Download Kovalev, Sinova, Tserkovnyak PRL 2010

Theory of the anomalous hall effect: from the metallic fully
ab-initio studies to the insulating hopping systems
JAIRO SINOVA
Texas A&M University
Institute of Physics ASCR
Institute of Physics ASCR
Texas A&M University
Jülich Forschungszentrum
Xiong-Jun Liu
UCLA
Yuriy Mokrousov, F. Freimuth, H. Zhang, J.
A. Kovalev
Weischenberg, Stefan Blügel
German Physical Society Meeting
March 26th, 2012
Berlin, Germany
Research fueled by:
Outline
1. Introduction
• Anomalous Hall effect phenomenology: more than meets the eye
2. AHE in the metallic regime
•
•
•
•
Anomalous Hall effect (AHE) in the metallic regime
Understanding of the different mechanisms
Full theory of the scattering-independent AHE: beyond intrinsic
ab-initio studies of simple ferromagnets
3. Scaling of the AHE in the insulating regime
•
•
•
•
Experiments and phenomenology
phonon-assisted hopping AHE (Holstein)
Percolations theory generalization for the AHE conductivity
Results
4. Summary
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Anomalous Hall effect: more than meets the eye
Spin Hall Effect
Anomalous Hall Effect
_
_
FS
O
FS
majority
_
_
I
O
FS
O
FS
I
O
minority
V
Inverse SHE
V
Wunderlich, Kaestner, Sinova,
Jungwirth PRL 04
Intrinsic
Mesoscopic Spin Hall Effect
Kato et al
Science 03
Extrinsic
Spin-injection Hall Effect
V
Intrinsic
Valenzuela et al
Nature 06
Brune,Roth, Hankiewicz, Sinova,
Molenkamp, et al Nature Physics 10
Wunderlich, Irvine, Sinova,
Jungwirth, et al, Nature Physics 09
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Anomalous Hall Effect: the basics
Spin dependent “force” deflects like-spin particles
_
__
majority
FSO
FSO
ρH=R0B ┴ +4π RsM┴
I
minority
V
Simple electrical measurement
of out of plane magnetization
InMnAs
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Anomalous Hall effect (scaling with ρ)
Co films
Dyck et al PRB 2005
Edmonds et al APL 2003
GaMnAs
Kotzler and Gil PRB 2005
Material with dominant skew scattering mechanism
Material with dominant scattering-independent mechanism
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Cartoon of the mechanisms contributing to AHE in the metallic regime
Intrinsic deflection B
independent of
impurity density
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
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.
Side jump scattering B
Vimp(r) (Δso>ħ/τ)
 λ*Vimp(r) (Δso<ħ/τ)
independent of impurity density
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.
Skew scattering
A
~σ~1/ni
Vimp(r) (Δso>ħ/τ)  λ*Vimp(r) (Δso<ħ/τ)
Asymmetric scattering due to the spin-orbit
coupling of the electron or the impurity.
Known as Mott scattering.
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Anomalous Hall Effect
Nagaosa, Sinova, Onoda, MacDonald, Ong
2
1
3
2
1
3
1. A high conductivity regime for σxx>106 (Ωcm)-1 in which AHE is skew dominated
2. A good metal regime for σxx ~104-106 (Ωcm) -1 in which σxyAH~ const
3. A bad metal/hopping regime for σxx<104 (Ωcm) -1 for which σxyAH~ σxyα with α=1.4~1.7
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Intrinsic AHE approach in comparing to experiment: phenomenological “proof”
• 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
AHE in GaMnAs
(2001), Fang et al Science 302, 92 (2003)
• Colossal magnetoresistance of manganites, Ye et~al Phys.
Rev. Lett. 83, 3737 (1999).
• CuCrSeBr compounds, Lee et al,
Science 303, 1647 (2004)
AHE in Fe
Experiment
AH  1000 (cm)-1
Theroy
AH 750 (cm)-1
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Scattering independent regime: towards a theory applicable to real materials
Q: is the scattering independent regime dominated by the intrinsic AHE?
Challenge: can we formulate a full theory of the ALL the scattering
independent contributions that can be coupled to ab-initio techniques?
