PHYSICAL REVIEW B 79, 195129 共2009兲
Transport theory for disordered multiple-band systems: Anomalous Hall effect and anisotropic magnetoresistance Alexey A. Kovalev,1,2 Yaroslav Tserkovnyak,1 Karel Výborný,3 and Jairo Sinova2,3 1
Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA 2 Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA 3Institute of Physics, ASCR, Cukrovarnická 10, 162 53 Praha 6, Czech Republic 共Received 16 February 2009; revised manuscript received 26 April 2009; published 28 May 2009兲
We present a study of transport in multiple-band noninteracting Fermi metallic systems based on the Keldysh formalism and the self-consistent T-matrix approximation 共TMA兲 taking into account the effects of Berry curvature due to spin-orbit coupling. We apply this formalism to a Rashba two-dimensional electron-gas ferromagnet and calculate the anomalous Hall effect 共AHE兲 and anisotropic magnetoresistance 共AMR兲. The numerical calculations of the AHE reproduce analytical results in the metallic regime revealing the crossover between the skew-scattering mechanism dominating in the clean systems and intrinsic mechanism dominating in the moderately dirty systems. As we increase the disorder further, the AHE starts to diminish due to the spectral broadening of the quasiparticles. Although for certain parameters this reduction of the AHE can be approximated as xy ⬃ xx, with varying around 1.6, this is found not to be true in general as xy can go through a change in sign as a function of disorder strength in some cases. Furthermore, the disordered region consistent with the TMA is relatively narrow and a theory with a wider range of applicability in strong disorder limit is called for. By considering the higher order skew-scattering processes, we resolve some discrepancies between the AHE results obtained by using the Keldysh, Kubo, and Boltzmann approaches. We also show that similar higher order processes are important for the AMR when the nonvertex and vertex parts cancel each other. We calculate the AMR in anisotropic systems properly taking into account the anisotropy of the nonequilibrium distribution function. These calculations confirm recent findings on the unreliability of common approximations to the Boltzmann equation. DOI: 10.1103/PhysRevB.79.195129
PACS number共s兲: 72.15.Eb, 72.20.Dp, 72.20.My, 72.25.⫺b
I. INTRODUCTION
Recently, the interest in transport calculations in multipleband systems1,2 has been rekindled in part due to the realization of diluted magnetic semiconductors 共DMSs兲 that have strong spin-orbit interactions, variable carrier densities, and ferromagnetic ordering. These properties imply the existence of the anomalous Hall effect 共AHE兲 共Ref. 3兲 and the anisotropic magnetoresistance 共AMR兲.4 Even though the mechanisms of the AHE and the AMR are different, they both have a similar description based on the multiple-band transport theory. In this paper, we formulate a relatively simple framework for doing such transport calculations. The AHE is usually described in terms of the anomalous Hall resistivity xy that measures the transverse voltage with respect to the transport direction and depends on the spontaneous magnetization M along the z direction. Theoretical studies of the AHE have a long history beginning with the work of Karplus and Luttinger.5 A number of papers on the AHE also appeared not so long ago6–12 after the interpretation of the AHE based on the Berry phase13 was proposed. Nevertheless, theoretical description of the AHE is far from being complete and it often involves cumbersome calculations without transparent interpretations.14 The difficulties appear due to the necessity to consider the off-diagonal elements in Bloch band indices 共the interband coherences induced by charge currents兲. There is a general trend to focus on particular simple models in order to overcome the common mistakes that are made in treating the AHE. A number of recent publications concentrate on the simpler but non1098-0121/2009/79共19兲/195129共19兲
trivial Rashba two-dimensional 共2D兲 electron system,1,15–23 yet arriving at contradictory predictions. Most of the disagreements have been finally resolved22–24 with some being addressed in this paper. In calculating the AHE for a given material, the usual approximations performed to leading order in ប / F can fail, where is the scattering time and F is the Fermi energy. The semiclassical description of the Hall conductivity within the usual Boltzmann equation leads to an AHE contribution due to the scattering asymmetry in the collision term usually labeled as skew scattering.25 Other terms, arising from subtle issues dealing with interband coherence during the collision and acceleration by the electric field between collisions, are usually introduced by hand through the so-called anomalous velocity26 and side jump.27 This approach however, is nonsystematic and prone to errors from missing terms and wrong interpretations, e.g., giving physical meaning to gauge dependent quantities. A more systematic way to derive the correct semiclassical equations is through the Keldysh formalism in which these interband coherence effects are taken into account automatically.1,24 The system under consideration also allows us to study the diagonal resistance as a function of the direction of the magnetization. The change in the resistance as a function of the magnetization direction relative to the current or crystallographic direction is called the AMR effect. The microscopic origin of the AMR in transition-metal ferromagnets is still elusive28–31 and detailed calculations require consideration of complicated band structures.32,33 A relatively simple host band structure in the DMS ferromagnets provides a pos-
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sibility for performing detailed microscopic calculations based on simple physical models.34 However, the relaxationtime approximation used in such calculations is not always reliable since it does not fully take into account the anisotropies of the system.35 The Kubo formula approach has been applied to the AMR calculations in Rashba systems and it has revealed the cancellation of the nonvertex and vertex parts,36 similar to the spin Hall effect 共SHE兲 and the AHE. In this paper, we apply the Keldysh formalism for transport calculations in multiple-band noninteracting Fermi systems. This treatment simultaneously takes into account the Berry curvature effects 共interband coherences兲 and scattering, allowing us to immediately account for such physical effects as side-jump scattering and skew scattering within the same footing. We calculate the AHE analytically and numerically for the Rashba model and find, in agreement with Onoda et al.,1,20 three distinct regimes: the skew-scattering regime, the disorder independent regime, and the dirty regime in which, although the basis of theory is not as well established, a distinct rapid reduction of the AHE is observed as the conductivity xx diminishes. Even though almost all ferromagnetic systems are three dimensional 共3D兲, the findings of this simple 2D model has been linked to higher dimensional systems arguing that most likely the major contributions to the AHE come from the band anticrossing regions1 similar to one observed in the Rashba model. We further analyze the scaling found in the dirty regime1,20 in which the AHE seems to diminish in a manner that can be approxi , with being close to 1.6. Some experimated as xy ⬃ xx mental results claim to confirm such scaling;37–41 however, treatment of some of these experimental results has to be done with extra care as the region of interest is often restricted to less than a single decade, the materials have strong mangetoresistances and in-plane anisotropies associated with them, and most of the data associated with the zero-field calculation are in fact at very high magnetic fields. Although at first sight our numerical results may seem to confirm this scaling, the closer analysis reveals that the selfconsistent T-matrix approximation 共TMA兲, which is the cornerstone of the formalism, fails when F ⬃ 1 leaving us with insufficiently wide range of applicability of our theory 共and others based on the TMA兲 for scaling claims. In addition, for the repulsive impurity potentials, the crossover from the disorder independent 共intrinsic兲 regime to the skew-scattering regime is always accompanied by the sign change of the AHE which can shrink the AHE reduction region even further. Although this simple model seems to capture qualitative aspects of the three regions, to make a quantitative link to 3D materials with much more complex behavior seems premature at this stage. In our calculations, we also identify the hybrid skew-scattering regime of the AHE 共Ref. 24兲 resulting from the higher order scattering processes. We take such processes into account in our AMR calculations and conclude their importance for the Rashba model in which nonvertex and vertex diagrammatic parts can cancel each other.36 Our results suggest that the relaxation-time approximation is not always reliable for the AMR calculations as it has been shown recently within the Boltzmann equation treatment.35 The paper is organized as follows. In Sec. II, we develop a general formulation of transport in multiple-band noninter-
acting Fermi systems with further generalizations in Appendix A. In Sec. III, we calculate the AHE in two-dimensional electron-gas 共2DEG兲 ferromagnet with spin-orbit interaction. The analytical and numerical results are followed by discussions and comparison to other works. In Sec. IV, we calculate the AMR in 2DEG ferromagnet with spin-orbit interaction. Finally in Sec. V, we present our conclusions. II. TRANSPORT IN MULTIPLE-BAND SYSTEMS
The method presented in this section can be applied to a ˆ + Vˆ共r兲 multiple-band system described by a Hamiltonian H 0 that is a matrix in the band 共chiral兲 index. In this section, we first derive general nonlinear equations using nonequilibrium diagrammatic technique, further restricting our consideration to a linear-response theory. A. Quantum kinetic equation
We start by defining the following Green’s functions:42 ˆ ⬅ − i具T ⌿共1 兲⌿†共1⬘ 兲典 = − i具Tជ ⌿共1 兲⌿†共1⬘ 兲典, G 11 c + + + + ˆ ⬅ − i具T ⌿共1 兲⌿†共1⬘ 兲典 = − i具⌿共1 兲⌿†共1⬘ 兲典, G 21 c − − + + ˆ ⬅ − i具T ⌿共1 兲⌿†共1⬘ 兲典 = i具⌿†共1⬘ 兲⌿共1 兲典, G 12 c + + − − ˆ ⬅ − i具T ⌿共1 兲⌿†共1⬘ 兲典 = − i具Tឈ ⌿共1 兲⌿†共1⬘ 兲典, G 22 c − − − −
共1兲
where Tc is the generalized time-ordering operator acting on the Keldysh contour which can be split in two time axes t+ 共forward兲 and t− 共backward兲, ⌿ is the vector in the band 共chiral兲 space corresponding to the Fermi field, and 1⫾ = 共r , t⫾兲 is the variable that describes the spatial variable r and the time variable t. The generalized time-ordering operator performs an ordinary time ordering Tជ for the time t+, an antitime ordering Tឈ for the time t−, and in the mixed case t− occurs always after t+ within the Keldysh time contour. We can now define the Green’s function in the Keldysh space
冉
ˆ ˜ = G11 G ˆ G 21
ˆ G 12 ˆ G 22
冊
.
共2兲
The scattering potential due to impurities in the Keldysh space has the form
冉
冊
ˆ 0 ˜V共1,1⬘兲 = V共r兲 ␦共1 − 1⬘兲, 0 − Vˆ共r兲
共3兲
where Vˆ共r兲 describes the potential in the band 共chiral兲 space formed by many scatterers which for current consideration can have any general matrix form. The negative sign arises here simply because the lower branch integration is taken from +⬁ to −⬁ while in the Keldysh loop the time goes from −⬁ to +⬁. The Green’s function in Eq. 共2兲 allows for a perturbation expansion relying on the Feynman rules. However, the four matrix elements of a so defined Green’s function are
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ˆ +G ˆ . Hence it is adˆ +G ˆ =G linearly dependent, i.e., G 21 11 22 12 vantageous to perform a linear transformation in the Keldysh space to eliminate one matrix element in Eq. 共2兲 ˇ = G
冉 冊冉 1
0
1 −1
ˆ ˆ G 11 G12 ˆ ˆ G 21 G22
冊冉 冊 冉 冊 1
0
−1 1
=
ˆ⬍ ˆR G G ˆA G
0
冉 冊冉 冊 冉 冊
1 0 ˆ 1 0 ˜ 1 0 V = V共r兲␦共1 − 1⬘兲, Vˇ = 0 1 1 1 1 −1 ˆA ˆ −G ˆ is the retarded Green’s function, G ˆ R=G where G 11 12 ⬍ ˆ ˆ ˆ ˆ = G12 − G22 is the advanced Green’s function, and G = G12. There are other choices for the linear transformation and our ˆ⬍ choice is dictated by the fact that the Green’s function G can be immediately related to the distribution function in the Boltzmann equation.43 As of now, it is assumed that Vˆ共r兲 describes some disordered potentials and all Green’s functions are averaged over this disorder. In the transformed Keldysh space, the Dyson equation42 becomes ˆ −1 − ⌺ ˆR G 0
ˆ⬍ −⌺
0
ˆA ˆ −1 − ⌺ G 0
冊冉 冊 丢
ˆR G ˆ⬍ G 0
ˆA G
= 1ˇ ,
共4兲
where R, A, and ⬍, respectively, stand for the retarded, advanced, and lesser components of the disorder averaged Green’s functions and self-energies. The symbol 丢 denotes a convolution 共in position, time, and band/spin兲. The diagonal components of Eq. 共4兲 yield the two equations for the retarded and advanced Green’s functions ˆ R,A兲 丢 G ˆ R,A = 1ˆ . ˆ −1 − ⌺ 共G 0
共5兲
The off-diagonal component of Eq. 共4兲 yields the kinetic equation 共sometimes called quantum Boltzmann equation兲 which contains the nonequilibrium information necessary to study transport ˆ⬍−⌺ ˆ⬍ 丢 G ˆ A = 0. ˆ R兴−1 丢 G 关G
ni
ni
+ ∨
+ ∨
∨
ni +
+ …..
