Disorder and temperature dependence of the Anomalous Hall Effect in thin ferromagnetic films: Microscopic model

Disorder and temperature dependence of the Anomalous Hall Effect in thin ferromagnetic films: Microscopic model

K. A. Muttalib Department of Physics, University of Florida, P.O. Box 118440, Gainesville, FL 32611-8440    P. Wölfle ITKM, Universität Karlsruhe, D-76128 Karlsruhe, Germany, INT, Forschungzentrum Karlsruhe, Postfach 3640, 76021 Karlsruhe, Germany
Abstract

We consider the Anomalous Hall Effect (AHE) in thin disordered ferromagnetic films. Using a microscopic model of electrons in a random potential of identical impurities including spin-orbit coupling, we develop a general formulation for strong, finite range impurity scattering. Explicit calculations are done within a short range but strong impurity scattering to obtain AH conductivities for both the skew scattering and side jump mechanisms. We also evaluate quantum corrections due to interactions and weak localization effects. We show that for arbitrary strength of the impurity scattering, the electron-electron interaction correction to the AH conductivity vanishes exactly due to general symmetry reasons. On the other hand, we find that our explicit evaluation of the weak localization corrections within the strong, short range impurity scattering model can explain the experimentally observed logarithmic temperature dependences in disordered ferromagnetic Fe films.

pacs:
73.20.Fz, 72.15.Rn, 72.10.Fk

I Introduction

It has been recognized since the 1950’s KL () that a Hall effect can exist in ferromagnetic metals even in the absence of an external magnetic field, hence the name Anomalous Hall Effect (AHE). There are several different mechanisms that might be responsible for the AHE observed in thin ferromagnetic films, namely the skew scattering smit () and side jump mechanisms berger () as well as Berry phase contributions niu (). All such mechanisms depend on spin-orbit interaction induced by the impurities and on the spontaneous magnetization in a ferromagnet which breaks the time reversal invariance and therefore gives rise to the AHE. For a disordered ferromagnetic film, AH conductivity due to the skew scattering and side jump mechanisms have been theoretically considered using a variety of methods within weak, short range impurity scattering luttinger (); lewiner (); sinitsyn (); dugaev (); LW (). However, a systematic calculation, starting from a microscopic Hamiltonian, of the longitudinal as well as the AH conductivities for different mechanisms for strong impurity scattering has been lacking. Recently, the effect of strong, short range impurity scattering on the longitudinal and Hall conductivities were considered for skew scattering as well as side jump mechanisms WM (), but quantum corrections, namely electron-electron (e-e) interaction corrections altshuler () or weak localization (WL) effects lee (), were not included.

Earlier experiments BY () have shown logarithmic temperature dependences of the longitudinal as well as Hall resistances highlighting the importance of such quantum corrections. However, the results were consistent with, and were interpreted as, vanishing interaction contributions to the AH conductivity, obtained theoretically within a weak impurity scattering model LW () and the absence of any weak localization effects. Recent experiments on the other hand clearly show non-vanishing contribution to the total quantum correction to the AH conductivity mitra (), which can arise in principle either from an interaction correction due to strong impurity scattering, or from a weak localization effect, or from a combination of both. It has been commonly believed that weak localization effects in ferromagnetic films would be cut off by the presence of large internal magnetic field among others, which suggests that the interaction corrections to the AH conductivity need to be revisited for strong impurity scattering as a source of difference between the two experiments.

In this paper we systematically develop a general formulation for the AHE for strong, finite range impurity scatterings starting from a microscopic model of electrons in a random potential of impurities including spin-orbit coupling. This generalizes an earlier work lewiner () which considered weak, short range impurity scattering only and did not include quantum corrections. We show on very general symmetry grounds that quantum correction to the AH conductivity due to (e-e) interaction effects vanish exactly, which shows that the previous weak scattering results LW () remain valid for arbitrary strengths of the impurity scattering. This forces us to consider the weak localization effects dugaev () as the only remaining source of the logarithmic temperature dependence in the above experiments despite the presence of large internal magnetic fields and spin-orbit scatterings in these ferromagnetic films. As we show below, the temperature independent cutoff of the weak localization effects in strongly disordered systems can be ineffective at higher temperatures if a temperature dependent contribution dominates the phase relaxation rate. It turns out that while the contribution from the e-e interaction to the phase relaxation rate is indeed too small for WL effects to be observed, a much larger contribution is obtained from scattering off spin waves tatara (), which should allow the observation of the WL effects within a reasonable temperature range. We find that the effects of strong impurity scatterings on the WL effects can be evaluated to obtain a very simple result, namely that the ratio of the WL corrections to the AH to the longitudinal conductivity can be written simply in terms of the eigenvalues of the impurity averaged particle-hole scattering amplitude for zero momentum transfer. This result, taken together with contributions to the AH conductivity from both the skew scattering and side jump mechanisms calculated within the same microscopic model, can explain both the earlier as well as the recent experiments on the disorder and temperature dependences of the AH conductivities of ultrathin Fe films mitra () mentioned above. This last result has been reported without details in combination with the recent experiment in a short letter mitra ().

