Coordinate shift in the semiclassical Boltzmann equation and the anomalous Hall effect

0.18 MB·English

by  N. A. Sinitsyn

Coordinate shift in the semiclassical Boltzmann equation and the anomalous Hall
effect
N.A. Sinitsyn,1 Q. Niu,1 and A.H. MacDonald1
1Department of Physics, University of Texas at Austin, Austin TX 78712-1081, USA
6 (Dated: February 2, 2008)
0
Electrons in a crystal generically experience an anomalous coordinate shift (a side jump) when
0
theyscatteroffadefect. Weproposeagaugeinvariantexpressionforthesidejumpassociated with
2
scatteringbetween particular Bloch states. Ourexpression for thesidejump follows from theBorn

n series expansion for the scattering T-matrix in powers of the strength of the scattering potential.
a Givenourgaugeinvariantsidejumpexpression,itispossibletoconstructasemiclassicalBoltzmann
J
theory of the anomalous Hall effect which expresses all previously identified contributions in terms

7 ofgaugeinvariantquantitiesanddoesnotreferexplicitlytooff-diagonaltermsinthedensity-matrix
2 response.

] PACSnumbers:
l
l
a
h I. INTRODUCTION. The rigorous quantum mechanical theory of the
-
s anomalousHalleffectinthiscasehasbeenconstructedby
e The near-equilibriumdynamicsofauniformsystemof Kohn and Luttinger1,2, who considered the equation of
m
classical charged particles in a weak electric field E is motion of the density-matrix in momentum space. They
t. described by the classical Boltzmann equation found that elements of the density matrix that are off
a diagonal in the band index (interband coherence contri-
-m ∂∂ftl +eE· ∂∂fkl =− ωll′(fl−fl′) (1) bbyutdioisnosr)daerre.iTndhuesceedcobuoptlhebtyoaonffedxitaegronnaalleleelcetmriecnfitselodfatnhde
d Xl′
velocity operator, thus contributing to the Hall current
n
o where l=(µ,k) is a combined index with k the momen- in addition to the skew scattering contribution3,4, that
c tum and µ the label for some discrete internal degree- can be completely explained in the framework of Eq.1.
[ of-freedom, and ωll′ is the scattering rate between unit The renewed semiclassical theory, based on wave
momentum space volumes. The momentum distribution packetequations5,providedasimpleexplanationofthose
2
v function fl can be written as the sum of its equilibrium contributions. InthistheorytheBerryphasechangesthe
0 value f0(ǫl), where ǫl is the energy dispersion, and a velocityofwavepacketsandleadstothesocalledintrin-
1 nonequilibrium correction g , i.e. f = f (ǫ )+g . Eq.1 sic contribution5,6,7,8,9,10,11 to the AHE. Another ingre-
l l 0 l l
3 can also be applied successfully to quantum systems in dientinthesemiclassicaltheoryistoconsiderthecharge
1 manyinstances. Inthesemi-classicaltheoryofelectronic transportnot onlybetween collisions with impurities but
1
transport in a crystal for example, µ becomes the Bloch also the transportduring collisions,namely the so called
5
bandindex,f canbeinterpretedasaprobabilitydensity side jump effect12.
0 l
/ inphasespacethathasbeencoarsegrainedbyconstruct- Thesidejumpisthecoordinateshiftacquiredbyapar-
t
a ing wave packets, and f0(ǫl) is the Fermi distribution ticle during the scattering event. Recently, it was shown
m function. that for a smooth impurity potential it can be found
InspiteoftheclassicalformofEq.1thescatteringrate by integrating wave packet equations over the scatter-
-
d ωll′ often has to be calculated purely quantum mechani- ing time10. After the gauge invariant expression for the
n cally and is given by the golden rule expression in terms side jump is found, one can calculate the related drift
o
of the T-matrix, which can in turn be written as a Born velocity and the anomalous contribution to the distribu-
c
series in powers of disorder strength. tion function, which in addition to the solution of the
:
v The semiclassical Eq.1 is very powerful since it auto- Eq.1 are sufficient to calculate the Hall current10.