∨
U U GU
lifted for the short-range disorder as it is shown in Appendix A兲. In this case, we can sum up the infinite series of diagrams in Fig. 1 arriving at the following expression for the selfenergy in the momentum representation 共for the sake of compact form we use the momentum representation here兲: ˇ 兩k⬘典 = n 具k兩Tˇ兩k典␦共k − k⬘兲, 具k兩⌺ i
共7兲
i
where ri describes the positions of random impurities of density ni and ˆ is some matrix in the band index 共e.g., in Sec. III, it is a unit matrix corresponding to scalar impurities and in Sec. IV, it is a combination of unit and Hermite matrices corresponding to charged and magnetic impurities兲. A common approximation to this problem is the self-consistent TMA which takes into account all the noncrossing scattering events from single impurities 共see Fig. 1兲. We assume here ˇ depends on the difference that the system is uniform and G of spatial variables 共r − r⬘兲 共however, this requirement can be
共8兲
with the following expression for the T-matrix operator of impurity placed in the origin: ˇ 丢 Vˇ + ¯兲, Tˇ ⬅ 共Vˇ + Vˇ 丢 G
共9兲
ˆ where Vˇ = 共 0 0ˆ 兲U共r兲␦共1 − 1⬘兲. Combining the T-matrix strucˇ 丢 Tˇ兴 and solving for the off-diagonal comture Tˇ = Vˇ 丢 关1ˇ + G ponent we obtain the equation for the lesser component of self-energy
ˆ ⬍ 丢 TˆA兩k典␦共k − k⬘兲. ˆ ⬍兩k⬘典 = n 具k兩TˆR 丢 G 具k兩⌺ i
共10兲
The retarded and advanced T matrices are given by the usual form ˆ R,A 丢 TˆR,A兲 = 共1 + TˆR,A 丢 G ˆ R,A兲 丢 Vˆ . TˆR,A = Vˆ 丢 共1 + G 共11兲 Equations 共6兲 and 共10兲 form a general closed set of equations ˆ ⬍. In order to solve these equations, we can further for G simplify them by looking for a solution of the form ˆ⬍+G ˆ ⬍, ˆ⬍=G G 2 1
共12兲
ˆA−G ˆR丢 n ˆ⬍=n 丢 G G F F 2
共13兲
where
共6兲
In order to solve Eq. 共6兲, one has to calculate the selfˆ ⬍ of the particular problem. Here we focus on scatenergy ⌺ tering by randomly distributed identical impurities at zero temperature with Vˆ共r兲 = 兺 ˆ U共r − ri兲,
ni
FIG. 1. 共Color online兲 The nonequilibrium self-energy calculated using the self-consistent T-matrix approximation in Keldysh space.
,
which leads to the following scattering potential:
冉
V
Σ =
and the operator nF is the Fermi distribution function. In the case of zero temperature, nF is the step function in the frequency representation nF共兲 = 共−兲 and nF共t , t⬘兲 = i / 关2共t − t⬘ + i0兲兴 in the time representation. Equations 共12兲 and 共13兲 will allow us to separate the Fermi sea and Fermi-surface components of the lesser Green’s function. By substituting ˆ ⬍, Eq. 共12兲 into Eq. 共6兲, we obtain the kinetic equation for G 1 ˆ⬍−⌺ ˆ⬍ 丢 G ˆ A = 关H ˆ 丢, n 兴 丢 G ˆ A, ˆ R兴−1 丢 G 关G 0 F 1 1
共14兲
ˆR ˆ ⬍ ˆA where with 具k兩⌺ˆ ⬍ 1 兩k⬘典 = ni具k兩T 丢 G1 丢 T 兩k典␦共k − k⬘兲, 丢 关. . . , . . .兴 stands for a commutator. In order to derive Eq. 共14兲, ˆ⬍ Eqs. 共5兲 and 共11兲 are used along with the fact that TˆR 丢 G 2 ˆ A = n 丢 TˆA − TˆR 丢 n and 丢T F F ˆR 丢 n , ˆ ⬍ = n 丢 ⌺ˆ A − ⌺ ⌺ F F 2 which is a consequence of Eqs. 共10兲, 共11兲, and 共13兲.
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The lesser Green’s function contains all the information about the transport properties of our system and the charge current density can be calculated as
共17兲
ˆ ⬍ in Eq. Applying the above Wigner transformation to G 2 共13兲, we obtain directly
e jx共y,z兲 = Tr具⌿†共1⬘兲ˆ x共y,z兲共1兲⌿共1兲典1=1⬘ + c.c. 2 =−
i Aˆ 丢 Bˆ ⬇ AˆBˆ − eE共kAˆBˆ − AˆkBˆ兲. 2ប
ˆ R兲 + i n eE共 G ˆA−G ˆA ˆR ˆ ⬍ = n 共G G F F k eq + kGeq兲, 共18兲 2 2ប
ie † ˆ ⬍共1,1⬘兲其 共1⬘兲兴G Tr兵关ˆ x共y,z兲共1兲 + ˆ x共y,z兲 1=1⬘ , 2 共16兲
ˆ 共1兲 / c兲 / m, where ˆ 共1兲 = 共−iប1 − eA ˆ †共1⬘兲 = 共iប1⬘ ˆ †共1⬘兲 / c兲 / m, and A ˆ 共1兲 is the generalized vector potential − eA matrix in the band index that also describes spin-orbit interactions; e = −兩e兩 stands for an electron charge. B. Linearized Fermi-surface contribution
The kinetic Eq. 共14兲 has not assumed linearity in electric field strength nor any particular temporal dependence. Higher order terms in the impurity density ni corresponding to noncrossed diagrams have been taken into account as the retarded and advanced Green’s functions in Eq. 共14兲 are calculated self-consistently. In the following, we solve the problem for linear-response theory of a uniform and stationary system in the presence of a uniform electric field. In the presence of slowly varying perturbations, it is useful to perform the Wigner transformation, viz., the center-ofmass coordinates 关X = 共R , T兲兴 and the Fourier transform with respect to the relative coordinates 关k = 共k , 兲兴. However, the Wigner coordinate k associated with the momentum operator −i is not gauge invariant and consequently it is not the correct choice for describing our system. On the other hand, the kinetic momentum k共T兲 = −i −eAE共T兲 / 共បc兲 is gauge invariant and, as it will be shown below, for the stationary case all time dependence can be conceived in k共T兲; here the vector potential AE共T兲 describes the external electric field. The time derivative within the canonical coordinates 共marked by wave兲 becomes a combination of time and momentum derivatives within the kinetic coordinates ˜T = T + Tk共T兲k, R˜ = R, ˜k = k, and ˜ = . In the Wigner representation with the kinetic momentum, the convolution of two operators is approximated as A B A B Aˆ 丢 Bˆ = expi共Xk −k X兲/2Aˆ共X,k兲Bˆ共X,k兲
ˆ R/A are the Green’s functions evaluated at equilibwhere G eq ˆ ⬍ solves the Kinetic Eq. 共6兲 up to zeroth rium, i.e., E = 0. G 2 order in the electric field E and therefore the expansion in E ˆ ⬍ and ⌺ˆ ⬍ starts from the linear in E terms. With this of G 1 1 knowledge, we apply the Wigner transformation to Eq. 共14兲 and find the self-consistent simple form of the kinetic equaˆ ⬍, tion for G 1 ˆA , ˆ R ⌺ˆ ⬍G ˆA ˆ R ˆG ˆ⬍=G G eq 1 eq 1 eq − ieE共nF兲Geq ⌺ˆ ⬍ 1 = ni
冕
d 2k ⬘ ˆ R ˆ ⬍共k⬘兲TˆA 共k⬘,k兲, T 共k,k⬘兲G 1 eq 共2兲2 eq
共20兲
ˆ / បk and TˆR/A are self-consistent T matrices where ˆ = H 0 eq evaluated at equilibrium. In the following section, we show how to solve the kinetic Eqs. 共19兲 and 共20兲 for a simple system described by the Rashba Hamiltonian. Whereas solving Eqs. 共19兲 and 共20兲 require only the equilibrium retarded and advance Green’s functions and T matrices, note that for ˆ ⬍ we need to solve these Green’s functions up to linear G 2 order in E 共see below兲. From the equations above, it is natural to decompose the ˆ ⬍ into the Fermi sea and Fermi-surface contributions to G 1 such that G ˆ ⬍=G ˆ ⬍+G ˆ ⬍=G ˆ ⬍+G ˆ ⬍ where contributions 1 2 I II ˆ ⬍ + i 共 n 兲eE共 G ˆA ˆR ˆ⬍=G G F k eq + kGeq兲, 1 I 2ប
共21兲
ˆ ⬍ = n 共G ˆA−G ˆ R兲. G F II
共22兲
Next, we linearize Eq. 共16兲 in E, carry out the Wigner transˆ ⬍, arriving at formation, and insert the two components of G the two corresponding components of the current density I jx共y,z兲 = − ie
i ⬇ AˆBˆ + 共XAˆkBˆ − kAˆXBˆ兲, 2 where we use the four vector notations Xk = Rk − ˜T and ˜T = T + eបE k. Here, we assume that a vector potential AE共T兲 = −cET which corresponds to a uniform electric field E. The first-order gradient expansion is sufficient for the linear-response theory, while the second-order gradient expansion may be necessary for time dependent problems and ˆ is spatially dependent in order to when the Hamiltonian H 0 account for the corresponding Berry curvature effects.2 Since we are seeking homogeneous solutions both in space and time with respect to the center-of-mass coordinates, the only surviving terms in the expansion are
共19兲
II = − ie jx共y,z兲
冕 冕
d 2k d ˆ ⬍ˆ Tr共G I x共y,z兲兲, 共2兲2 2
共23兲
d 2k d ˆ ⬍ˆ Tr共G II x共y,z兲兲, 共2兲2 2
共24兲
I II where the Fermi surface 共jx共y,z兲 兲 and Fermi sea 共jx共y,z兲 兲 contributions are identical to ones defined within Kubo-Streda formalism.44 Eqs. 共19兲–共21兲 are the main results of this subsection.