The paper is organized in the following way: A microscopic model Hamiltonian is introduced in Section II, and a general formulation in two dimensions for strong, finite range impurity scatterings is developed in Section III. Section IV reviews the results on the conductivity tensor in the absence of interactions. In Sections V and VI we consider the e-e interaction corrections and the weak localization corrections, respectively, to both longitudinal and AH conductivities within the general strong, finite range impurity scattering formulation. We then consider the special case of a short range, but still strong, impurity scattering model in Section VII. In Section VIII we collect all the results and compare them with recent experiments. Section IX summarizes the paper. For the sake of completeness, we include models of small and large angle scatterings in the Appendix.

Ii Hamiltonian

The single particle Hamiltonian of a conduction electron in a ferromagnetic disordered metal, including spin-orbit interaction induced by the disorder potential , is given in its simplest form by (throughout the paper we use units with )

 H1 = [−∇22m+Vdis(r)]δσσ′−Mτzσσ′ (1) − iλ2c(4π)2[τσσ′⋅(∇Vdis×∇)], (2)

where is the Compton wavelength of the electron, and is the Zeeman energy splitting caused by the ferromagnetic polarization. Here is a matrix in spin space with being spin indices and is the vector of Pauli matrices. The above model is only a crude approximation of the bandstructure of Fe, which has been determined by several authors (see e.g. ref singh ()). We model the energy band crossing the Fermi surface by a single isotropic band. As will be discussed below, the quantum corrections to the conductivity exhibit certain qualitative features, which do not depend sensitively on the details of the band structure. The disordered potential in (2.1) will be modelled as randomly placed identical impurities, . We will later average over the impurity positions .

The matrix elements of in the plane wave (or Bloch state) representation are given by

 ⟨k′σ′|H1|kσ⟩=∫d2re−ik′⋅rH1e−ik⋅r (3) = (4) + Vso(k′σ′;kσ) (5)

where is the Fourier transform of the single impurity potential, and the spin-orbit interaction part is given by

 Vso(k′σ′;kσ) = −iλ2c(4π)2∑jV(k−k′)e[i(k−k′)⋅Rj] (6) × τσσ′⋅(k×k′) (7)

Here we have used

 −i∫d2rexp(−ik′⋅r)(∇Vdis×∇)exp(−ik⋅r) (8) = −i∫d2r∫d2q(2π)2ei(k−k′−q)⋅r(−iq)V(q)×(ik) (10) = −iV(k−k′)(k×k′) (11)

The many-body Hamiltonian is given in terms of electron creation and annihilation operators as

 H = ∑kσ(εk−Mσ)c+kσckσ (12) + ∑kσ,k′σ′∑jV(k−k′)ei(k−k′)⋅Rj (13) × {δσσ′−i¯gsoτσσ′⋅(ˆk×ˆk′)}c+k′σ′ckσ (14)

where we have defined a dimensionless spin-orbit coupling constant , . Note: An estimate of the spin-orbit coupling constant , using a typical Fermi wave number , shows that it is rather small, of order . However, in transition metal compounds the coupling is substantially enhanced by interband mixing effects berger (), so that the renormalized coupling constant is of order unity: , where  is a measure for the atomic spin-orbit energy,    is a typical energy splitting of d-bands, and the constant . In the following we will replace by the phenomenological spin-dependent parameter .

Iii Impurity scattering: General Formulation

In this section, we will develop a general formulation for strong, finite range impurity scattering in two dimensions using standard field theory techniques at finite temperature AGD (). For simplicity, we will need to make approximations for short range impurity scattering later. However, keeping the formulation general as long as possible will allow us e.g. to check if the anisotropic scattering can have a large impact on our final results.