i
X matically takes care of the summation of various infinite The advantage of the semiclassical approach is in its
series of Feynman diagramsthat appear in quantum lin- simplicity. It operates only with gauge invariant quanti-
r
a ear response theory, and it keeps the physical meaning ties, such as the side jump, the scattering rate, anoma-
of all terms transparent. However, since the only role of lous and usual velocities and the distribution function.
theelectricfieldinEq.1istoacceleratewavepacketscon- All of them have a clear physical meaning. In contrast,
structed from states within a single band, and the only the approach by Kohn and Luttinger is rather obscure.
role of impurities is to produce incoherent instantaneous The main reason is that the off-diagonal elements of the
scatterings, it appears clear that this approach must be density matrix or velocity operatorare not gauge invari-
often insufficient. The best known example where Eq.1 ant. Many of the individual contributions to the Hall
apparentlyfailsisinevaluatingtheanomalousHalleffect effect in the Kohn-Luttinger approach are expressed in
(AHE)inspin-orbitcoupledferromagneticmetals,which terms of gauge-dependent quantities which cannot have
can be dominated by an inter-band coherence response. separate physicalmeaning. This may be one reasonwhy
2
manyauthorscitethisarticle,buttrytoinventtheirown shift. Below we construct the theory that enables us
way to calculate the Hall current13,14,15,16. Another al- to calculate the coordinate shift in the lowest nonzero
ternative approach can be found in work by Adams and Born approximation and add it to the contributions to
Blount14, and Nozieres15 and proceeds by projecting all the Hall conductance captured by calculations based on
operators to a single band of interest, or in the case of Eq.1. For simplicity, we will consider first transport in
band-degeneracies in semiconductors, to a subsystem of a single band and later generalize results to the multiple
two degenerate bands15. For electrons in the conduction bandsituation. Letψk(r,t)=(1/(2π)D/2)eikr−iǫ(k)t|uki
band of common semiconductors, the disorder potential be the Bloch state with a momentum-dependent peri-
then acquires form V(r)→V′ =V(r)+ασ·k×∇V(r) odicspinor|uki. Thenaiveexpressionforthecoordinate
Projectiontothesubsystemalsomodifiesthe coordinate shift, δrk′,k = huk′|i∂∂k′|uk′i−huk|i∂∂k|uki is gauge de-
operatorr→i∂/∂k+A,whereAistheBerryconnection pendent, i.e. it changes under an arbitrary momentum
oftheband(s)(see17,18 forreviews). Onethencandefine dependent phase change for the periodic spinors |uki,
gauge-dependent side-jump velocities −i[i∂/∂k,V′] and and cannot be correct in general. We derive the correct
−i[A,V′]. Although such a technique can lead to the form for this expression in the weak potential scatter-
correct answer it is framed in terms of non-commuting ing limit. To find the correct expression, we prepare the
coordinates, and gauge-dependent velocities. The pro- wave packet that approaches the impurity. The wave
jected theory is in practice useful only for semiconduc- function of the wave packet is a superposition of eigen-
tors with sufficiently smooth disorder potential. Hence states of the unperturbed Hamiltonian ψk(r,t) with the
its applicability is strongly restricted. real-valuedGaussianenvelopefactorw(k−k0),centered
Unfortunately, the gauge invariant approach to calcu- near the average momentum k0.
late the anomalous shift proposed in Ref.[ 10] is also re-
setvreinctoebdvoionulystwohveethryersmitoisotehqudiivsaolrednetrtpoottheentBiaolrsn;iatpipsrnooxt- Ψk0(r,t)=Z dkw(k−k0)ψk(r,t) (3)
imation in the weak potential limit, because the Born
We assume a vanishing width of the wave packet in mo-
and the adiabatic approximations often have very differ-
mentum space in the usual way. Hence, when multiplied
ent domains of validity. Hence it would be valuable to
byasmoothfunctionofmomentumtheenvelopefunction
find an alternative approach that applies to weak disor- w(k−k0) can be treated as a delta-function. However,
der potentials of arbitrary range.
whenmultipliedbyatruedelta-function,itisconsidered
In the present work we propose a gauge invariant ex-
smooth, reflecting the finite width of the wave packet.
