Anomalous Hall effect in multiband disordered systems: from the metallic to the hopping regime Workshop on Spintronics and Low Dimensional Magnetism June

Anomalous Hall effect in multiband disordered systems: from the metallic to the hopping regime

Workshop on Spintronics and Low Dimensional Magnetism June 17th, 2010

Fudam University, Shanghai, China

Research fueled by:

JAIRO SINOVATexas A&M University

Institute of Physics ASCR

Hitachi CambridgeJ. Wunderlich, A. Irvine, et al

Institute of Physics ASCRTomas Jungwirth, et al

UCLA A. Kovalev, Y. Tserkovnyak

Texas A&M UniversityXiong-Jun Liu, Xin Liu (Nankai)

Workshop on Spintronics and Low Dimensional Magnetism, Fudam University, Shanghai, China

Valenzuela et al Nature 06

Inverse SHE

Anomalous Hall effect: more than meets the eye

Wunderlich, Kaestner, Sinova, Jungwirth PRL 04

Kato et al Science 03

IntrinsicExtrinsic

V

Mesoscopic Spin Hall Effect

Intrinsic

Brune,Roth, Hankiewicz, Sinova, Molenkamp, et al Nature Physics 10

Wunderlich, Irvine, Sinova, Jungwirth, et al, Nature Physics 09

Spin-injection Hall Effect

Anomalous Hall Effect

I

_ FS

OFS

O

_ _majority

minority

V

Spin Hall Effect

I

_ FS

OFS

O

_ _

V

0/1


Simple electrical measurement of out of plane magnetization

InMnAs

Spin dependent “force” deflects like-spin particles

ρH=R0B ┴ +4π RsM┴

Anomalous Hall Effect: the basics

I

_ FSO

FSO

_ __

majority

minority

V

1/1


Anomalous Hall effect (scaling with ρ)

Dyck et al PRB 2005

Kotzler and Gil PRB 2005

Co films

Edmonds et al APL 2003

GaMnAs

Material with dominant skew scattering mechanismMaterial with dominant scattering-independent mechanism

Although a more correct scaling was demonstrated recently by Tian, Ye, and X.F. Jin PRL 09

1/2


Phenomenological regimes of AHE

Review of AHE (RMP 2010), Nagaosa, Sinova, Onoda, MacDonald, Ong

1. A high conductivity regime for σxx>106 (Ωcm)-1 in which AHE is skew dominated2. A good metal regime for σxx ~104-106 (Ωcm) -1 in which σxy

AH~ const3. A bad metal/hopping regime for σxx<104 (Ωcm) -1 for which σxy

AH~ σxyα with α>1

Skew dominated regime

2/3


Scattering independent regime

3/5


Hopping conduction regime

•Approximate scaling seen as a function of T•Diagonal hopping conductivity for most systems showing approximate scaling•phenomenology gives σxy~σxx

1.5-1.7 over a few decades

5/8

•1880-81: Hall discovers the Hall and the anomalous Hall effect

The tumultuous history of AHE

•1970: Berger reintroduces (and renames) the side-jump: claims that it does not vanish and that it is the dominant contribution, ignores intrinsic contribution. (problem: his side-jump is gauge dependent)

Berger

Luttinger

•1954: Karplus and Luttinger attempt first microscopic theory: they develop (and later Kohn and Luttinger) a microscopic theory of linear response transport based on the equation of motion of the density matrix for non-interacting electrons, ; run into problems interpreting results since some terms are gauge dependent. Lack of easy physical connection.

Hall

•1970’s: Berger, Smit, and others argue about the existence of side-jump: the field is left in a confused state. Who is right? How can we tell? Three contributions to AHE are floating in the literature of the AHE: anomalous velocity (intrinsic), side-jump, and skew contributions.

•1955-58: Smit attempts to create a semi-classical theory using wave-packets formed from Bloch band states: identifies the skew scattering and notices a side-step of the wave-packet upon scattering and accelerating. .Speculates, wrongly, that the side-step cancels to zero.