C. Linearized Fermi sea contribution
In order to calculate the Fermi sea contribution using Eqs. 共22兲 and 共24兲, we expand the retarded 共advanced兲 Green’s
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function and self-energy up to the first order in E following the procedure of Onoda et al.:1
Ω(k)
(i)
ˆ R,A = G ˆ R,A + eEG ˆ R,A + O共E2兲, G eq E
Ω=h
(ii)
ˆ R,A + eE⌺ ˆ R,A + O共E2兲, ˆ R,A = ⌺ ⌺ eq E
共25兲
ˆ R,A 兩 , ˆ R,A 兩 , ˆ R,A = 1 G ⌺ˆ ER,A = 1e E⌺ and where G E=0 E=0 E e E R,A R,A ˆ 兲 are the Green’s functions 共self-energies兲 evaluˆ 共⌺ G eq eq ated at equilibrium, i.e., E = 0. The Fermi sea lesser Green’s ˆ ⬍ calculated up to the first order in the electric function G II field E becomes ˆ R 兲. ˆA −G ˆ R 兲 + n eE共G ˆA−G ˆ ⬍ = n 共G G F F eq eq E E II
共26兲
We now substitute Eq. 共25兲 into Eqs. 共5兲 and 共11兲 only retaining linear terms in E in order to arrive at the following self-consistent equations: ˆ R,A⌺ ˆ G ˆ R,A i ˆ R,A ˆ + ⌺ ˆ R,A ˆ R,A ˆ R,A = G G E eq − 关Geq 共 បk eq 兲Geq E eq 2 ˆ R,A共ˆ + ⌺ ˆ R,A ˆ R,A − G បk eq 兲Geq 兴, eq ˆ R,A = n ⌺ i E
冕
共27兲
d2k⬘ ˆ R,A ˆ R,A共k⬘兲TˆR,A共k⬘,k兲, 共28兲 T 共k,k⬘兲G E eq 共2兲2 eq
where in Eq. 共5兲 we also performed the gradient expansion and linearized form of Eq. 共11兲 was substituted in 具k兩⌺ˆ R,A兩k⬘典 = ni具k兩TˆR,A兩k典␦共k − k⬘兲. Equations 共26兲–共28兲 are the main results of this subsection.
Ω = -h
(iii)
FIG. 2. Electronic band dispersions of the Rashba model; throughout the paper, F is the Fermi energy measured from the bottom of the lower band while F is the Fermi energy measured from the middle of the gap 关region 共ii兲兴.
ˆ = 1ˆ 共បk˘ 兲2/2m + ␣˘ k˘ · ˆ ⫻ z − h˘ˆ z + 1ˆ V共r˘ 兲, H R
V共r兲 = V˘0 兺 ␦共r˘ − r˘ i兲,
共30兲
i
where r˘ i describes the positions of randomly distributed impurities of density n˘i. We rewrite the Hamiltonian in dimensionless quantities ˆ 1 H R ˆ ⫻ z − hˆ z + 1ˆ V0 兺 ␦共r − ri兲, 共31兲 = 1ˆ k2 + ␣k · 2 F i where F is the Fermi energy measured from the minimum of energy and k = k˘l0 is the dimensionless momentum. The dimensionality can be restored by substituting expressions for the dimensionless units into the final formulas l0 =
冑
ប2 , ␣ = ␣˘ mF
h=
A. Calculational procedure
We restrict ourselves here to 2DEG Rashba Hamiltonian with an exchange field h˘ 共breve accent here means that h is in dimensional units兲 in order to obtain simple analytical results that connect directly with other microscopic linearresponse calculations19,22,46
共29兲
ˆ are Pauli where ␣˘ is the strength of spin-orbit interaction, ˘ matrices, បk = −iប −eA / c, A共t兲 = −cEt describes the external electric field, and V共r兲 describes the impurities. From symmetry considerations, the most general form of the Hamiltonian in Eq. 共29兲 should treat the coordinate r as an ˆ ⫻ k originating from the operator r + rˆ so共k兲, with rˆ so共k兲 = projection procedure onto the band under consideration.47 The spin-orbit interaction can also include higher, e.g., cubic terms relevant for the bulk InSb and the HgTe quantum wells with an inverted band structure.48,49 Here, only linear terms with Rashba symmetry are considered with rˆ so共k兲 being disregarded as we expect effect of Hso = rˆ so共k兲 V共r兲 on the AHE to be small for wide band semiconductors in which is relatively small.50 The disorder in the system is modeled by impurity delta scatterers
III. AHE IN RASHBA SYSTEMS
In this section, we apply the above formalism to 2DEG with exchange field and spin-orbit interaction. A general numerical procedure is followed by analytical results valid in the metallic regime in the limit of small impurity scattering broadening ប / with respect to the Fermi energy F. We end the section with a discussion of the numerical and analytical results comparing them to other approaches. For convenience and in order to keep the expressions more concise, we introduce here the dimensionless units that can easily be transformed into dimensional units by following equations at the beginning of this section. Note that our formalism cannot be used close to the energies = ⫾ h in Fig. 2, as kFl 共l is the mean-free path兲 can become very small and the noncrossing approximation in Fig. 1 may fail. Nevertheless, we do not expect large corrections to our results around these singularities as the nondiagonal conductivity seems not to be strongly affected by including the crossed diagrams.45
k
冑
m mV˘0 , V = , 0 ប 2 F ប2
h˘ , ni = n˘il20, k = k˘l0 . F
Also note that whereas F is measured from the bottom of the lower band, in the notation below, we introduce F which is the Fermi energy measured from the middle of the gap 关region 共ii兲 in Fig. 2兴. In the following, we solve Eqs. 共19兲 and 共20兲 in order to find the nonequilibrium Green’s
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ˆ ⬍ describing processes at the Fermi surface and function G 1 Eqs. 共27兲 and 共28兲 for the nonequilibrium Green’s function ˆ ⬍-primarily Fermi sea contribution. G 2 ˆ R,A using ˆ R,A and the Green’s functions G We calculate ⌺ eq eq the self-consistent TMA, i.e., diagonal components of Eq. 共8兲 共Refs. 1 and 20兲 R,A = V0共1ˆ − V0␥ˆ R,A兲−1 , Tˆeq
共32兲
R,A R,A R,A R,A 共兲 = niTˆeq 共兲 = ⌺eq0 ˆ 0 + ⌺eqz ˆ z , ⌺ˆ eq
共33兲
=
W−
冊
k2 ˆ 0 + ␣kyˆ x − ␣kxˆ y − Hˆ z 2 , k2 2 2 2 2 W− −H −␣ k 2
冉
R
Σ^
<
=
+
=
+
+
冊
+
共34兲
R R , H = h − ⌺eqz , and ␥ˆ R,A = 兰d2k / where W = − ⌺eq0 ˆ R,A共k , 兲 ⬅ ␥R,Aˆ + ␥R,Aˆ . We calculate self共2兲2G 0 z eq z ˆ R,A共兲 for each by consistent value of the self-energy ⌺ eq
performing sufficient number of iterations in Eq. 共33兲 in order to achieve the prescribed accuracy 共see Appendix B for details兲. With the knowledge of the equilibrium Green’s function ˆ R 共k , 兲, we can calculate the local densities of states G eq D共兲 ⬅ −
1
冕
F
dD共兲.
ˆ =
ˆ ER,A =
共35兲
The number of electrons changes as we increase the disorder and following Eq. 共35兲, F is always adjusted so that the total number of electrons is constant. ˆ ⬍ and ⌺ˆ R,A from The same TMA is also used to calculate ⌺ E Eqs. 共20兲 and 共28兲, respectively,
冕 冕
ˆ R,A = n ⌺ i E
+ ….. b)
d 2k ˆ ⬍ G 共k, 兲, 共2兲2 1
共38兲
冕
d2k ˆ R,A G 共k, 兲. 共2兲2 E
共39兲
冕
d 2k ˆ R ˆ R A ˆA G T 共兲ˆ 共兲Tˆeq 共兲G eq 共2兲2 eq eq
− inFeE
−⬁
⌺ˆ ⬍ 1 = ni
+
The elements of matrices ˆ and ˆ E satisfy a system of linear equations obtained by integrating in momentum space the left- and right-hand sides of Eqs. 共19兲 and 共27兲, respectively,
d 2k ˆ R 共k, 兲兴其 Im兵Tr关G eq 共2兲2
冕
+
a)
冕
ˆ ER,A共兲 ⬅
and the total number of electrons N=
+
+ …..
FIG. 3. 共Color online兲 An infinite set of diagrams representing the self-consistent TMA in calculating 共a兲 the retarded 共advanced兲 ˆ R,A and 共b兲 the lesser component of self-energy ⌺ˆ ⬍ in self-energy ⌺ eq Eqs. 共33兲 and 共36兲, respectively.
ˆ 共兲 ⬅ ˆ R 兲† , ˆ A = 共G G eq eq
+
self-consistently calculated retarded 共advanced兲 Green’s function. For the delta scatterers, T matrix does not depend on momentum k which allows us to perform momentum integrations in Eqs. 共36兲 and 共37兲. It is then useful to introduce the following 2 ⫻ 2 matrices:
ˆ −⌺ ˆ R 兲−1 ˆ R = 共1ˆ − H G 0 eq eq
冉
Σ^
d 2k ˆ R ˆ ⬍共k, 兲TˆA 共兲, T 共兲G 1 eq 共2兲2 eq
共36兲
d2k ˆ R,A ˆ R,A共k, 兲TˆR,A共兲. T 共兲G E eq 共2兲2 eq
共37兲
The TMA with self-consistent calculation of the equilibrium ˆ R,A described in Appendix B allows us to Green’s functions G eq take into account higher order noncrossed diagrams in the concentration of impurities ni, with weak localization diagrams being disregarded. The procedure of calculating the retarded 共advanced兲 and nonequilibrium self-energies in Eqs. 共33兲 and 共36兲 is represented graphically in Fig. 3. In this graphical representation, the bold arrow corresponds to the
冕 −
冕
d 2k ˆ R ˆ A G ˆ G , 共2兲2 eq eq
共40兲
d2k ˆ R,A ˆ R,A R,A ˆ R,A G T 共兲ˆ ER,A共兲Tˆeq 共兲G eq 共2兲2 eq eq i 2
冕
d2k ˆ R,A ˆ R,A − G ˆ R,A ˆ G ˆ R,A兲. 共41兲 共G ˆ G eq eq eq 共2兲2 eq
The momentum integrations in the right-hand side of Eqs. 共40兲 and 共41兲 are done analytically using the general form of ˆ R,A共k , 兲 in Eq. 共34兲. Without loss of the Green’s functions G eq generality, we take the electric field E along the y axis E = 共0 , Ey兲 and solve the system of linear Eqs. 共40兲 and 共41兲 for the elements of matrices ˆ and ˆ E in Appendixes C and D, respectively. With this, we calculate the current from Eqs. 共23兲 and 共24兲, respectively, with a use of Eqs. 共19兲, 共21兲, 共26兲, and 共27兲, I = − ie jx共y兲
195129-6
冕
再
d 2k d ˆA ˆ ˆ R TˆR ˆ TˆA G Tr G x共y兲 eq eq eq eq 共2兲2 2
冋
册 冎
ˆ R 兲 ˆ ˆ A − 1 共G ˆA +G ˆ R ˆ G ˆ A ˆ G ˆ R ˆ G − ieEnF G x共y兲 , eq eq eq eq 2 eq eq 共42兲
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TRANSPORT THEORY FOR DISORDERED MULTIPLE-BAND…
II jx共y兲 = ie
冕
冋
d 2k d ˆR ˆ ˆ R TˆR ˆ R TˆR G eEnF Tr G x共y兲 eq eq E eq eq 共2兲2 2
册
i ˆR ˆR − G ˆ R 兲ˆ ˆ R ˆG − 共G ˆ G eq eq x共y兲 + c.c., eq 2 eq
共43兲
ˆ R,A = G ˆ R,Aˆ G ˆ R,A, kG eq eq eq
where we use which holds for the model of delta impurities. In Eq. 共42兲, we perform analytical integrations over momentum k and energy while in Eq. 共43兲, we only perform analytical integration over momentum. The results of these integrations are given in Appendixes E and F for Eqs. 共42兲 and 共43兲, respectively. B. Analytical results in the metallic regime
In the metallic regime, we are able to obtain analytical results as it is sufficient to consider only finite number of terms in the expansion with respect to the strength of impurity in Fig. 3. For the same reason, we are also able to generalize the disorder in Eq. 共30兲 共generalization of the theory is given in Appendix A兲 as follows: V共r兲 = 兺 Vi0␦共r − ri兲,
x =
b) y
x + y
c) y
x +
d) y
x with
+
x
All combinations of two skew scatterings Third order correction
R =
x y
e) y
+
+
x y
x
FIG. 4. 共Color online兲 Different diagrammatic contributions to Ixy within the Kubo formula formalism: 共a兲 the ladder diagram 共vertex兲 contribution ⬃1, 共b兲 the skew-scattering contribution ⬃1 / 共niV0兲 ⬃ V3 / V22, 共c兲 the double skew-scattering contribution ⬃1 / ni ⬃ V23 / V32, 共d兲 the skew-scattering contribution in which the retarded 共advanced兲 self-energy is calculated up to the third order ⬃1 / ni ⬃ V23 / V32, and 共e兲 the fourth-order skew-scattering contribution ⬃1 / ni ⬃ V4 / V22.