pressionforthesidejumpforanarbitrarytypeofascat-
Correspondingly, in coordinate space, the wave packet
tering that can be treated in the Born approximation.
should be considered as large in comparison with a lat-
This allows to evaluate all the major contributions to
tice constant,but smallcomparedto other lengthscales.
the anomalous Hall effect that have been identified in
We can evaluate its charge center as follows:
work by Kohn and Luttinger, but now using only classi-
calconcepts,withoutexplicitreferencetoelementsofthe
density matrix or Green functions that are off diagonal rc(k0)=Z drΨk0(r,t)∗rΨk0(r,t) (4)
in band index or to non-commuting coordinates.
We substitute (3) into (4), then notice that reikr =
−i( ∂ eikr) and integrate by parts. Using the orthogo-
∂k
II. GAUGE INVSAHRIIFATN.T COORDINATE nality of plane waves, (2Ωπ0)D R~ ei(k1−k2)r =δ(k1−k2),
andassumingthattheperiodPicfunctionsarenormalized,
huk|uki = 1, and then the delta-function-like properties
To define the coordinate shift (side jump) in terms of of envelope functions we finally find that, before scat-
well defined quantities that can be evaluated using scat- tering, the center of mass of such an unperturbed wave
teringtheory,weassumethatalongtimebeforethescat- packet moves according to the law
teringeventthe center ofmassofthe wavepacketmoves
wfreheelrye avckcoisrdtihnegvteolocthiteyloafwthrec(frt)ete→w−a∞ve=pacδkre−t∞an+dvtkist rc(k0,t)t→−∞ =vk0t+δr−∞ = ∂∂ǫ(kk00)t+huk0|i∂∂k0uk0i.
time. Suppose the wave packet scatters from an impu- (5)
rity with the center at r0 = 0. Then, if the momentum (Ω0 is the unit cell volume and R~ is a sum over lat-
changes to k′, a long time after the scattering event the tice vectors.) Now consider how tPhe state which initially
coordinate of the outgoing state center of mass should coincides with the Bloch state ψk(r,t) moves under the
behave for t → +∞ as rc(t)t→+∞ = δr+∞ +vk′t. We influence of a weak potential of an impurity V(r). The
define the scattering induced coordinate shift as solution of the Schr¨odinger equation can be written in
terms of the eigenvectors of the unperturbed Hamilto-
δrk′,k =δr+∞−δr−∞ (2) nian ψk′(r,t) as
ThenaivetreatmentoftheBoltzmannequation(1)based
only on the scattering rate disregards this coordinate ψkout(r,t)=Z dk′C(k′,t)ψk′(r,t) (6)
3
To lowest order in the strength of the potential, pertur-
bation theory leads to the following expression for time-
∂
idnepReenfd.(en1t9)c)o:efficients C(k′,t) (see for example Eq. 19.9 r+∞II =Z dk′(c(k′,k0))∗i∂k′c(k′,k0) (14)
t
C(k′,t)=−iVk′,kZ−∞ei(ǫ(k′)−ǫ(k))t′dt′+δ(k′−k) (7) r+∞III =k1,kli2m→k0(cid:18)i∂∂k1c(k1,k2)−i∂∂k2c∗(k2,k1)(cid:19)
(15)
where Vk′,k = hψk′(r)|Vˆ|ψk(r)i is the matrix element r III originates from the delta-function in Eq. 8. In
of the disorder potential between two eigenstates of the +∞
whatfollows,wewill restrictour calculationsto the low-
unperturbedHamiltonian. Higherordertermscanbein-
corporatedintotheaboveformulabysubstitutingtheT- est nonzero order in the potential Vk′,k.
The perturbation expansion of the T-matrix is well
matrix instead of the disorder potential matrix elements
known
(see for example Eq. 19.10 in Ref.( 19)).