The repeated scattering of an electron off a single impurity may be described symbolically in terms of the scattering amplitude as

 f=V+VGV+VGVGV+.... (15)

where is the single particle Green’s function

 Gkσ(iωn)=[iωn−εkσ−Σkσ(ωn)]−1, (16)

with the single particle self energy . Here is the fermion Matsubara frequency with being the temperature and is the density of states at the Fermi level of spin species . (We use units of temperature such that Boltzmann’s constant is equal to unity). V is the bare interaction with one impurity at and includes the spin-orbit scattering

 Vk,k′;σ=V(k−k′)[1−igστzσσ(ˆk×ˆk′)], (17)

where we have used the fact that is diagonal in spin space. In the case of finite range, or even long-range correlated scattering potentials, we may still use the model of individual impurities or scattering centers, but now of finite spatial extension. This is reasonable as long as the scattering centers do not overlap too much. If they overlap, a more statistical description in terms of correlators of the impurity potential should be used. Within our model, the nonlocal character of scattering is described in terms of the momentum dependence of the Fourier Transform of the potential of a single impurity (assuming only one type of impurity) , which for an isotropic system depends only on the angle between and ,   In 2d we may expand in terms of eigenfunctions , where is the polar angle of vector , . Adding the skew scattering potential we may write

 Vk,k′σ=∑mVmσχm(ˆk)χ∗m(ˆk′) (18)

where is a sum of the normal and skew scattering parts

 Vmσ=Vnsm+Vssmσ. (19)

Time reversal invariance and rotation symmetry in the case of potential scattering implies

 Vns−m=(Vnsm)∗=Vnsm. (20)

Equation (3.3) then yields

 Vssmσ=12gστzσσ(Vnsm−1−Vnsm+1) (21)

iii.1 Scattering amplitude

For diagonal in spin space, the scattering amplitude   obeys the integral equation

 fsk,k′σ = Vk,k′σ+∑k1Gk1σ(iωn)Vk,k1σfsk1,k′σ (22) = Vk,k′σ−isπNσ⟨Vk,k1σfsk1,k′σ⟩k1, (23)

where and denotes averaging over the direction of wavevector . Defining dimensionless potential and the dimensionless scattering amplitude and expanding , we find

 ¯fsmσ=¯Vmσ1+is¯Vmσ. (24)

For notational simplicity, we will always use a bar on a symbol to represent the corresponding dimensionless quantity.

iii.2 Single particle relaxation rate

The single particle relaxation rate is given by the imaginary part of the self energy

 12τσ ≡ −sImΣkσ(iωn) (25) = −snimpIm(fskσ,kσ)=nimpπNσγσ (26)

where is a dimensionless parameter characterizing the scattering strength, , and is the density of states at the Fermi energy of spin species . Note that are all real.

iii.3 Particle-hole propagator

The particle-hole propagator  is an important ingredient of vertex corrections of any kind. Here  are the initial , the final momenta and    are the Matsubara frequencies of the particle and the hole line, respectively. In terms of the particle-hole scattering amplitude , satisfies the following Bethe-Salpeter equation (we have defined dimensionless quantities by multiplying both with a factor )

 ¯Γkk′(q;iϵn,iΩm)=¯tkk′(q;iϵn,iΩm) (28) + (2πNστσ)−1∑k1¯tkk1(q;iϵn,iΩm)Gk1+q/2,σ(iϵn) (29) × Gk1−q/2,σ(iϵn−iΩm)¯Γk1k′(q;iϵn,iΩm) (30)

The (dimensionless) impurity averaged particle-hole scattering amplitude (we consider only the case of equal spin of particle and hole) is given in terms of the (dimensionless) scattering amplitudes by the equation

 ¯tss′kk′(q;iϵn,iΩm)=2τσnimpπNσ¯fsk+q/2,σ;k′+q/2σ(iϵn) (31) × ¯fs′k′−q/2,σ;k−q/2,σ(iϵn−iΩm). (32)

We will later need the limit of small of this expression,

 ¯tss′kk′(q;iϵn,iΩm)=¯tss′kk′(q=0)+Δ ¯tss′kk′(q). (33)

It is useful to represent the operator  in terms of its eigenvalues . Assuming isotropic band structure, the eigenfunctions are those of the angular momentum operator component . The eigenvalue equation is