The physical interpretation of the cancellation is based on a gauge dependent object!!

•1972-87: Detailed quantitative by Fert and Levy understanding of resonant skew scattering: paramagnetic systems with imbedded magnetic impurities (d and f). 5/8

The tumultuous history of AHE: last three decades

•2004’s: Spin-Hall effect is revived by the proposal of intrinsic SHE (from two works working on intrinsic AHE): AHE comes to the masses, many debates are inherited in the discussions of SHE.

•1980’s: Ideas of geometric phases introduced by Berry; QHE discoveries

•2000’s: Materials with strong spin-orbit coupling show agreement with the anomalous velocity contribution: intrinsic contribution linked to Berry’s curvature of Bloch states. Ignores disorder contributions.

•2004-10’s: Linear theories in simple models treating SO coupling and disorder finally merge: full semi-classical theory developed and microscopic approaches are in agreement among each other in simple models.

•2010: More detailed ab-initio studies and formalism developed for metals; hopping regime begins to be understood 8/13


Cartoon of the mechanisms contributing to AHEindependent of impurity density

Electrons have an “anomalous” velocity perpendicular to the electric field related to their Berry’s phase curvature which is nonzero when they have spin-orbit coupling.

Electrons deflect to the right or to the left as they are accelerated by an electric field ONLY because of the spin-orbit coupling in the periodic potential (electronics structure)

E

SO coupled quasiparticles

Intrinsic deflection B

Electrons deflect first to one side due to the field created by the impurity and deflect back when they leave the impurity since the field is opposite resulting in a side step. They however come out in a different band so this gives rise to an anomalous velocity through scattering rates times side jump.

independent of impurity density

Side jump scatteringVimp(r) (Δso>ħ/τ) ∝ λ*∇Vimp(r) (Δso<ħ/τ)

B

Skew scattering

Asymmetric scattering due to the spin-orbit coupling of the electron or the impurity. Known as Mott scattering.

~σ~1/niVimp(r) (Δso>ħ/τ) ∝ λ*∇Vimp(r) (Δso<ħ/τ) A

13/21


Microscopic vs. Semiclassical

• Boltzmann semiclassical approach: easy physical interpretation of different contributions (used to define them) but very easy to miss terms and make mistakes. Must be confirmed microscopically. How one understands but not necessarily computes the effect.

• Kubo approach: systematic formalism but not very transparent.

• Keldysh approach: also a systematic kinetic equation approach (equivalent to Kubo in the linear regime). In the quasi-particle limit it must yield Boltzmann semiclassical treatment.

21/34


Kubo microscopic approach to transport: diagrammatic perturbation theory

Averaging procedures: = 1/ τν0 = ν0τ

= +

Bloch ElectronReal Eigenstates

Need to perform disorder average (effects of scattering)

n, q

Drude Conductivityσ = ne2 τ /m*~1/ni

Vertex Corrections∼ 1-cos(θ)

Perturbation Theory: conductivity

n, q

34/55


“Skew scattering”

“Side-jump scattering”

Intrinsic AHE: accelerating between scatterings

n, q

n, q m, p

m, pn’, k

n, q

n’≠n, q

Early identifications of the contributions

Vertex Corrections∼ σIntrinsic ~ τ0 or n0

i

Intrinsic∼σ0 /εFτ~ τ0 or n0

i

Kubo microscopic approach to AHE

n, q

n, q m, p

m, pn’, k

matrix in band index

m’, k’

55/89


Defining intrinsic anomalous Hall effect in materials where AHE is dominated by scattering-independent mechanisms

n, q

n’≠n, q

89/144


Scattering independent regime

Q: is the scattering independent regime dominated by the intrinsic AHE?

144/233


intrinsic AHE approach in comparing to experiment: phenomenological “proof”

n, q

n’≠n, q

• DMS systems (Jungwirth et al PRL 2002, APL 03)• Fe (Yao et al PRL 04)• layered 2D ferromagnets such as SrRuO3 and

pyrochlore ferromagnets [Onoda et al (2001),Taguchi et al., Science 291, 2573 (2001), Fang et al Science 302, 92 (2003)

• colossal magnetoresistance of manganites, Ye et~al Phys. Rev. Lett. 83, 3737 (1999).