Vk′,k′′Vk′′,k
C(k′,t)=−iTk′,k t ei(ǫ(k′)−ǫ(k))t′dt′+δ(k′−k) (8) Tk′,k =Vk′,k+Xk′′ ǫ(k′′)−ǫ(k)+iη +... (16)
Z
−∞
Itwillbeusefultorepresentthedisorderpotentialmatrix
The time integral in (7) is formally divergent, reflecting elements in the form
the fact that for infinite interactiontime the initial state
is completely destroyed. We add a regularizing factor in Vk′,k =|Vk′,k|eiArg(Vk′,k). (17)
the exponent ei(ǫ(k′)−ǫ(k))t′ → ei(ǫ(k′)−ǫ(k))t′−(ηsign(t′))t′
to limit the effective finite time of interaction. Perform- Then, taking into account that dk′∂∂k′|c(k′,k0)|2 = 0,
ing the integration in (7) taking the limit η → 0 after the Eq. 14 can be rewritten as R
t→+∞ we thus find that at large positive time (see for
∂
example Eq. 19.60 in Ref.( 19)) r+∞II =−Z dk′|c(k′,k0)|2∂k′Arg(Vk′,k0) (18)
C(k′,+∞)=c(k′,k)+δ(k′−k) (9)
Substituting (16) into (15) and noting that
where −2πiVk′,kδ(ǫ(k′) − ǫ(k)) ≈ c(k′,k) we find to the
second order in Vk′,k that
c(k′,k)=−2πiTk′,kδ(ǫ(k′)−ǫ(k)) (10) ∂
For k′ 6= k the square of the amplitude |c(k′,k)|2 is the r+∞III =−Z dk′|c(k′,k0)|2∂k0Arg(Vk′,k0)+f(k0)vk0
(19)
scattering probability from the state with momentum
k into the one with momentum k′. Due to the delta- where f(k0) is some function, whose exact expression
function in (10), the expression|c(k′,k)|2, should be un- will not be needed. The last termin (19) does not break
any symmetry and can be interpreted as renormalizing
derstood in the regularized sense i.e. assuming that η is
the normal velocity of the part of wave packet that did
smallbut finite. Giventhese standardresultsfromtime-
notchangethe directionofmotionafter interactingwith
dependent perturbation theory, we can reconstruct the
impurity. In what follows, we will ignore it as it has
state of the wave packet after scattering
no influence on the Hall current at the leading order of
perturbationtheory. Combiningtheremainingnontrivial
Ψout(r,t)= dkw(k−k0)ψkout(r,t) (11) terms from (13), (18) and (19) we find the coordinate of
Z
the wave packet center of mass is
where ψout(r,t) is given in (6). The average coordinate
of the ceknter of mass of the final state can be calculated rc(t)t→+∞ = dk′ |C(k′,+∞)|2(vk′t+
by the same steps as used for the ingoing wave packet. R (20)
We find that +huk′|i∂∂k′uk′i−Dˆk′,k0Arg(Vk′,k0))
rc(t)t→+∞ = dr Ψout(r,+∞) ∗rΨout(r,+∞) where Dˆk′,k0 = ∂∂k′ + ∂∂k0 The coefficient |C(k′,+∞)|2
Z canbeinterpretedasthescatteringprobabilityintostate
(cid:0) (cid:1)
= r I +r II +r III (12) k′fromtheinitialstatek0. ThusEq. 20hasasemiclassi-
+∞ +∞ +∞
calmeaning suchthat the averagefinalcoordinateis the
where sum over probabilities of final states multiplied the cor-
respondingcoordinateshifts. Combiningthis resultwith
r+∞I =Z dk′|C(k′,+∞)|2(cid:18)vk′t+huk′|i∂∂k′uk′i(cid:19) t(h5)eoenxeprceasnsiorneafdorthineietixaplrceossoirodninfaotrethoef tthoetawl aavneompaaclokuest
(13) shift corresponding to the scattering of the wave packet
4
from the state with average momentum k into the one
with k′ in the lowest nonzero Born approximation.
∂ ∂
δrk′,k =huk′|i∂k′uk′i−huk|i∂kuki−Dˆk′,kArg(Vk′,k)
(21)
Generalization to the multiple band case is simple.