⫾ = 冑共␣k⫾兲2 + h2, F = 冑2F␣2 + h2 ,
共44兲
i
where ri is random, the strength of each impurity has the same arbitrary distribution, and all strength distributions are independent leading to the first four cumulants 具Vi0典dis = 0, ni具共Vi0兲2典dis = V2, ni具共Vi0兲3典dis = V3, and ni具共Vi0兲4典dis = V4, where ni is the concentration of impurities. For the disorder described in Eq. 共30兲, we have 冑V2 / ni = 冑3 V3 / ni = 冑4 V4 / ni = V0 and for the telegraph white-noise disorder we have V3 = 0 as it is mentioned in Appendix A. In this section, we first expand the retarded 共advanced兲 self-energy in Eq. 共33兲 up to the third order in V0 关or up to the terms V3 in Eq. 共A4兲兴. The lesser component of the selfenergy in Eq. 共36兲 has to be expanded up to the fourth order in V0 关or up to the terms V4 in Eq. 共A5兲兴, which corresponds to the four legged diagrams in Fig. 3共b兲. This ensures that the expansion of the conductivity Ixy following from Eqs. 共E3兲 and 共E5兲 captures all possible terms proportional to 1 / V0 and 1. The expansion of IIxy following from Eq. 共F3兲 is somewhat simpler as it only contains the terms proportional to 1 and its calculation requires consideration of only one bare bubble diagram 共e.g., summation of vertices leads to higher order corrections兲. In our discussion, we thus concentrate on the diagrams for calculating Ixy and also present the result for the bare bubble diagram of IIxy. Note that in the expansion of I共II兲, it is important to properly consider the branch cut of the “ln” function taken as 共−⬁ , 0兴. The diagrams in Fig. 3 have direct correspondence to the Kubo formalism diagrams in Fig. 4 used in Ref. 22. This allows us to separate the conductivity into terms that directly relate to each diagram in Fig. 4. We distinguish three regimes for the position of the Fermi energy with respect to the gap of the size 2h: 共i兲 F ⬎ h, 共ii兲 −h ⬍ F ⬍ h, and 共iii兲 F ⬍ −h 共see Fig. 2兲. To simplify formulas, we introduce the following notation:
a) y
⫾ = 冑共␣k⫾兲2 + 4h2 ,
⫾ = k
冏 冏 d共k兲 dk
−1
=
冦
⫾ , F ⬎ h ⫾ ⫾ ␣2 − , − h ⬍ F ⬍ h − − ␣2 ⫾ , F ⬍ − h, 兩⫾ − ␣2兩
冧
where ⫾ is the density of states at the Fermi level and k⫾ are the two Fermi wave numbers for regimes 共i兲 and 共iii兲. In regime 共ii兲, k+ becomes pure imaginary and only k− has the meaning of the Fermi wave number. Further, we introduce the following parameter: ⌳=
V3 i V4 ␥z + 2 共3␥r␥zi + ␥i␥zr兲, V22 V2
ˆ R ⬅ ␥ˆ + ␥ ˆ , with ␥ = ␥r + i␥i and where ␥ˆ = 兰d2k / 共2兲2G 0 z z eq r i ␥z = ␥z + i␥z. Note that the two-dimensional integral over momentum diverges and ␥ˆ is calculated by introducing the momentum cutoff 共see Appendix B兲. By expanding the result of Appendix B up to the zeroth order in the strength of impurities, we obtain
2 k⫾ = 2共F + ␣2 ⫿ 冑h2 + 2F␣2 + ␣4兲,
冏 冏
共k−2 − 2F兲ln
␥r =
k20 − k−2
冏 冏
− 共k+2 − 2F兲ln
2共k−2 − k+2兲
␥zr = 195129-7
k−2
冏
冏
k+2共k20 − k−2兲 h ln , 共k+2 − k−2兲 k−2共k20 − k+2兲
k+2
k20 − k+2
,
PHYSICAL REVIEW B 79, 195129 共2009兲
KOVALEV et al.
␥i =
冦
␥zi =
− + + , F ⬎ h 4 − − , − h ⬍ F ⬍ h 4 −
−
k−2 + k+2 − 4F 2共k−2
冦
−
k+2兲
冉
, F ⬍ − h,
冊
h + − − , F ⬎ h 4 + − h − , − h ⬍ F ⬍ h 4 −
− −
2h k−2
−
, k+2
F ⬍ − h,
冧
冧
the diagrams in Figs. 4共c兲 and 4共d兲 lead to the terms in Eq. 共46兲 proportional to V23 / V32. II共ii兲 is calculated from a bare xy bubble contribution given by Eq. 共F3兲 and also corresponds to the intrinsic contribution. Finally for the region 共iii兲 共F ⬍ −h兲, we obtain
I共iii兲 xy =
+
I共i兲 xy =
2e2␣2 V4 ⌳=− 2 ប V2
冏
II共i兲 xy
k+2共k20 − k−2兲 k−2共k20 − k+2兲
ប2共k+2 − k−2兲
冏
,
冉
⫻
8h共2h2 + 2F␣2 + k−2␣2兲
2hk−4␣2 V23
−4
V32
II共ii兲 xy =
冊
−2
␥zi + 共k−2 − k+2兲␥i
册
,
冉
冊
e2 h 1− 4 冑␣ + F2 , 4ប
共46兲
where the diagrams in Fig. 4共a兲 lead to the first three disorder independent terms in Eq. 共46兲 共the intrinsic, the side-jump, and the disorder independent skew-scattering terms, respectively兲,24 the skew-scattering diagrams in Figs. 4共b兲 and 4共e兲 lead to the term in Eq. 共46兲 proportional to ⌳ and
册
冊
h␣2共k−2 − k+2兲3 V23 , 4共h2 + ␣4兲2 V32 h共− − +兲 e2 , 2 4ប 共␣ − −兲共␣2 − +兲
共47兲
where the diagrams in Fig. 4共a兲 lead to the disorder independent term in Eq. 共47兲 共it includes the intrinsic, the side-jump, and the disorder independent skew-scattering contributions兲, the skew-scattering diagrams in Figs. 4共b兲 and 4共e兲 lead to the term in Eq. 共47兲 proportional to ⌳, and the diagrams in Figs. 4共c兲 and 4共d兲 lead to the terms in Eq. 共47兲 proportional is again calculated from a bare bubble conto V23 / V32. II共ii兲 xy tribution given by Eq. 共F3兲. The diagonal conductivities can also be calculated by expanding Eqs. 共E4兲 and 共E6兲,
共45兲
= 0,
e2 h␣2− 4hk−2␣2 3hk−4␣2 8k−4␣2−2 I共ii兲 − + 4 + ⌳ xy = 4ប −2 −−2 − − −4−2
冋
h共h2F + 2␣2h2 − 3F␣4兲 i ␥z + ␥i 共h2 − F␣2兲共h2 + ␣4兲
II共iii兲 = xy
which reproduces result of Ref. 24 in the limit of large cutoff k0. In reference to the Kubo formula formalism, we can claim the following: the diagrams in Fig. 4共a兲 vanish after summation 共the intrinsic and side-jump contributions defined in Ref. 46 cancel each other兲,24 the diagrams in Figs. 4共b兲–4共d兲 are all proportional to ++ − −− ⬅ 0 and also vanish, and the diagrams in Fig. 4共e兲 lead to the result in Eq. 共45兲. II共i兲 xy is zero as the corresponding bare bubble contribution in Eq. 共F3兲 vanishes. Repeating the same procedure for the region 共ii兲 共−h ⬍ F ⬍ h兲, we obtain
+
冋
⫻
where k0 is the cutoff in the momentum integration. As it follows from the Appendixes E and F, the nondiagonal conductivities I共II兲 xy can be calculated by properly choosing the “ln” branch that corresponds to the regimes 共i兲, 共ii兲, or 共iii兲, respectively. The result of expanding Eqs. 共E3兲 and 共E5兲, and Eq. 共F3兲 for conductivities Ixy and IIxy, respectively, in the region 共i兲 共F ⬎ h兲 becomes e2h␣2 ln
冉
32hF2 ␣4 ␣2共k−2 − k+2兲4 e2 + ⌳ 4ប 共h2 + ␣4兲2共k−2 − k+2兲 32共h2 + ␣4兲2
yy =
冦
e2 F + ␣2 , F ⬎ h ប V2 e2 k−2−2 , − h ⬍ F ⬍ h ប V2−2−2 e2 共F + ␣2兲共␣4 + F2 兲 , F ⬍ − h, ប V 2共 ␣ 4 + h 2兲
冧
where we only present the dominant nonvanishing terms V−1 2 as the higher order terms are quite cumbersome. C. Numerical results and discussions
Here, we present results of our numerical calculations based on the formalism developed in Sec. III A. Figures 5–8 show the numerical results for the anomalous Hall conductivity as a function of the Fermi energy F and the first Born scattering amplitude ␥Born = niV20. The strength of the spinorbit interaction is chosen to be the same as in Ref. 1, 2␣2 / h = 35.9 共2␣2 / Eres = 3.59, Eres = 10h兲 and the strengths of impurity are V0 = 0.1, 0.3, −0.1, and −0.3. For the retarded 共advanced兲 self-energy, the cutoff in the momentum integration is k0 = 12 which corresponds to the energy cutoff of Ref. 1, c = 3Eres. The Born scattering amplitude is varied by changing the impurity concentration ni. In the clean limit, when ␥Born → 0, we observe skewscattering behavior 关xy ⬃ 1 / 共niV0兲兴 in which 兩xy兩 rapidly increases. For repulsive scatterers 共V0 ⬎ 0, see Figs. 5 and 6兲, the negative conductivity diminishes as we increase the Fermi energy until the point F = −h is reached. At this point, the conductivity suddenly increases without a change of sign,
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Σxy 2
0
2eΠ
Σxy e2
0.2
9
2 Π
0.15
18 0.1
5
ΩF h
2 Π
0
ΩF h
5
5
30 0.2
0 10
ΓBorn h
0.15 0.1
5
0.05
0
5
2 2eΠ 15
0.15 0.1
0.05
0
subbands are partially occupied兲 does not change its sign as we change the sign of disorder.24 Comparing Figs. 7 and 8, we again see that the hybrid skew scattering is more pronounced for larger impurity strength. As we increase the disorder by increasing ␥Born, the skew scattering becomes less important while the other mechanisms, such as intrinsic and side jump, become more important. The intrinsic conductivity only gradually decreases with the disorder because the only effect of disorder on the intrinsic component comes from broadening of Green’s functions used in the calculation of the intrinsic component. For repulsive scatterers 共V0 ⬎ 0兲, the skew scattering has sign opposite to the sign of intrinsic and side-jump contributions in the region −h ⬍ F ⬍ h 共see, e.g., Refs. 22 and 24兲. This explains the sign change we observed in Figs. 5, 6, and 10 in the region −h ⬍ F ⬍ h as we increase ␥Born 共more detailed plots are presented in Appendix G兲. The positions of points in which the AHE vanishes can be estimated by comparing the Fermi sea intrinsic term IIxy with the skew-scattering term in Eq. 共46兲 as those two are the major contributions. Physically, the AHE vanishes because the intrinsic deflection of electrons between the scattering events can be balanced by the skew-scattering events 共in the cross-over region between intrinsic and extrinsic mechanisms兲. As the former does not rely on impurities and the latter does 共and changes sign with impurities changing sign兲,
0.2
60
ΩF h
5
FIG. 6. 共Color online兲 Identical to Fig. 5 plot but for larger strength of impurity V0 = 0.3.