• CuCrSeBr compounts, Lee et al, Science 303, 1647 (2004)

Experiment σAH ∼ 1000 (Ωcm)-1

TheroyσAH ∼750 (Ωcm)-1

AHE in Fe

AHE in GaMnAs

233/377


Contributions understood in simple metallic 2D models

Semi-classical approach:Gauge invariant formulation

Sinitsyn, Sinvoa, et al PRB 05, PRL 06, PRB 07

Kubo microscopic approach:in agreement with semiclassical

Borunda, Sinova, et al PRL 07, Nunner, JS, et al PRB 08

Non-Equilibrium Green’s Function (NEGF) microscopic approach

Kovalev, Sinova et al PRB 08, Onoda PRL 06, PRB 08

610/987


Comparing Boltzmann to Kubo (chiral basis) for “Graphene” model

Kubo identifies, without a lot of effort, the order in ni of the diagrams BUT not so much their physical interpretation according to semiclassical theory

Sinitsyn et al 2007

ϕ


Generalization to 3DGeneral band structure in the presence of

delta-correlated Gaussian disorderN-band projected Hamiltonian expressed via envelope fields

In the presence of Gaussian disorder

To test our theory we will consider band structures of a 2D Rashba and 3D Luttinger models:

6-band Luttinger model will also be consideredKovalev, Sinova, Tserkovnyak PRL 2010

1/4


Idea behind this calculation

1. Use Kubo-Streda formalism or linearized version of Keldysh formalism to obtain

where

2. Integrate out sharply peaked Green’s functions which leads to integrals over the Fermi sphere and no dependence on disorder

3. In order to identify the relevant terms in the strength of disorder it is convenient to use diagrams (Synistin et al PRB 2008)

Kovalev, Sinova, Tserkovnyak PRL 20102/4


AHE conductivity expressed through band structure

Well known intrinsic contribution

Side jump contribution related to Berry curvature (arises from unusual disorder broadening term usually missed)

Remaining side jump contribution (usual ladder diagrams)

Kovalev, Sinova, Tserkovnyak PRL 2010

3/4


Results for Luttinger model: simplest model of GaMnAs


4/4


Nagaosa, Sinova, Onoda, Ong, MacDonald et al RMP 10


5/4


Anomalous Hall effect in the hopping regime

Three regimes of anomalous Hall effect

1. Intrinsic regime:

2. Skew scattering:

3. Hopping regime:

This scaling relation has been seen in many experiments:B. A. Aronzon etal., phys. stat. sol. (b) 218, 169 (2000) .W. Allen,etal., Phys. Rev. B 70, 125320 (2004) .H. Toyosaki etal., Nature, 3, 221 (2004) K. Ueno etal., Appl. Phys. Lett. 90, 072103 (2007) T. Miyasato etal., Phys. Rev. Lett., 99, 086602 (2007) D. Venkateshvaran etal., Phys. Rev. B 78, 092405 (2008) A. Fernández-Pacheco etal., Phys. Rev. B 77, 100403(R) (2008)

Previous Theories: S. H. Chun etal., Phys. Rev. Lett. 84, 757 (2000), A. A. Burkov and L. Balents, Phys. Rev. Lett. 91, 057202 (2003),

Cannot explain the scaling relation in the experiment.1/9


Ueno et al 2007

It is an approximate scaling in several materials

2/9


A small warning on “guides to the eye”

2/9

Workshop on Spintronics and Low Dimensional Magnetism, Fudam University, Shanghai, China 2/9

A small warning on “guides to the eye”


Phonon-assisted hopping between localized states

The Hamiltonian describing localized states and e-ph interaction (Holstein 1961, Burkov etal, 2003)

with

i j k

phononLocalization:

phonon-assisted

hopping

Electric current between two sites:

Ri

RjRk

: direct conductance due to two-site hopping.