One should introduce the combined index l = (µ,k) in
Eqs. 6, 7, 16 and so on. Repeating the analogous steps
wefindthattheexpressionfortheanomalouscoordinate
shift after scattering from the state l into the state l′ is
∂ ∂
δrl′l =hul′|i∂k′ul′i−hul|i∂kuli−Dˆk′,kArg(Vl′,l) (22)
Equation (22) is the main result of this work. It pro-
vides a gauge invariant expression for the wave packet
coordinate shift (side jump) which has a clear semiclas-
sicalinterpretationandisvalidforanarbitraryimpurity
potentialthatcanbetreatedintheBornapproximation.
III. RELATION TO PANCHARATNAM PHASE
Under the gauge transformation |u i →
l FIG.1: Closedpathinthemomentumspacerepresentingthe
exp(iφµ(k))|uli the argument of the poten- hoppingamplitude of Eq. 26.
tial operator matrix element in (22) changes as
Arg(Vl′,l) → Arg(Vl′,l) + φµ(k) − φµ′(k′), which com-
pensates noninvariance of other two terms. To make
approximation. In the smooth potential limit the adia-
more incite in symmetries and gauge invariance of the
batic and Born approximation results coincide.
side jump expression we consider the case when the
One can check that (23) is relatedto the gaugeinvari-
periodic part of the Bloch function is only momentum
dependent. For simplicity we consider scatterings in ant Pancharatnamphase Φk′′,k,k′
the same band only, from radial spin-independent
ipmotpeunrtiitayl bpeoctoemnteialV.k′,kT=henV0m(ka′tr−ixke)lheumk′e|nutksi,owf htehree δrk′,k =−(cid:18)∂Φ∂kk′′,′k′,k′(cid:19)k′′→k−(cid:18)∂Φ∂kk′′,′k′,k′(cid:19)k′′→k′
V0(k′−k) ∼ drexp(−i(k′−k)·r)V(r) and for radial (25)
impurityDˆk′,kRArg(V0(k′−k))=0. Thus(21)simplifies where
Φk′′,k,k′ =Arg(huk′′|ukihuk|uk′ihuk′|uk′′i) (26)
∂ ∂
δrk′,k =huk′|i∂k′uk′i−huk|i∂kuki−Dˆk′,kArg(huk′|uki) The Pancharatnam phase Φk′′,k,k′ is the phase that
(23) would appear after the hopping over the closed path
Interestingly, in this case the side jump does not depend in the momentum space over the contour k′′ → k →
on the form of the scattering potential explicitly. In the k′ → k′′ shown in Fig. 1. One can demonstrate that
case of a very small scattering angle20 (|k′−k| << |k|) the phase (26) is also responsible for the skew scattering
one can (for non-degenerate bands) disregard interband contribution. Taking the expression for T-matrix (16)
scattering and make additional approximations valid up and calculating the scattering rate via the golden rule
to the first order in the small parameter |k′ − k|, for ωk′,k = 2π|Tk′,k|2δ(ǫk −ǫk′), we find for its asymmet-
example |uk′i≈|uki+(k′−k)|∂∂ukki and |huk′|uki|≈1. ric part the following expression (see1 for the detailed
Substituting this in (23) we find that up to 1st order in derivation)
|k′−k| the anomalous shift is
δrk′k ≈F×(k′−k) (24) ωk(3,ka)′ =−(2π)2 δ(ǫk−ǫk′′)δ(ǫk−ǫk′)Im(Vk,k′Vk′,k′′Vk′′,k)
Xk′′
(27)
where Fk = ǫijkIm(cid:16)h∂∂ukkj|∂∂ukkii(cid:17). By definition, F is the here the superscript (3a) means that this is the anti-
gaugeinvariantmomentum space Berrycurvature of the symmetricpartofthescatteringratecalculateduptothe
Bloch band. The result for the anomalous shift (24) co- order V3 in the disorder potential. The nonzero value of
incides with the onederivedinRef.( 10)in the adiabatic the phase (26) is crucial to make the product of three
5
potential matrix elements in (27) nonzero. collisiontermf =f +ga insteadoff ,inthestationary
l 0 l 0
state we find the equation that determines ga
l
Im(Vk,k′Vk′,k′′Vk′′,k)∼Im(huk|uk′ihuk′|uk′′ihuk′′|uki)
(28) −∂f
ωll′(cid:18)gla−gla′ + ∂ǫ0eE·δrl′l(cid:19)=0 (32)
Xl′
IV. APPLICATION TO THE AHE
As in the case of the side jump velocity, we can find
an analog of this equation in Luttinger’s classic paper1,
We now apply the side-jump expression (22) to the however Luttinger split the correction ga into nongauge
l
AHEusingideasthathavetheirrootsinearlierworkthat invariant parts. One find that Eqs. 3.21, 3.22, 3.23 and
startedfromadiabatic approximations10,15. The average 3.16 of his work are equivalent to (32).