ΓBorn h
FIG. 7. 共Color online兲 Identical to Fig. 5 plot but for negative strength of impurity V0 = −0.1.
Σxy
30
0.1
5
0.05
0
ΩF h
0.15
0
in contrast to Ref. 1 where the sign change has been observed but in agreement with Refs. 22 and 24 关note that Fig. 5 is calculated for exactly the same parameters as Fig. 5共c兲 in Ref. 1兴. As we increase the Fermi energy further, the conductivity increases again around F = h acquiring a very small negative value. In this regime, both subbands are partially occupied and only the higher order skew scattering22,24 共hybrid skew scattering兲 contributes to the anomalous Hall effect. Relatively large hybrid skew scattering is present in Fig. 6 compared to Fig. 5 as the hybrid skew-scattering contribution is proportional to 1 / ni ⬃ V20 / ␥Born and should be larger for greater impurity strength.24 The same is true for the conventional skew scattering proportional to 1 / 共V0ni兲 ⬃ V0 / ␥Born, which can be immediately seen from Figs. 5–8. For attractive scatterers 共V0 ⬍ 0, see Figs. 7 and 8兲 the sign of the ordinary skew scattering dominating in the clean limit is opposite to the sign of the ordinary skew scattering for the repulsive scatterers. The conductivity now increases until we reach the point F = −h in which we observe a sudden drop. One more drop happens around the point F = h where the anomalous Hall conductivity changes sign 共see Figs. 7 and 8兲. This change of sign is consistent with the fact that the higher order 共hybrid兲 skew scattering 共prevailing when both
e2
0.2
7
ΓBorn h
FIG. 5. 共Color online兲 The anomalous Hall conductivity xy as a function of the Fermi energy F and the Born scattering amplitude ␥Born. The parameters are chosen as 2␣2 / h = 35.9, k0 = 12, and V0 = 0.1. The Fermi energy F corresponds here to the clean system and it is renormalized according to Eq. 共35兲 in the presence of disorder.
Σxy
14
ΓBorn h
0.05
0 5
FIG. 8. 共Color online兲 Identical to Fig. 5 plot but for negative strength of impurity V0 = −0.3.
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we can have full cancellation of the two by choosing the proper sign and strength of impurities. As it can be seen from Figs. 5–8, the anomalous Hall effect is resonantly increased around the band anticrossing which suggests that for more general band structures, the major contribution to the AHE also comes from the band anticrossings that happened to be in the vicinity of the Fermi level.1 This view is well justified for the intrinsic AHE in the metallic regime 共F Ⰷ 1兲 as it follows from the ThoulessKohmoto-Nightingale-Nijs formula51 applied to the AHE.1 This leads to the intrinsic AHE conductivity of the order of e2 / 共4ប兲 within the region 共ii兲 in Fig. 2. The full conductivity that includes the intrinsic, side-jump, and skew-scattering contributions seems to also have the resonant behavior around the anticrossing for the Rashba model as it follows from our analysis. Whereas our analysis justifies focusing the calculations on simplified phenomenological models near the anticrossing locations, we emphasize that it is unlikely that these would be characterized universally by the Rashba geometry rather than by a combination of Rashba and Dresselhauss symmetry. In the regime of strong disorder, where this resonant behavior is not pronounced, the disorder broadening of the Green’s functions becomes more dominant and the xy has dependence that can no longer be expanded correctly in powers of . This expectation can be easily seen from the expressions for xy in our formulation or the Kubo formulation, in which
xy ⬀ 兺
␣,
具␣兩vˆ x兩典具兩vˆ y兩␣典 , 共E␣ − E兲2
共48兲
where 兩␣典 are the exact eigenstates in the presence of disorder and the major contribution for xy in the dirty limit comes from interband matrix elements. When expanding things in the momentum basis, the denominator is often approximated as 关En共kជ 兲 − En⬘共kជ 兲兴2 + 共ប / 兲2 while the matrix elements are evaluated within the disorder-free eigenstates. Hence, in the limit of large disorder broadening, the denominator is simply replaced by 共ប / 兲2 and xy ⬃ 2 共this is different for xx as the contribution from interband matrix elements vanishes and xx ⬃ 兲. This of course gives an upper bound for the xy ⬃ scaling and in intermediate regimes one would expect to be lower than 2. In Figs. 9 and 10, we study the AHE calculated in the anticrossing region in order to examine in detail the universal anomalous Hall-effect regimes that could be valid for more general band structures. We now plot in the logarithmic scale xy as a function of xx tuned via ni while all other parameters are kept constant. In the clean limit, we recover the skew-scattering behavior 关xy ⬃ 1 / 共niV0兲 ⬃ xx / V0兴 and our numerical results 共bold line兲 agree well with the analytical results 共dashed line兲 obtained in Sec. III B. In the moderately dirty limit, we observe the intrinsic-side-jump regime 共xy = const., this regime is more pronounced for smaller V0兲 in which the side-jump and intrinsic mechanisms are dominant. All analytical curves 共dashed lines兲 asymptotically reach this regime when xx is very small. In the stronger disorder regime, as reported in Ref. 1, the numerical curves have down-
FIG. 9. 共Color online兲 The absolute value of the anomalous Hall conductivity 兩xy兩 versus the conductivity xx for the spin-orbit interaction strength 2␣2 / h = 35.9. Dimensionality of quantities displayed in this plot is restored.
turn for smaller xx approaching the third regime in which xy ⬃ xx , with ⬇ 1.6 in Fig. 9. However, a universal scaling cannot be claimed since for large and positive strength of impurities in Fig. 10 we only observe the reduction of the AHE. One should keep in mind that the TMA is not fully justified close to the line F = 1 and our results are meaningful only for F ⬎ 1. Furthermore, since in this regime the resonant behavior is strongly diminished, in realistic threedimensional systems, the result could be more accurately expressed via the averaged matrix elements with some appropriate treatment of the disorder broadening. Although some experimental works claim to confirm the , with around 1.6,37–41 comparison of scaling xy ⬃ xx theory and experiments has to be done with care since determining a scaling exponent over a single decade is often difficult and has led to many errors in the past. For example, in DMS ferromagnets 共mentioned in Ref. 1 to support the scaling hypothesis兲 the change of doping will cause change in the impurity concentration, in the magnetization, and even in the band structure. The theoretical calculations only take into account the change in the impurity concentration and further assume a Rashba symmetry at the crossing points.20,1 Note also that within the theoretical treatment, the Hall conductivity changes its sign for repulsive impurities 共V0 ⬎ 0兲 in Fig.
FIG. 10. 共Color online兲 Identical to Fig. 9 plot except for the disorder which is repulsive here 共V0 ⬎ 0兲. Note that the conductivity xy changes sign around the cusps.
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10 which is expected as the skew scattering dominating in the clean limit has the sign opposite to the sign of the intrinsic contribution dominating in the dirty limit 关see Eq. 共46兲兴. These types of changes of signs have also been observed in experimental systems, e.g., DMS,52 and of course at that stage scaling is not justified.
