Disorder effects in the AHE induced by Berry curvature

0.25 MB·English

by  N. A. Sinitsyn

Disorder effects in the AHE induced by Berry curvature
N.A. Sinitsyn,1 Qian Niu,1 Jairo Sinova,2 and Kentaro Nomura1
1Department of Physics, University of Texas at Austin, Austin TX 78712-1081, USA
2Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA
6 WedescribethechargetransportinferromagnetswithspinorbitcoupledBlochbandsbycombin-
0 ingthewave-packet evolution equations with theclassical Boltzmann equation. This approach can
0 be justified in the limit of a smooth disorder potential. Besides the skew scattering contribution,
2 wedemonstratehowothereffectsofdisorderappearwhicharecloselylinkedtotheBerrycurvature
n of the Bloch states associated with the wavepacket. We show that, although being of the same
a orderofmagnitudeasthecleanlimitcontribution,generally disordercorrectionsdependdifferently
J on various parameters and can lead to the sign reversal of the Hall current as the function of the
8 chemical potential in systems with a non-constant Berry curvature in momentum space. Earlier
2 conclusions on the effects of disorder on the anomalous Hall effect depended stricly on the lack of
momentum dependence of the Berry curvature in the models studied and generalizations of their

] findingsto othersystems with more complicated band structures were unjustified.
l
l
a PACSnumbers: 73.20.Fz,72.10.Fk,72.15.Rn
h
-
s
e I. INTRODUCTION interpretation of the Hall current in terms of the Berry
m phase.5,17 The Berry phase contribution to the Hall cur-
. The theory of the anomalous Hall Effect (AHE) has a rent was shown to be in good quantitative agreement
t
a long history The appearanceof the Hall currentrequires with experiment in many different materials with strong
m spin-orbitcouping in their band structure, giving weight
breaking of some basic symmetries. It was proposed by
- Karplus and Luttinger in early 50s1 that the anomalous to the theory involving this contribution alone.17,18,19
d Hall effect in ferromagnets results from the interplay of The Luttinger’s theory of the intrinsic AHE, however,
n
the exchange field, which breaks the time-reversal sym- predicts that other contributions should arise due to the
o
c metry, and the spin-orbitcoupling, that violates the chi- scattering from a disordered potential even if the disor-
[ ral symmetry. Interestingly, at the same time a similar der potential itself is spin independent. The asymmetry