: off-diagonal conductance due to three-site hopping.

3/9


Recall the charge current:

Perturbation method for the present triad:

a) First step: applying the external electric field, neglecting the Hall current, then calculating the voltage of each node according to the Kirchoff’s equation:

b) Second step: turning off the electric field, and considering only above calculated Hall current flowing through each node. One can then calculate the Hall current between each two sites, and the corresponding Hall voltage, with which one can finally obtain the Hall conductivity.

The Hall current through each node is then calculated with the voltage obtained in the first step:

Perturbative method to calculate the Hall conductivity

4/9


For example, we calculate the Hall current flowing between site 2 and 3. For this we first derive the voltage difference between them due to the present current situation:

2

3

1

The corresponding field is:

Then the Hall current flowing between site 2 and site 3 is given by:

Recall the original current when the external electric field is applied:

Perturbative method to calculate the Hall conductivity (II)

5/9


Linear response to the electric field

The charge current in the presence of external electric field is given by:

with Responsible for the normal conductivity

Responsible for the Hall conductivity: arises from interference of two phonon hopping process

Transverse voltage

yields zero when the magnetization is zero

and

6/9


Percolation theory

For each two site, a bond that can conduct electricity establishes when the direct conductance between them is no less than a critical value (Ambegaokar et al., 1971):

This gives the condition:

Ri Rj connected

Ri Rj disconnected

For a site with energy , the average number of sites connecting to it satisfying above condition is given by:

7/9

Critical paths appear when the average number :


The macroscopic AH conductivity/resistivity should be averaged along the critical path:

Anomalous Hall conductivity via percolation theory in hopping regime

Instead of exactly calculating the above extremely complicated integral, we shall evaluate its two limits: the lower limit and upper limit, which will be more meaningful. This is because, in the insulating regime, the system is very complicated and the relation between normal and Hall conductivity cannot be uniquely determined by a few parameters. Therefore, in this regime it is more reasonable to consider the scaling relation between them has a range.

Xiong-Jun Liu, Sinova 2010

8/9


The lower and upper limits are given by (note the differences in the constrain)

For the Mott variable range hopping, we find

Physically, the upper limit (γ=1.34) corresponds to the situation that most triads in the system are equilateral triangles (this is actually a regular distribution), while for the lower limit (γ=1.68) the geometry of triads is randomly distributed. Since the impurity sites are randomly distributed, we expect the realistic result is usually close to the lower limit. This is true for the experiment.

Xiong-Jun Liu, Sinova, 2010

9/9


SUMMARY

10/9

AHE general theory for metallic multi-band systems which contains all scattering-independent contributions developed: useful for ab-initio studies(Kovalev, Sinova, Tserkovnyak PRL 2010)

AHE hopping regime approximate scaling arises directly from a generalization of the Holstein theory to AHE (Xiong-Jun Liu, Sinova 2010)

AHE hopping regime scaling remains even when crossing to different types of insulating hopping regimes, only algebraic pre-factor changes

36

Nanoelectronics, spintronics, and materials control by spin-orbit coupling

Allan MacDonald U of Texas

Tomas JungwirthTexas A&M U.

Inst. of Phys. ASCRU. of Nottingham

Joerg WunderlichCambridge-Hitachi

Laurens MolenkampWürzburg

Xiong-Jun LiuTexas A&M U.

Mario BorundaTexas A&M Univ.

Harvard Univ.Nikolai SinitsynTexas A&M U.

U. of TexasLANL

Alexey KovalevTexas A&M U.

UCLA

Liviu ZarboTexas A&M Univ.

ASCR

Xin LiuTexas A&M U.

Ewelina Hankiewicz(Texas A&M Univ.)

Würzburg University

Principal Collaborators

Gerrit BauerTU Delft

Bryan GallagherU. of Nottingham

Y. TserkovnyakUCLA

Documents

Anomalous Hall effect in multiband disordered systems: from the metallic to the hopping regime Workshop on Spintronics and Low Dimensional Magnetism June