a strong SHE it is necessary to have at
the same time large absolute values of ∆δ1/2(EF),
∆δ3/2(EF) and ∆δ1/2(EF) - ∆δ3/2(EF).
Let us consider first the heavy impurities.
According to Fig. 2, Bi is the only strong p
scatterer of them. In addition, the effective spin-
orbit splitting, |∆δ1/2(EF) - ∆δ3/2(EF)|, depends strongly on the host. For Au this quantity is reduced
in comparison to the Ag and Cu hosts because of a stronger spin-orbit coupling of the host. As a
consequence, σ s yx caused by the Bi impurities is largest in copper. The larger value of α in the Ag
host in comparison to Cu is due to the smaller longitudinal conductivity (see Table 1). The situation
with the light p scatterer-impurities (B, C, and N) is opposite. They do not have a strong spin-orbit
interaction by themselves. That is why the largest effective spin-orbit scattering (Fig. 2) and, as a

30 Trends in Magnetism
result, the strongest SHE (Table 1) are present in Au. This mechanism is similar to the strong spinflip
scattering at light impurities [13]. As it was discussed already in Ref. [14], the extremely large
α induced by the C impurities in the Au host leads to an alternative explanation of the gigantic SHE
measured in gold [17].
Summary
We have performed ab initio calculations of the extrinsic spin Hall effect caused by different
substitutional impurities in the Cu, Ag, and Au hosts. Depending on the chemical nature of the
impurity atom, the SHE can show opposite sign. A strong SHE is provided by the p scatterer-
impurity. They have to be either light, such as B, C, and N, in a heavy host like Au, or heavy, such
as Bi, in a light host like Cu. We have shown that the obtained results for the noble metals can be
explained by the consideration of the differences of the scattering phase shifts between the impurity
and host atoms.
References
[1] R. Karplus and J.M. Luttinger, Phys. Rev. Vol. 95 (1954), p. 1154.
[2] J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth, and A.H. MacDonald, Phys. Rev.
Lett. Vol. 92 (2004), p. 126603.
[3] L. Berger, Phys. Rev. B Vol. 2 (1970), p. 4559.
[4] L. Berger, Phys. Rev. B Vol. 5 (1972), p. 1862.
[5] N.F. Mott and H.S. Massey, The Theory of Atomic Collisions (Clarendon Press, Oxford, 1965).
[6] L.D. Landau and E.M. Lifshitz, Quantum Mechanics (Pergamon, New York, 1965).
[7] J. Smit, Physica (Amsterdam) Vol. 21 (1955), p. 877.
[8] J. Smit, Physica (Amsterdam) Vol. 24 (1958), p. 39.
[9] M.I. Dyakonov and V.I. Perel, Phys. Lett. A Vol. 35 (1971), p. 459.
[10] S. Takada, Prog. Theor. Phys. Vol. 36 (1966), p. 224.
[11] Y. Onodera and M. Okazaki, J. Phys. Soc. Jpn. Vol. 21 (1966), p. 1273.
[12] M. Gradhand, M. Czerner, D.V. Fedorov, P. Zahn, B.Yu. Yavorsky, L. Szunyogh, and
I. Mertig, Phys. Rev. B Vol. 80 (2009), p. 224413.
[13] M. Gradhand, D.V. Fedorov, P. Zahn, and I. Mertig, Phys. Rev. B Vol. 81 (2010), p.
020403(R).
[14] M. Gradhand, D.V. Fedorov, P. Zahn, and I. Mertig, Phys. Rev. Lett. Vol. 104 (2010), p.
186403.
[15] N.A. Sinitsyn, J. Phys. Condens. Matter Vol. 20 (2008), p. 023201.
[16] I. Mertig, Rep. Prog. Phys. Vol. 62 (1999), p. 237.
[17] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. Nitta, and K.
Takanashi, Nature Materials Vol. 7 (2008), p. 125.

Trends in Magnetism
doi:10.4028/www.scientific.net/SSP.168-169
Skew Scattering Mechanism by an Ab Initio Approach: Extrinsic Spin Hall Effect in
Noble Metals
doi:10.4028/www.scientific.net/SSP.168-169.27