side-jumpvelocitycanbeexpressedintermsoftherateof In 2D the selfconsistent approachto calculate ga from
l
transitionsandtheside-jumpassociatedwithaparticular (32) is to look for a solution in the form22 ga(φ) =
l
transition: (ga)(n)einφ. For the case of isotropic bands and
n l
vlsj = l′ωl′lδrl′l = irPseonttroapticzersocattetmerpeersratounree wmiathyouctalficunldaitneg tthhee eHxparllescsiuorn-
P
for the full distribution function. Multiplying (32) by
= l′2πNδ(ǫl−ǫl′)[|Vl′l|2(hul′|i∂∂k′ul′i− (29) evµsin(φ), where vµ is the Fermi velocity in the µ-th
P band and summing over k, then taking into account
−hul|i∂∂kuli)−Im Vll′Dˆk′kVl′l ]. that the Hall current contribution from the band µ is
(cid:16) (cid:17) I = e gav sin(φ), we arrive at the set of algebraic
µ k l µ
Here we have used lowest Born approximation expres- equatioPns
sion ωl′l =2πNδ(ǫl−ǫl′)|Vl′l|2, where N is the impurity
concentration. A similar expression can be found in the Iµ/τµ− Iµ′/τµc,µ′ +iµ =0 (33)
second partof Eq. 2.38 in Ref.( 1). Luttinger called this Xµ′
velocity the off-diagonal velocity because its calculation
involved interband matrix elements of the velocity oper- where 1/τµ ≡ l′ωll′, 1/τµc,µ′ ≡ kωll′cos(φ −
ator. Our result (29) is more generalbecause we did not φ′)vµ/vµ′ and iµ ≡P k −∂∂ǫfl0e2vµsin(φ)Pl′ωll′E·δrl′l.
assume, as in Ref.( 1), that Bloch bands are nondegen- The final Hall currenPt from the anomaloPus distribution
erate and that disorder potential is spin-independent. is Jadist = I . The above result can be simplified
y µ µ
The side-jump velocity does not produce any contri- if directtranPsitions between different bands are for some
bution to the total current from the equilibrium dis- reasons forbidden i.e. ωll′ = δµµ′ωkµ,k′. Then, employ-
tribution, but in an external electric field a nonequi- ingthe symmetryoftheproblemonecanderiveasimple
librium correction to the distribution function ap- result
pears. This correction is well known1,21 g =
(−∂f /∂ǫ )eE |v|τ||cos(φ), where τ|| is the translport Jyadist =Jysj (34)
0 l x l µ µ
φlif′e))tiamneddφefiisnetdheasa1n/gτleµ||b=etPwel′eωnllt′h(1e−ve(l|ovcl′i|t/y|val|n)dcosth(φe−xˆ wbeheenrenoJtysijceidsifnouanldessinge(n30er).al cTohnitsexetquinalRityef.h(a1s0)a.lready
direction, which we chose to be along the electric field. InthesemiclassicaltheoryoftheAHEthesidejumpef-
With this correction to the distribution, the side jump fectisnottheonlydisordereffectcontributingtotheHall
velocity leads to the current current. Intheweakdisorderlimit,the dominantcontri-
butiontothe anomalousHalleffectisratherduetoskew
Jsj =e glvlsj (30) scattering3,4 which appears in the semiclassical Boltz-
Xl mann equation through1,2,4 the antisymmetric part of
A second effect follows from the change of energy of the the Boltzmann equation collisionterm kernel, ωl′l−ωll′.