 ⟨¯tkk′(q=0)χm(ˆk′)⟩k′=λmχm(ˆk). (34)

The operator may be represented as

 ¯t+−kk′(q=0) = ∑mλmχm(ˆk)χ∗m(ˆk′) (35) ¯t−+kk′(q=0) = [ ¯t+−k′k(q=0)]∗ (36)

In general, using the definitions

 tss′k,k′σ = nimp(πNσ)2¯fsk,k′σ¯fs′k′,kσ=(2πNστσ)−1¯tss′k,k′σ (37) ¯tss′k,k′σ = ∑m¯tss′mσχm(ˆk)χ∗m(ˆk′) (38)

we have

 ¯tss′mσ=γ−1σ∑m′¯fsm′σ¯fs′m′−m,σ. (39)

We will consider for the special case of strong short range impurity scatterings in section VII-A.

The energy integral over the product of Green’s functions in the integral equation for may be done first, after expanding the G’s in   and ,

 ∫ dε1 Gk1+q/2,σ(iϵn)Gk1−q/2,σ(iϵn−iΩm) (40) = 2πτ[1+iτ(iΩm−q⋅vk1) (41) − τ2(q⋅vk1)2] (42)

with and , where . Expanding and in terms of eigenfunctions ,   and using one obtains ()

 ¯Γss′mm′ = λmδmm′+λm{[1−τ(|Ωn|+D0q2)]¯Γss′mm′ (43) − i2vFqτs[¯Γss′m−1,m′χ∗1(ˆq)+¯Γss′m+1,m′χ1(ˆq)] (44) − 14(vFqτ)2[¯Γss′m−2,m′χ∗2(ˆq) (45) + ¯Γss′m+2,m′χ2(ˆq)]} (46)

For    the solution is

 ¯Γmm=λm1−λm+O(q)≡˜λm+O(q), (47)

where we have defined . The (and therefore ) are complex valued and depend on the spin projection . Using conventional notation, we will denote the real and imaginary parts of by and , respectively, and similarly the real and imaginary parts of by and , respectively.

The case needs special consideration, because particle number conservation causes to have a pole in the limit , here expressed by . Solving the above equation for  in lowest order in q, one finds

 ¯Γ00=1/τ|Ωm|+Dq2, (48)

where the renormalized diffusion constant is defined as

 D = D0(1+ ˜λ′1),D0=12v2Fτ (49) ˜λ′1 ≡ Re˜λ1=12(˜λ1+˜λ−1); (50)

This is found by solving the following equations for small

 ¯Γss′00 = 1+[1−τ(|Ωm|+D0q2)]¯Γss′00 (51) − i2vFqτs[˜Γss′−1,0χ∗1(ˆq)+¯Γss′1,0χ1(ˆq)] (52) ¯Γss′−1,0 = λ−1{¯Γss′−1,0−i2vFqτs¯Γss′0,0χ1(ˆq)} (53) ¯Γss′1,0 = λ1{¯Γss′1,0−i2vFqτs¯Γss′0,0χ−1(ˆq)} (54)

Substituting into the equation for one finds:

 ¯Γss′00{|Ωm|+D0q2[1+12(˜λ1+˜λ−1)]}=1τ (55)

The leading singular dependence on is obtained from:

 ¯Γss′0,±1 = [1−τ(|Ωm|+D0q2)]¯Γss′0,±1 (56) − i2vFqτs[¯Γss′−1,±1χ∗1(ˆq)+¯Γss′1,±1χ1(ˆq)] (57) ¯Γss′−1,1 = −i2˜λ−1vFqτs¯Γss′0,1χ1(ˆq) (58) ¯Γss′1,1 = ˜λ1−i2˜λ1vFqτs¯Γss′0,1χ∗1(ˆq) (59)

The complete particle-hole propagator in the regime is given by

 ¯Γkk′=1τγk˜γk′|Ωm|+Dq2+ ∑m≠0˜λmχm(ˆk)χ∗m(ˆk′) (60)

with

 γk = 1−i2vFqτs∑m=±1˜λmχm(ˆk)χm∗(ˆq) (61) = 1−i2vFτs∑m=±1˜λmχm(ˆk)q−m (62)

and

 ˜γk=1−i2vFτs∑m=±1˜λmχ∗m(ˆk)qm. (63)

The vertex corrections of the density and current vertices and (for the incoming and outgoing current) are obtained by