ˆ R = 共1ˆ − H ˆ −⌺ ˆ R 兲−1 G 0 eq eq
=
IV. AMR IN RASHBA SYSTEMS
−
In Sec. III, we showed how the formalism developed in Sec. II can be applied to calculations of the anomalous Hall effect in multiple-band systems on the example of a Rashba system. In this section, we perform calculations of the AMR in 2DEG with the in-plane exchange field, spin-orbit interaction, and magnetic impurities following the same formalism. A general numerical procedure allows us to rigorously perform AMR calculations in multiple-band anisotropic systems. Within the Boltzmann equation approach, such calculations are usually performed by using the relaxation-time approximation in which the transport relaxation time is calculated from the scattering amplitudes without fully taking into account the asymmetries.30,34 This approach was improved in Ref. 53 by introducing the perpendicular relaxation time ⬜. However, in some cases this improvement is yet not sufficient and some of the present authors formulated a procedure for finding an exact solution to the Boltzmann equation in Ref. 35. Here we propose an alternative approach for AMR calculations in multiple-band anisotropic systems to the one proposed in Ref. 35. We consider here a 2DEG Rashba Hamiltonian with additional in-plane exchange field hx directed along the x axis without any loss of generality ˆ = k2/2 + ␣k · ˆ ⫻ z − hxˆ x − hˆ z + Vˆ共r兲, H R
共49兲
where now Vˆ共r兲 describes the disorder corresponding to dilute charged magnetic impurities34,54,55 Vˆ共r兲 = V0共aˆ 0 + ˆ x兲 兺 ␦共r − ri兲,
冉
共50兲
i
where ri describes the positions of random impurities and we assume that the magnetic impurities are magnetized along the exchange field. The quantity a describes the relative strength of the electric part of impurity with respect to the magnetic part. Note that the AMR is measured by changing the direction of electric field E which is equivalent to changing the direction of the exchange field. For the AMR, we only need the diagonal conductivities, thus the Fermi sea contribution given by Eq. 共43兲 vanishes. The AMR can be calculated from Eq. 共42兲 and we only need to calculate Green’s functions at the Fermi level. We calcuˆ R,A using the self-consistent ˆ R,A and Green’s functions G late ⌺ eq eq TMA R,A = V0共1ˆ − V0␥ˆ R,A兲−1 , Tˆeq
共51兲
ˆ R,A = n TˆR,A共 兲 = ⌺R,Aˆ + ⌺R,Aˆ + ⌺R,Aˆ , ⌺ i eq F eq eq0 0 eqx x eqz z
共52兲
冉
冊
k2 ˆ 0 + ␣kyˆ x − ␣kxˆ y − Hˆ z 2 k2 2 W− − H2 − ␣2k2 + 2Hx␣ky 2
W−
冉
冊
冊
2 2
W−
k 2
Hxˆ x
,
− H2 − ␣2k2 + 2Hx␣ky
ˆ R 兲† , ˆ A = 共G G eq eq
共53兲
R R R , H = h − ⌺eqz , Hx = hx − ⌺eqx , and ␥ˆ R,A where W = − ⌺eq0 R,A R,A 2 2 ˆ R,A R,A ˆ = 兰d k / 共2兲 Geq 共k , 兲 ⬅ ␥ 0 + ␥x ˆ x + ␥z ˆ z. We calcuˆ R,A共 兲 by late the self-consistent value of the self-energy ⌺ F eq iterating Eq. 共52兲 until the prescribed accuracy is reached. As soon as we know the T matrix, we can substitute it into Eq. 共40兲 and find the matrix ˆ by performing the momentum integrations in the right-hand side. Finally, by substituting ˆ into Eq. 共42兲 we can calculate the conductivity. Note that throughout this section, the angular part of the momentum integrations is calculated analytically while the radial part is calculated numerically. The anisotropic resistance in our system is defined as follows:
AMR = −
xx − yy , xx + yy
and it describes the relative difference in conductivity for current flowing parallel or perpendicular to the magnetization 共represented by the exchange field and/or impurity magnetization兲. First, we calculate the anisotropic magnetoresistance in Rashba system with in-plane exchange field and nonmagnetic delta scatterers 关see Eq. 共30兲, the magnetic scatterers are absent in this model兴. Kato et al.36 found vanishing AMR in the regime 共i兲 共see Fig. 2兲 when both subbands are partially occupied due to the cancellation of the nonvertex and vertex parts in the Kubo formulation. In Fig. 11, we observe the nonvanishing AMR in the regime 共i兲 and this suggests the importance of the higher order diagrams 关such as plotted in Fig. 4共e兲兴 not only for the AHE but also for the AMR. The AMR effect resulting from the higher order diagrams is more pronounced for the larger strength of impurities, similar to the AHE. The AMR approaches its maximum around the point at which the exchange energy is comparable to the spin-orbit energy, 2hx ⬃ ␣. We note that the nonzero but comparatively weak magnitude of the AMR here in the Rashba system is reminiscent of the results in three-dimensional DMS ferromagnets.34 This agrees with physical intuition. Under comparison of two mechanisms by which AMR can arise—carrier polarization/anisotropy in wave functions and impurity polarization/anisotropy in scattering operator 共see Fig. 1 of Ref. 34兲—the former implies a competition between the exchange and spin-orbit terms 共in the Hamiltonian兲 resulting in reduced anisotropy strength.
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tering potential.35 As soon as the spatial dependences of the electric and magnetic parts cease to be identical 关in Eq. 共50兲, they correspond both to delta scatterers兴 the divergence of the yy is removed 共causing AMR⬍ 1兲. V. CONCLUSIONS
FIG. 11. The AMR 共100% corresponds to AMR= 1兲 as a function of the dimensionless spin-orbit interaction strength ␣ / 冑F 共F is counted from the middle of the gap in Fig. 2兲. The parameters of the model are V0 = −0.6 and −0.9, ni / F = 0.01, and hx / F = 0.3.
Consequently, even though observation of the AMR effect is deemable in the absence of magnetic scatterers, we expect much more pronounced effects when the magnetic scatterers are present. Our numerical results in Fig. 12 共plotted together with the analytical results from Ref. 35兲 confirm this. For the case when the Fermi level crosses only one band 关region 共ii兲 in Fig. 2兴, it was found in Ref. 35 that AMR= 1 / 共2 − a2兲 when 兩a兩 ⬍ 1 and AMR= 1 / a2 when 兩a兩 ⬎ 1, provided the exchange fields are small. For the case when the Fermi level crosses two bands 关region 共i兲 in Fig. 2兴 it was found in Ref. 35 that AMR= a2 when 兩a兩 ⬍ 1 and AMR= 1 / a2 when 兩a兩 ⬎ 1, in the limit of large Fermi energy 共compared to the spin-orbit and exchange splittings兲. We observe a perfect agreement between our numerical results and the analytical results from Ref. 35. The result in Fig. 12共a兲 cannot be reproduced within the common approximate approaches30,34,53 based on the relaxation-time approximation as it was pointed out in Ref. 35. The nonphysical divergence in yy at the point a = 1 in Fig. 12 is caused by the special choice of the scat-
1.0
AMR
0.6 0.4
0.8
b)
0.6
ACKNOWLEDGMENTS
AMR
0.8
1.0 a)
We have developed a framework for transport calculations in multiple-band noninteracting Fermi systems. By applying this framework to Rashba 2DEG, we have resolved some recent discrepancies related to the AHE in such systems. The findings of this simple 2D model have been linked to higher dimensional systems arguing that most likely the major contributions to the AHE come from the band anticrossing regions similar to one observed in the Rashba model. Our analytical and numerical results reveal the crossover between the skew-scattering-dominated regime in clean systems 共xy ⬃ V0 / ␥Born ⬃ xx兲 and the intrinsic deflection dominated regime in moderately dirty systems 共xy ⬃ const.兲. In dirty systems, we observe the third distinct regime also dominated by the intrinsic contribution. In this regime, the AHE diminishes , with being close to 1.6. in a manner similar to xy ⬃ xx This, however, cannot be called by scaling as the theory is not meaningful in a sufficiently wide range of xy and xx due to breakdown of the TMA when F ⬃ 1. For the repulsive impurities, we observe that the intrinsic and skew anomalous Hall effects have opposite signs. As a result, the crossover between those two is also accompanied by the change of sign of the AHE. We suggest to engineer samples with repulsive impurities in order to see this change of sign in the AHE and a possible effect on the scaling. We have resolved some discrepancies between the AHE results obtained by using the Keldysh, Kubo, and Boltzmann approaches by considering the higher order skew-scattering processes. We have included similar higher order processes in our AMR calculations and shown their importance for the Rashba model in which nonvertex and vertex diagrammatic parts cancel each other. We have calculated the AMR in anisotropic systems properly taking into account the anisotropy of the nonequilibrium distribution function. These calculations confirm recent findings on the unreliability of common approximate approaches to the Boltzmann equation.
0.4
0.2
0.2
0.0
0.0 0 2 4 6 8 10 0 2 4 6 8 10 a a
FIG. 12. The AMR as a function of the relative strength a of the electric and magnetic parts of impurity potential. By solid line we plot analytical results and dots represent numerical results: 共a兲 Fermi level crosses only one band 共F = 0兲 with the following dimensionless parameters: V0 = 0.05, ␣ = 1.4, ni = 0.0015, hx = 0.0015, and h = 0.015; 共b兲 Fermi level crosses both bands V0 = 0.05, ␣ = 0.03, ni = 0.002, hx = 0.002, and h = 0.001.
We gratefully acknowledge fruitful discussions with E. I. Rashba, V. Dugaev, J. Inoue, T. Jungwirth, A. H. MacDonald, G. E. W. Bauer, N. Nagaosa, and S. Onoda. This work was supported by the Alfred P. Sloan Foundation 共Y.T.兲, by ONR under Grant No. ONR-N000140610122, by NSF under Grant No. DMR-0547875, by SWAN-NRI and Czech Grants No. KJB100100802, No. LC510, and No. AV0Z10100521. J.S. is a Cottrell Scholar of the Research Corporation. APPENDIX A: GENERALIZATIONS FOR SHORT-RANGE DISORDER