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Contributions understood in simple metallic 2D models
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
Semi-classical approach:
Gauge invariant formulation
Sinitsyn, Sinvoa, et al PRB 05, PRL 06, PRB 07
Kovalev, Sinova et al PRB 08,
Onoda PRL 06, PRB 08
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Comparing Boltzmann to Kubo (chiral basis) for “Graphene” model
Sinitsyn et al 2007
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
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Generalization to 3D
General band structure in the presence of
delta-correlated Gaussian disorder
N-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 phenomenological models applicable to DMSs:
Kovalev, Sinova, Tserkovnyak PRL 2010
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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 2010
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Scattering independent 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
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Application to simple ferromagnets
15
16
17
18
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Outline
1. Introduction
• Anomalous Hall effect phenomenology: more than meets the eye
2. AHE in the metallic regime
•
•
•
•
Anomalous Hall effect (AHE) in the metallic regime
Understanding of the different mechanisms
Full theory of the scattering-independent AHE
ab-initio studies of simple ferromagnets
3. Scaling of the AHE in the insulating regime
•
•
•
•
Experiments and phenomenology
phonon-assisted hopping AHE (Holstein)
Percolations theory generalization for the AHE conductivity
Results
4. Summary
20
AHE in hopping conduction regime
This scaling has been confirmed in many experiments. Below are some examples:
H.Toyosaki etal (2004)
S. Shen etal (2008)
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In magnetite thin films:
A. Fernández-Pacheco etal (2008)
Deepak Venkateshvaran etal (2008)
•Microscopic mechanisms?
•Why is irrespective of material?
•Why doesn’t it depend on type of
conduction?
S. H. Chun et al., PRL 2000; Lyanda-Geller et al., PRB 2001 (theory for manganites;
A. A. Burkov and L. Balents, PRL 2003;
)
22
Phonon-assisted hopping
The minimal Hamiltonian for the AHE in insulating regime:
represents the local on-site total angular-momentum state.
localization
i
j
k
phonon
Longitudinal hopping charge transport
1. Two-site direct hopping with one-phonon process :
i
j
k
: responsible for longitudinal conductance.
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Phonon-assisted hopping: Hall charge transport
2. How to capture the Hall effect? Three-site hopping (Holstein, 1961)
The transition must break the time-reversal (TR) symmetry
Two-site direct hopping
preserves the TR symmetry.
Need three site hopping
Interference
term
Hall transition rate
Geometric phase: break TR symmetry
m: the number of real phonons included in the whole transition.
Including these triads the electric current between two sites is:
: direct conductance due to two-site hopping.
: off-diagonal conductance due to hopping via three-sites.
Challenging: Macroscopic anomalous Hall conductivity/resistivity?
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Percolation theory for AHE: the resistor network
1. Map the random impurity system to a random resistor network based on direct
conductance:
Treated as perturbation
2. Introduce the cut-off to redefine the connectivity (Ambegaokar etal., 1971):
connected
disconnected
Cut-off
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Percolation theory for AHE: the resistor network
3. Percolation path/cluster
For a site with energy
, the average number of sites connecting to it under the
condition (Pollak, 1972)
:
the probability that the n-th smallest resistor connected to the i site has the
conductance larger than
.
Percolating path/cluster appears when the averaging connectivity (G.E. Pike, etal 1974):
Percolation path/cluster
4. Configuration averaging of general m-site function along the critical path/cluster, with
the i-th site has at least
sites connected to it:
Transverse resistivity/conductivity: each site has at least three sites connected to it?
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Macroscopic anomalous Hall conductivity/resistivity
Percolation path/cluster appears when
(G.E. Pike, etal 1974):
The averaging transverse voltage is given by:
In the thermodynamic limit, we get the AHC:
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The approach
However, instead of an exact calculation, one may find the upper and lower limits of AHC
by imposing different restrictions for the configuration integrals in it. Once we obtain the
two limits of the AHC, we can determine the range of the scaling relation between the AHC
and longitudinal conductivity. Note:
What bonds in a triad play the major role for the charge current flowing through it is
determined by the optimization on both the resistance magnitudes and spatial
configuration of the three bonds.
Let:
Extreme situation (I):
In each triad of the whole percolation cluster, it
is the two bonds with larger conductance (
,
) that dominate the charge transport.
The lower limit of the AHC.
Extreme situation (II):
In each triad of the whole percolation cluster, it
is the two bonds with smaller conductance (
,
) that dominate the charge transport.
The upper limit of the AHC.
Limits of distributions: final result
where
Direct numerical calculation gives 1.6
•Depends weakly on the type of hopping!
•Generic to hopping conductivity
Xiong-Jun Liu, Sinova PRB 2011
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SUMMARY
 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 ab-initio theory of of simple ferromagnetic metals of the scattering
independent contribution (Weischenberg, Freimuth, Sniova, Blügel, Mokrousov,
PRL 2011)
 AHE hopping regime approximate scaling arises directly from a generalization
of the Holstein theory to AHE (Xiong-Jun Liu, Sinova, PRB 2011)
 AHE hopping regime scaling remains even when crossing to different types of
insulating hopping regimes, only algebraic pre-factor changes
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