 Tk(q)≡1+⟨¯Γkk′⟩k′=1+1/τ|Ωm|+Dq2γk (64)

and

 jkα(q) = vkα+⟨vk′α¯Γk′k⟩k′ (66) = vkα+∑m=±1˜λmχ∗m(ˆk)⟨vk′αχm(ˆk′)⟩k′ (67) + ⟨vk′αγk′⟩k′1/τ|Ωm|+Dq2˜γk (68) ˜jkα(q) = vkα+⟨vk′α¯Γkk′⟩k′ (70) = vkα+∑m=±1˜λmχm(ˆk)⟨vk′αχ∗m(ˆk′)⟩k′ (71) + ⟨vk′α˜γk⟩k′1/τ|Ωm|+Dq2γk. (72)

Note that  , as the eigenvalues are in general complex valued. Using

 χ−1(ˆk)χ∗1(ˆk′)+χ1(ˆk)χ∗−1(ˆk′) = 2(ˆk⋅ˆk′) (73) χ−1(ˆk)χ∗1(ˆk′)−χ1(ˆk)χ∗−1(ˆk′) = 2i(ˆk×ˆk′) (74)

and

 ⟨ˆk′α(ˆk⋅ˆk′)⟩=12ˆkα;⟨ˆk′α(ˆk×ˆk′)⟩=−12(ˆeα×ˆk) (75)

and defining , , we have

 jkα = vF[(1+ ˜λ′1)ˆkα+ ˜λ′′1(ˆeα×ˆk)z] (76) ˜jkα = vF[(1+ ˜λ′1)ˆkα− ˜λ′′1(ˆeα×ˆk)z]. (77)

More explicitly, for , the incoming and outgoing current vertices and have the forms

 jkx = vF[(1+ ˜λ′1)ˆkx+ ˜λ′′1ˆky] (78) = 12vF[(1+ ˜λ∗1)ˆk++(1+ ˜λ1)ˆk−] (79) jky = vF[(1+ ˜λ′1)ˆky− ˜λ′′1ˆkx] (80) = −i12vF[(1+ ˜λ∗1)ˆk+−(1+ ˜λ1)ˆk−] (81) ˜jkx = vF[(1+˜λ′1)ˆkx− ˜λ′′1ˆky] (82) = 12vF[(1+ ˜λ1)ˆk++(1+˜λ∗1)ˆk−] (83) ˜jky = vF[(1+ ˜λ′1)ˆky+ ˜λ′′1ˆkx] (84) = −i12vF[(1+ ˜λ1)ˆk+−(1+ ˜λ∗1)ˆk−] (85)

where we have defined .

iii.4 Particle-particle propagator

The integral equation for the particle-particle propagator or Cooperon reads (again multiplying the Cooperon and the particle-particle scattering amplitude by the factor to define dimensionless Cooperon and dimensionless particle-particle scattering amplitude )

 ¯Ckk′(Q;iϵn,iΩm)=¯tpkk′(Q;iϵn,iΩm) (87) + (2πNστσ)−1∑k1¯tpkk1(Q;iϵn,iΩm)Gk1,σ(iϵn) (88) × GQ−k1,σ(iϵn−iΩm)¯Ck1k′(q;iϵn,iΩm) (89)
 tp,ss′k,k′σ = nimp(πNσ)2¯fsk,k′σ¯fs′−k,−k′,σ (90) = (2πNστσ)−1γ−1σ¯fsk,k′σ¯fs′−k,−k′,σ (91)
 ¯tp,ss′k,k′σ=2πNστσtp,ss′k,k′σ=∑m¯tp,ss′mσχm(ˆk)χ∗m(ˆk′) (92)
 ¯tp,ss′mσ=γ−1σ∑m′¯fsm′σ¯fs′m−m′,σ (93)

If rotation invariance or time reversal invariance is broken,  , where .

The energy integral over the product of Green’s functions in the integral equation for may be done first, after expanding the ’s in and ,

 ∫ dε1 Gk1,σ(iϵn)GQ−k1,σ(iϵn−iΩm) (94) = 2πτ[1+iτ(iΩm−Q⋅vk1) (95) − τ2(Q⋅vk1)2], (96)

with , , where  . Expanding and in terms of eigenfunctions , and denoting one obtains ()

 ¯Cmm′=λpm{δmm′+[1−τ(|Ωn|+D0Q