In Sec. II A, we derive the kinetic equation with the selfenergy expression that is valid for uniform systems. Here, we
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generalize this self-energy to nonuniform systems in the presence of a short-range disorder postulated by the following infinite set of correlators: 具VV典 = 具V典具V典 + V2␦r1−r2 , 具VVV典 = 兺 具VV典具V典 + V3␦r1−r3 , 具VVVV典 = 兺 共具VVV典具V典 + 具VV典具VV典兲 + V4␦r1−r4, etc. 共A1兲 where we decouple the correlators into a product of two lower order correlators and sum all such products, ␦r1−rN = 兿i=1..N−1␦共ri − ri+1兲. Note that usually the averaged impurity potential is zero, 具V典 = V1 = 0. After performing the averaging procedure for the Green’s function, we again arrive at the kinetic Eq. 共6兲 with the selfenergy given by the following formal expression: ˇ 丢 Vˇ + . . .兲兩 n ˇ = 共Vˇ + Vˇ 丢 G ⌺ 0 0 0 V →Vn ,
Finally, we would like to present several examples in which the disorder given by Eq. 共A1兲 is realized. The simplest example is given by Eq. 共30兲 and in this case Vn = niVn0. For the disorder given by Eq. 共44兲, we have Vn = ni具共Vi0兲n典dis. For the Gaussian white-noise disorder, only V2 is nonzero and Vn 兩n⫽2 = 0. For the telegraph white-noise disorder all odd correlators vanish, V2n+1 = 0. APPENDIX B: CALCULATION OF SELF-CONSISTENT ˆ R,A SELF-ENERGY ⌺ eq
The following relations can be calculated by a direct analytical integration of Eq. 共34兲:
␥ˆ R,A =
␥R =
共A2兲
0
ˆ R,A = EˆR,A兩 n ⌺ V →Vn , 0
ˆ ⬍ 丢 EˆA兲兩 n ˆ ⬍ = 共EˆR 丢 G ⌺ V →Vn ,
where the notation 兩Vn→Vn is formal and it means that EˆR,A 0 has to be fist expanded with respect to V0 and then after grouping V0’s together the substitution has to be applied. Equations 共14兲 and 共15兲 can now be rederived for nonuniform systems with the disorder given by Eq. 共A1兲 and ⌺ˆ ⬍ 1 ˆ ⬍ 丢 EˆA兲 兩 n . = 共EˆR 丢 G V0→Vn 1 Nevertheless, for the purposes of this paper, it is sufficient to consider the uniform and stationary case. This leads to substantial simplifications outlined in Secs. II B and II C. Results of Secs. II B and II C also hold for the disorder given by Eq. 共A1兲 with the exception of Eqs. 共20兲 and 共28兲 that should be replaced by the following equations:
R,A ⌺ˆ ER,A = Eˆeq
冊 册
d 2k ⬘ ˆ ⬍ A G 共k⬘兲 Eˆeq 共2兲2 1
, Vn0→Vn
冊 册
d2k⬘ ˆ R,A R,A G 共k⬘兲 Eˆeq 共2兲2 E
␥zR =
共K− − 2W兲关ln共K0 − K−兲 − ln共− K−兲兴 , 2共K− − K+兲
ln共K0 − K+兲 − ln共− K+兲 ln共K0 − K−兲 − ln共− K−兲 − , 共K− − K+兲/H 共K− − K+兲/H A R ␥A = 共␥R兲ⴱ ; ␥x共y,z兲 = 共␥x共y,z兲 兲ⴱ ; ␥Ry共z兲 = 0,
R R where W = − ⌺eq0 , H = h − ⌺eqz , K⫾ = 2共W 2 + ␣2 ⫿ 冑H2 + 2W␣2 + ␣4兲, and K0 = k0 describes the cutoff k0 in momentum integration. R R 共兲 and ⌺eqz 共兲 are calculated by For each energy, ⌺eq0 performing a number of iterations with the consequent iteration according to
共A4兲
0
冋 冉冕 冋 冉冕
共K+ − 2W兲关ln共K0 − K+兲 − ln共− K+兲兴 2共K− − K+兲
共A3兲
which, in analogy with the self-energy, also has retarded 共advanced兲 ER,A and lesser E⬍ components. Equation 共A3兲 can be rewritten in the form of T-matrix equation, Eˇ = Vˇ 丢 关1ˇ ˇ 丢 Eˇ兴, which leads to the expressions for the self-energies +G
ˆ ⬍ = EˆR ⌺ eq 1
ˆ R,A共k, 兲 ⬅ ␥R,Aˆ + ␥R,Aˆ , d2k/共2兲2G 0 z eq z
−
ˆ where Vˇ0 = V0共 0 0ˆ 兲␦共1 − 1⬘兲 and in the term of nth order pron portional to V0 we replace Vn0 by Vn which ensures that the correlators in Eq. 共A1兲 are properly considered. It is convenient to introduce the notation
ˇ 丢 Vˇ + ¯兲, Eˇ = 共Vˇ0 + Vˇ0 丢 G 0
冕
. Vn0→Vn
共A5兲
1 R,A = Tr关niV0共1ˆ − V0␥ˆ R,A兲−1ˆ 0兴, ⌺eq0 2 1 R,A = Tr关niV0共1ˆ − V0␥ˆ R,A兲−1ˆ z兴. ⌺eqz 2 The iterations are performed until the prescribed accuracy is reached. APPENDIX C: CALCULATION OF THE MATRIX ˆ ()
For the electric field E along the y axis E = 共0 , Ey兲, we solve here the linear Eq. 共40兲 for the elements of the matrix ˆ 共兲 by performing analytically the momentum integrations ˆ R,A共k , 兲 关given by Eq. 共34兲兴 in the of the Green’s functions G eq right-hand side. For each energy , we obtain the following expressions that also depend on the self-consistent values of R R 共兲 and ⌺0z 共兲: ⌺00
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+−共兲 = inFeEy␣ +
再
共K− − K+兲共K−ⴱ − K+兲共K+ − K+ⴱ 兲
再 冋
关K−ⴱ2 + 4共H + W兲共Hⴱ − Wⴱ兲兴ln共− K−ⴱ 兲 共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲
共K− − K+ⴱ 兲共K−ⴱ − K+ⴱ 兲共K+ − K+ⴱ 兲
共2H − K− + 2W兲共2Hⴱ + K− − 2Wⴱ兲ln共− K−兲 共K− − K−ⴱ 兲共K− − K+兲共K− − K+ⴱ 兲
再
−
共K− − K+兲共K−ⴱ − K+兲共K+ − K+ⴱ 兲
−
共K− −
共K− − K+ⴱ 兲共K−ⴱ − K+ⴱ 兲共K+ − K+ⴱ 兲 K+ⴱ 兲
共2H + K+ − 2W兲共2Hⴱ − K+ + 2Wⴱ兲ln共− K+兲
−
册
ⴱ niT++T−−
冎
−1
,
共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲
关K+ⴱ2 + 4共H − W兲共Hⴱ + Wⴱ兲兴ln共− K+ⴱ 兲
− K+兲共K− −
共K− − K+兲共K−ⴱ − K+兲共K+ − K+ⴱ 兲
共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲
关K−ⴱ2 + 4共H − W兲共Hⴱ + Wⴱ兲兴ln共− K−ⴱ 兲
共2H + K− − 2W兲共2Hⴱ − K− + 2Wⴱ兲ln共− K−兲 K−ⴱ 兲共K−
共2H − K−ⴱ + 2W兲共2Hⴱ + K−ⴱ − 2Wⴱ兲ln共− K−ⴱ 兲
共K− − K+ⴱ 兲共K−ⴱ − K+ⴱ 兲共K+ − K+ⴱ 兲
共K− − K−ⴱ 兲共K− − K+兲共K− − K+ⴱ 兲 −
−
冎
共2H − K+ⴱ + 2W兲共2Hⴱ + K+ⴱ − 2Wⴱ兲ln共− K+ⴱ 兲
关K−2 + 4共H − W兲共Hⴱ + Wⴱ兲兴ln共− K−兲
关K+2 + 4共H − W兲共Hⴱ + Wⴱ兲兴ln共− K+兲
再 冋
−
关K+ⴱ2 + 4共H + W兲共Hⴱ − Wⴱ兲兴ln共− K+ⴱ 兲
共K− − K+兲共K−ⴱ − K+兲共K+ − K+ⴱ 兲
⫻ −1+ +
−
共2H − K+ + 2W兲共2Hⴱ + K+ − 2Wⴱ兲ln共− K+兲
−+共兲 = inFeEy␣ +
共K− − K−ⴱ 兲共K− − K+兲共K− − K+ⴱ 兲
关K+2 + 4共H + W兲共Hⴱ − Wⴱ兲兴ln共− K+兲
⫻ −1+ +
关K−2 + 4共H + W兲共Hⴱ − Wⴱ兲兴ln共− K−兲
−
冎
共2H + K−ⴱ − 2W兲共2Hⴱ − K−ⴱ + 2Wⴱ兲ln共− K−ⴱ 兲 共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲
共2H + K+ⴱ − 2W兲共2Hⴱ − K+ⴱ + 2Wⴱ兲ln共− K+ⴱ 兲 共K− − K+ⴱ 兲共K−ⴱ − K+ⴱ 兲共K+ − K+ⴱ 兲
册
ⴱ niT−−T++
冎
−1
,
R 共兲. where T⫾⫾ corresponds to the elements of the matrix Tˆeq
APPENDIX D: CALCULATION OF THE MATRIX ˆ RE()
For the electric field E along the y axis E = 共0 , Ey兲, we solve here the linear Eq. 共41兲 for the elements of the matrix ˆ ER,A共兲 ˆ R,A共k , 兲 关given by Eq. 共34兲兴 in the rightby performing analytically the momentum integrations of the Green’s functions G eq R 共兲 and hand side. For each energy , we obtain the following expressions that also depend on the self-consistent values of ⌺00 R ⌺0z共兲:
ERy−−共兲 = ERy++共兲 = 0,
R R ERy+−共兲 = 4i␣关H共− 1 + ⌺00 兲 − W⌺0z 兴兵K−2 − K+2 + 2K−K+关− ln共− K−兲 + ln共− K+兲兴其
⫻兵共K− − K+兲关K−3K+ + K+3K− + K−2K+niT−−T++ − 4K+niT−−T++共H2 − W2兲 + K+2K−niT−−T++ − 4K−niT−−T++共H2 − W2兲 − 2K+2K−2 − 8K+K−niT−−T++W兴 + 2K−K+niT−−T++关4H2 + 共K− − 2W兲共− K+ + 2W兲兴关ln共− K−兲 − ln共− K+兲兴其−1 ,
ERy−+共兲 = − ERy+−共兲, R where T⫾⫾ corresponds to the elements of the matrix Tˆeq 共兲.
APPENDIX E: CALCULATION OF THE FERMI-SURFACE CONDUCTIVITY
For the electric field E along the y axis E = 共0 , Ey兲, we perform momentum k and frequency integrations in Eq. 共42兲. It is convenient to divide the resultant conductivity into two parts: the bare bubble part Ib xy共yy兲 that corresponds to calculating only 195129-14
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the second line in Eq. 共42兲 effectively assuming that ˆ 共F兲 = 0 and self-consistent part Isc xy共yy兲 that corresponds to calculating the first line in Eq. 共42兲 that takes into account correction due to self-consistent calculation of ˆ 共F兲, Isc Ixy = Ib xy + xy ,
共E1兲
Isc Iyy = Ib yy + yy .
共E2兲
Ib We arrive at analytical expressions for the bare bubble contributions to the conductivities Ib xy and yy that depend on the R R self-consistent values of ⌺00共F兲 and ⌺0z共F兲 at the Fermi surface
Ib 2i␣2关− Hⴱ共K− + 2W兲 + H共K− + 2Wⴱ兲兴ln共− K−兲 2i␣2关Hⴱ共K−ⴱ + 2W兲 − H共K−ⴱ + 2Wⴱ兲兴ln共− K−ⴱ 兲 xy − = − e2/ប 共K− − K−ⴱ 兲共K− − K+兲共K− − K+ⴱ 兲2 共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲2 −
2i␣2关− Hⴱ共K+ + 2W兲 + H共K+ + 2Wⴱ兲兴ln共− K+兲 共K− − K+兲共K−ⴱ − K+兲共K+ − K+ⴱ 兲2
−
2i␣2关Hⴱ共K+ⴱ + 2W兲 − H共K+ⴱ + 2Wⴱ兲兴ln共− K+ⴱ 兲 共K− − K+ⴱ 兲共K−ⴱ − K+ⴱ 兲共K+ − K+ⴱ 兲2
,
共E3兲
Ib K−2共K− − 3K+兲K+ⴱ ln共− K−兲 K−3关K−共K− + K−ⴱ 兲 − 共K− + 3K−ⴱ 兲K+ + 2K+2兴ln共− K−兲 K−K−ⴱ2共K−ⴱ − 3K+ⴱ 兲ln共− K−ⴱ 兲 yy = − − + e2/ប 4共K− − K+兲3共K− − K+ⴱ 兲2 4共K− − K−ⴱ 兲共K− − K+兲3共K− − K+ⴱ 兲2 4共K− − K−ⴱ 兲共K−ⴱ − K+ⴱ 兲32 + − + − − + − + −
K−ⴱ3关K−ⴱ 共K−ⴱ + K+兲 − 共K−ⴱ + 3K+兲K+ⴱ + 2K+ⴱ2兴ln共− K−ⴱ 兲 4共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲32 K+3关2K−2 + K+共K−ⴱ + K+兲 − K−共3K−ⴱ + K+兲兴ln共− K+兲 4共K−ⴱ
− K+兲共K− − K+兲 共K+ − 3
+
K+2共3K− − K+兲K+ⴱ ln共− K+兲 4共K− − K+兲3共K+ − K+ⴱ 兲2
K−K+ⴱ2共3K−ⴱ − K+ⴱ 兲ln共− K+ⴱ 兲 4共K− − K+ⴱ 兲共K−ⴱ − K+ⴱ 兲32
K+ⴱ3关2K−ⴱ2 + K+ⴱ 共K+ + K+ⴱ 兲 − K−ⴱ 共3K+ + K+ⴱ 兲兴ln共− K+ⴱ 兲 4共K− − K+ⴱ 兲共K−ⴱ − K+ⴱ 兲3共K+ − K+ⴱ 兲2 K−K+共K−ⴱ2 + K+ⴱ2兲 − K−2共K−ⴱ2 − K−ⴱ K+ⴱ + K+ⴱ2兲 − K+2共K−ⴱ2 − K−ⴱ K+ⴱ + K+ⴱ2兲 2共K− − K+兲2共K−ⴱ − K+ⴱ 兲22 兵4WWⴱ␣2 + 2HHⴱ共K− − 2␣2兲 − K−2共W + Wⴱ + ␣2兲 + 2K−关WWⴱ + 共W + Wⴱ兲␣2兴其ln共− K−兲 共K− − K−ⴱ 兲共K− − K+兲共K− − K+ⴱ 兲2 兵4WWⴱ␣2 + 2HHⴱ共K−ⴱ − 2␣2兲 − K−ⴱ2共W + Wⴱ + ␣2兲 + 2K−ⴱ 关WWⴱ + 共W + Wⴱ兲␣2兴其ln共− K−ⴱ 兲 共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲2 兵4WWⴱ␣2 + 2HHⴱ共K+ − 2␣2兲 − K+2共W + Wⴱ + ␣2兲 + 2K+关WWⴱ + 共W + Wⴱ兲␣2兴其ln共− K+兲 共K− − K+兲共K−ⴱ − K+兲共K+ − K+ⴱ 兲2 兵4WWⴱ␣2 + 2HHⴱ共K+ⴱ − 2␣2兲 − K+ⴱ2共W + Wⴱ + ␣2兲 + 2K+ⴱ 关WWⴱ + 共W + Wⴱ兲␣2兴其ln共− K+ⴱ 兲 共K− − K+ⴱ 兲共− K−ⴱ + K+ⴱ 兲共− K+ + K+ⴱ 兲2 − 2H2 + 共K− + K+ − 2W兲W 关共K− + K+兲共4H2 + K−K+兲 − 8K−K+W − 4共K− + K+兲W2兴␣2 − 共K− − K+兲22 2K−共K− − K+兲2K+2
− 关ln共− K−兲 − ln共− K+兲兴 −
K+ⴱ 兲2
−
W关− 2K−K+ + 共K− + K+兲W兴 + 关− K−K+ + 2共K− + K+兲W + 4W2兴␣2 + H2共K− + K+ − 4␣2兲 共K− − K+兲32
− 2Hⴱ2 + 共K−ⴱ + K+ⴱ − 2Wⴱ兲Wⴱ 共K−ⴱ − K+ⴱ 兲22
−
关共K−ⴱ + K+ⴱ 兲共4Hⴱ2 + K−ⴱ K+ⴱ 兲 − 8K−ⴱ K+ⴱ Wⴱ − 4共K−ⴱ + K+ⴱ 兲Wⴱ2兴␣2 2K−ⴱ 共K−ⴱ − K+ⴱ 兲2K+ⴱ 2
− 关ln共− K−ⴱ 兲 − ln共− K+ⴱ 兲兴 ⫻
Wⴱ关− 2K−ⴱ K+ⴱ + 共K−ⴱ + K+ⴱ 兲Wⴱ兴 + 关− K−ⴱ K+ⴱ + 2共K−ⴱ + K+ⴱ 兲Wⴱ + 4Wⴱ2兴␣2 + Hⴱ2共K−ⴱ + K+ⴱ − 4␣2兲 共K−ⴱ − K+ⴱ 兲32
,
共E4兲
R R 共F兲, H = h − ⌺0z 共F兲, and K⫾ = 2共W where in this appendix all parameters are taken at the Fermi surface W = F − ⌺00 2 冑 2 2 4 + ␣ ⫿ H + 2W␣ + ␣ 兲.