The firstnonzerocontributionto the asymmetric partof
scattered particle under side-jump in the presence of an
external electric field. Since total energy is conserved, ωll′ appearsfromthegoldenrulealreadyintheorderV3,
however, that contribution is parametrically very differ-
the scattered particle acquires additional kinetic energy
∆chǫaln′lg=e inǫl′th−eǫplo=tenetEia·lδernl′elrginyoinrdtehreteoleccotmricpefinesladt.eTthhee eansytmfrmometroicthperasrtasnodfoωnlel′suhpoutlodVpr4o,loinncglutdoincaglctuhleatleoctahle-
ization correction, since corrections to Hall current due
equilibrium distribution would then become unstable,
to ωll′ in this order are parametrically similar with the
∂∂ftl =−Pl′ωll′(f0(ǫl)−f0(ǫl′))= (31) stiiodne-sjuismwpeclolnktnroibwunt1io.nsT.hTehaesytemcmhneitqruicesocfastutecrhincgallceualdas-
=− l′ωll′(−∂∂fǫ0∆ǫl′l)6=0, tgoω.thOenaesycmanmfientrdicitcofrrormectaiosntatnodtahreddsiesltfrciobnustiisotnenftunpcroticoen-
P l
unless compensated by an additional anomalous correc- dure,describedindetails,forexample,in21andthecorre-
tion ga to the distribution function. Substituting in the sponding currentcontribution is Jω =e gω|v |sin(φ).
l y l l l
P
6
Thus skew scattering can be totally understood and cal- general gauge invariant expression for this shift and re-
culated with the semiclassical Boltzmann equation. latedittothephasesofthescatteringT-matrixelements.
Finally the important contribution to the AHE is the We demonstrated that when equipped with this expres-
Berry-phase contribution5,6,7,8,9,10,11. This contribution sion, the semiclassical Boltzmann equation correctly re-
is completely independent of scatterers and is now often produces all contributions to the AHE, that have been
referred to as the intrinsic contribution to the anoma- derived by Luttinger with a purely quantum mechanical
lous Hall effect. The intrinsic anomalous Hall effect has approach. The existing alternative techniques inevitably
been evaluated explicitly in recent years for a variety of havetodealeitherwithadiabaticapproximationsorwith
different ferromagnetic materials using relativistic first non-gauge invariant quantities like nondiagonal density
principles electronic-structure methods23,24,25. It can be matrixelements. Instead,thegoldenrulewithourgauge
non-perturbative in characterbecause of band crossings, invariant expression for the side jump and the semiclas-
a property that partially explains the fact that it is of- sicalBoltzmannequationaresufficienttoderivetheHall
ten quantitatively important. Although it is really an currentfor arbitrarytype of disorder. Such calculations,
interband coherence effect, it can be captured in a semi- though tedious, usually can be well automated with sci-
classical theory by working with modified Bloch bands5 entific software packages.
that include band-mixing by the electric field to leading
order. The end resultin this approachis the appearance
of an anomalous velocity proportional to |E| in addition
to the usual velocity v = ∂ǫ /∂k. The anomalous ve-
l l
locity v(a) = F ×eE captures changes in the speed at
l l
whichawavepacketmovesbetweenscatteringeventsun- Our conclusions about the role of the coordinate shift
der the influence of the external electric field only. Fl is in the semiclassical Boltzmann equation are rather gen-
the Berry curvature of the band5. The corresponding eral and might be important beyond the physics of the
correction to the current is Jintrinsic =e f v(a). anomalous Hall effect. Recently, Coulomb interactions
l l l
Finally the total Hall current in the trPansverse to the and interactions with phonons and magnetic fields be-
electric field y-direction is yondconventionalapproximationshavebeendiscussedin
the context of the Boltzmann equation26,27,28. It would
Jtotal =Jintrinsic+Jsj +Jadist+Jω (35) be interesting to trace the role of the coordinate shift in
y y y y y
similar interacting systems.
Acknowledgments. The authors are grateful for useful
V. CONCLUSIONS.
discussionswithKentaroNomuraandJairoSinova. This
workwassupportedbyWelchFoundation,byDOEgrant
In this work we demonstrated the importance of the DE-FG03-02ER45958andbyNSFunderthegrantDMR-
coordinate shift at a scattering event. We found the 0115947.
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