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(a)
(b)
(c)
(d)
(e)
(f)
Ib Isc II Ib Isc II FIG. 13. 共Color online兲 The anomalous Hall conductivity Tot xy = xy + xy + xy and its components 共xy , xy , xy 兲 vs the averaged A 2 relaxation rate 1 / = 2 Im⌺00 共defined in Appendix B兲. The spin-orbit interaction strength is 2m␣ / Eres = 3.59 共Eres = 10h兲; the strength of impurities V0 = 0.01, 0.1, 0.2, 0.3; the Fermi energy F / Eres = 0.9 for F = 0, F / Eres = 0.5 for F = −4h, and F / Eres = 1.5 for F = 6h. The Fermi energy F corresponds here to the clean system and it is renormalized according to Eq. 共35兲 in the presence of disorder. Dimensionality of quantities displayed in this plot is restored.
Isc The analytical expressions for the self-consistent contributions to the conductivities Isc xy and yy become ⴱ ⴱ Isc ␣ni兵+−T++T−− 关K−ⴱ2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 − −+T−−T++ 关K−ⴱ2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K−ⴱ 兲 xy = e2/ប 2共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲2eEy
−
ⴱ ⴱ ␣ni兵+−T++T−− 关K−2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 − −+T−−T++ 关K−2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K−兲 2共K− − K−ⴱ 兲共K− − K+兲共K− − K+ⴱ 兲2eEy
+
ⴱ ⴱ ␣ni兵+−T++T−− 关K+ⴱ2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 − −+T−−T++ 关K+ⴱ2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K+ⴱ 兲 2共K− − K+ⴱ 兲共− K−ⴱ + K+ⴱ 兲共− K+ + K+ⴱ 兲2eEy
−
ⴱ ⴱ ␣ni兵+−T++T−− 关K+2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 − −+T−−T++ 关K+2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K+兲 , ⴱ 2共K− − K+兲共K− − K+兲共K+ − K+ⴱ 兲2eEy
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(a)
(b)
(c)
(d)
(e)
(f)
FIG. 14. 共Color online兲 Identical to Fig. 13 plot with attractive disorder 共V0 = −0.01, −0.1, −0.2, and −0.3兲. ⴱ ⴱ Isc 关K−2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 + −+T−−T++ 关K−2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K−兲 i␣ni兵+−T++T−− yy = e2/ប 2共K− − K−ⴱ 兲共K− − K+兲共K− − K+ⴱ 兲2eEy
− + −
ⴱ ⴱ 关K−ⴱ2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K−ⴱ 兲 关K−ⴱ2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 + −+T−−T++ i␣ni兵+−T++T−−
2共K− − K−ⴱ 兲共K−ⴱ − K+兲共K−ⴱ − K+ⴱ 兲2eEy ⴱ ⴱ 关K+2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K+兲 关K+2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 + −+T−−T++ i␣ni兵+−T++T−−
2共K− − K+兲共K−ⴱ − K+兲共K+ − K+ⴱ 兲2eEy ⴱ ⴱ 关K+ⴱ2 + 4共H + W兲共Hⴱ − Wⴱ兲兴 + −+T−−T++ 关K+ⴱ2 + 4共H − W兲共Hⴱ + Wⴱ兲兴其ln共− K+ⴱ 兲 i␣ni兵+−T++T−−
2共K− − K+ⴱ 兲共− K−ⴱ + K+ⴱ 兲共− K+ + K+ⴱ 兲2eEy
,
共E6兲
where again all parameters are calculated at the Fermi surface and T⫾⫾ and ⫾⫿ correspond to the elements of the matrices R 共F兲 and ˆ 共F兲, respectively. Tˆeq APPENDIX F: CALCULATION OF THE FERMI SEA CONDUCTIVITY
For the electric field E along the y axis E = 共0 , Ey兲, we perform momentum integrations in Eq. 共43兲 arriving at the following expressions for conductivities IIxy and IIyy: 195129-17
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IIb xy =− e2/ប −
IIsc xy = e2/ប
冕
dnF
冕
dnF
再
IIsc IIxy = IIb xy + xy ,
共F1兲
IIyy = 0,
共F2兲
R R 4i␣2关H共1 − ⌺00 兲共K− + K+兲 + ⌺0z 关K+W + K−共K+ + W兲兴兴 K−K+共K− − K+兲22
冎
R R 共K− + K+ + 4W兲兴关ln共− K−兲 − ln共− K+兲兴 兲H + ⌺0z 2i␣2关4共1 − ⌺00 + c.c., 共K− − K+兲32
␣共ERy−+ − ERy+−兲共4H2 + K−K+ − 4W2兲共K−2 − K+2 + 2K−K+关− ln共− K−兲 + ln共− K+兲兴兲 2K−K+共K+ − K−兲32/共T−−T++兲
R where T⫾⫾ corresponds to the elements of the matrix Tˆeq 共兲. IIsc 2 The fact that xy = 0 follows from the identity 4H + K−K+ − 4W2 ⬅ 0. As one can see, IIyy and IIsc xy contributions to the Fermi sea Hall conductivity vanish and the nonvanishing contribuR tion IIb xy depends on the self-consistently calculated ⌺00共兲 R and ⌺0z共兲 that are functions of energy. Calculation of IIb xy from Eq. 共F3兲 requires numerical integration over .
APPENDIX G: DETAILED RESULTS FOR THE HALL CONDUCTIVITY
In order to gain more insight into the behavior of the anomalous Hall effect, in Figs. 13 and 14 we plot different components of the AHE conductivity, particularly the Fermi sea contribution IIxy, the bare bubble contribution Ib xy 共this corresponds to Iint xy in Ref. 1兲, and the self-consistent contriext bution Isc xy 共this corresponds to xy in Ref. 1兲. In Fig. 13, we take the same parameters as in Figs. 7 and 8 of Ref. 1 and we find disagreement with Ref. 1 in the results for the contribuIsc Ib II tion ext xy 共xy 兲. The contributions xy and xy perfectly agree with Ref. 1. In the clean limit → ⬁, we see that Isc xy and thus the total diverge. This divergence 关Isc Hall conductivity Tot xy xy
1 S.
Onoda, N. Sugimoto, and N. Nagaosa, Phys. Rev. B 77, 165103 共2008兲. 2 R. Shindou and L. Balents, Phys. Rev. B 77, 035110 共2008兲. 3 E. H. Hall, Philos. Mag. 10, 301 共1880兲. 4 W. Thomson, Proc. R. Soc. London 8, 546 共1856兲. 5 R. Karplus and J. M. Luttinger, Phys. Rev. 95, 1154 共1954兲. 6 Y. Taguchi, Y. Oohara, H. Yoshizawa, N. Nagaosa, and Y. Tokura, Science 291, 2573 共2001兲. 7 T. Jungwirth, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 88, 207208 共2002兲. 8 M. Onoda and N. Nagaosa, J. Phys. Soc. Jpn. 71, 19 共2002兲. 9 Y. Yao, L. Kleinman, A. H. MacDonald, J. Sinova, T. Jungwirth,
共F3兲
+ c.c. = 0,
共F4兲
⬃ 1 / 共niV0兲 in regions 共ii兲 and 共iii兲 and Isc xy ⬃ 1 / ni in region 共i兲, see Fig. 2兴 is due to the skew scattering. The conductivity Isc xy also contains the side-jump contribution which can be best seen in Fig. 13共a兲 in the sharp peak in the Isc xy conductivity for small 1 / . The skew-scattering contribution decays much faster compared to the side-jump and/or intrinsic mechanisms as we go to larger 1 / . As a result, we can expect a crossover between the region dominated by the skew scattering and the region dominated by the side-jumpintrinsic mechanisms. When both subbands are partially occupied 关see Figs. 13共f兲 and 14共f兲兴, the higher order skew scattering is still present. However, we do not expect a wellpronounced crossover as the intrinsic contribution cancels the side-jump contribution in the metallic regime 关see Eq. 共45兲兴. By comparing Figs. 13共f兲 and 14共f兲, one can see that the higher order skew scattering 共hybrid skew scattering兲 共Ref. 24兲 does not change sign when we change the sign of impurities. When the side-jump-intrinsic and the skew-scattering components have opposite signs, as in Fig. 13, we observe the AHE sign change instead of the crossover. In Figs. 13共a兲–13共d兲, the skew scattering is negative in the clean limit while the side-jump-intrinsic part is positive. This inevitably leads to the sign change of the conductivity xy as we increase the disorder.
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