effectwaspredictedandexplainedingeometricaloptics2, in scatterings becomes transparent in the basis related
3
however, its relation to the AHE was revealed only to the Bloch states. As an example, Luttinger consid-
v
6 recently3,4afterthemoderninterpretationofbotheffects ered a rather simplified model which demonstrated that
2 in terms of the Berry phase, which affects the motion of suchcorrectionsmustreversethesignoftheHallcurrent
4 wavepackets,hadbeenconstructed.5,6,7 Luttinger8 built in the dc-limit, in comparison to the clean limit, though
2 a detailed theory of the AHE based on the high order in the high frequency AC-case the clean limit contribu-
0 quantumBoltzmannequationcalculations.9Inthatwork tionshoulddominate. Thesameresultshavebeenshown
5
Luttinger identified various contributions, known today by severalother works utilizing different approachesand
0
asBerryphasecontribution,theskewscattering,theside focusing on this simplified model.20,21,22
/
at jump on the impurity potential and a contribution that The resultsfromthese simplified modelsseemto be in
m involves interference from many scatterers. acontradictionwithrecentworkwhichdidnotfind such
Since then, a number of theoretical works appeared achangeofsigninnumericalsimulationsorinacompar-
-
d that extended the theory. However untill recent time ison with the experimental results. We show here that a
n most of them had been devoted to what is today called possibleresolutionofthisdiscrepancyisthesimplicityof
o the extrinsic AHE. In the extrinsic AHE a simple Bloch the early models from which many generalizations were
c
bandstructureofthe systemisassumedandthe spinor- stated without justification and whose results depended
:
v bit interaction is localized on the impurity potential via drastically on the simple momentum depencence of the
Xi terms of the form λSOσˆzˆzk×∇Vdis(x) e.t.c. It was rec- Berry curvature of the Bloch states.
ognized by Smit, Berger and others10,11,12,13,14,15,16 that The work by Luttinger and works of other authors,
r
a in this case the main contributions to the Hall current related to the disorder contribution to the AHE, are
will be those from impurities via the side jump and the rather involved. Generally various contributions sepa-
skew scattering mechanisms. In contrast to the extrin- rately turned out to be not gaugeinvariantand only the
sic AHE, the intrinsic one assumes that the spin orbit final result was physically meaningful. Because of these
coupling is already present in the band structure of the shortcomings,we reformulatethe basicargumentsof the
system and generally cannot be considered as weak in previousauthorsintermsofwavepacketdyanamicequa-
comparison even with the Fermi energy. Recently the tions whicharefully gaugeinvariantandconsistentwith
interest toward the intrinsic AHE has grown up consid- prior results.6 Being gauge invariant, the wave packet
erably due to new applications in the diluted magnetic equations allow one to identify the physical meaning of
semiconductors(DMS)andduetotheinterestingmodern various contributions. We should note, however, that
2
the wave packet equations are valid only in the limit of The second term in (2) is responsible for the so called
smooth potentials. This restricts the applicability of our anomalous velocity. For the reader not familiar with the
conclusions and generalizationsof our results should not notionoftheanomalousvelocityweprovidetheappendix
be madetoregimeswherethesemi-classicaltreatmentis with a simple example from classical physics that gives
not justified without carefulchecks;therefore we did not an intuitive explanation of the physical meaning of the
make the goal to construct the final theory of the disor- anomalous velocity in Rashba coupled 2DEG. For the
der in the AHE but rather to construct a simple formal- rigorous theory we refer to the original papers6.
ism that demonstrates basic features of the problem and The anomalous velocity is orthogonal to the direction
highlightsabasickeyingredientmissedbypriortheories of the electric field1,23 (which we chose to be along the
studying the effects of disorder, namely the importance x-axes).
of the momentum dependence of the Berry curvature.
We have found that the change of the sign of the Hall v(a) =F dkx (4)
current due to the disorder, found in the early works, is y z dt
not universaland arises in the models where bands have
a constant Berry curvature in the momentum space. In Thesemiclassicalequations(1)and(2)mapthe quan-
diluted magnetic semiconductors the Berry curvature is tum mechanical problem to a classical one where par-
stronglymomentumdependent. Thisseparatesparamet- ticles have the electric charge e and move according to
rically the clean contribution from the others and makes equations of the wavepacketdynamics. We assume that
the sign reversalnot universal. Rather generally we find at equilibrium the distribution of such classical particles
the samesignofthe totalHallcurrentforrealisticbands is the same as the Dirac distribution of electrons in the
except in extreme situations where one of the bands be- sample. This mapping to a classicalsystemconsiderably
comes depleted. simplifiesthe treatmentofthe problembothanalytically
Weorganizetherestofthepaperasfollows. InSec. II andnumerically. Analytically,onecanapplytheclassical
wereviewthewavepacketdynamicstheoryofSundaram Boltzmann equationapproachto calculate the transport
and Niu6. In Sec. III we analyze the model for constant coefficients, numerically the molecular dynamics simula-
Berry curvature obtaining in a physically clear way the tionofthemotionofclassicalparticlesissimpleandmay
reversal of the sign in the presence of smooth disorder. not be restricted to a small system size as, for example,
InSec. IVweapplythetheorytothe caseofthe Rashba in the case of a numerical diagonalization of a quantum
model and show the non-trivial dependence of the con- mechanical Hamiltonian.
tributions from disorder scattering and the clean Berry Unfortunately equations (1) and (2) are valid in the
phasecontributionasafunctionofthe Fermienergyand adiabatic limit only. They cannot be applied to the case
in Sec. V we present our conclusions. of scatterings on a short range delta-function like po-
tential. Only scatterings on impurities, whose potential
variesappreciablyonlyondistancesmuchlargerthanthe
II. THE WAVE PACKET EQUATIONS size of the wave packet can be calculated this way6. In
spite of this restriction of its applicability the limit of a
The motion of wave packets formed by Bloch states is smooth potential is a very interesting one to investigate
governedby the following equations.6 theinfluenceofdisorderontheanomalousHalleffect. In
realisticapplicationsasystemwithlongrangeimpurities
dk=eE−∇V(r) (1) is realized,for example, in the highmobility 2D electron
dt gas24 and the out of plane Zeeman field can be induced
there by polarizing nuclear spins or by introducing addi-
tional magnetic impurities.
d ∂ǫ(k) dk
r= − ×F (2) Recently another related effect, namely the intrinsic
dt ∂k dt spinHalleffect,wasintroduced25. Anumberoftheoreti-
where ǫ(k) is the energy dispersion in the band, E is cal papers have exploredthe importance of the disorder.
the externalelectric fieldactingonawavepackethaving The debates on this topic are ongoing (see e.g.26,27,28).
the electric charge e, V(r) is the local potential in the Understanding the AHE may shed light on the disorder
sample, for example the potential of impurities and F role in the spin Hall effect.
is the Berry curvature of the Bloch band. For the two-
dimentional motion only the out of plane component of
F is nonzero III. CONSTANT BERRY CURVATURE
∂us ∂us A. Clean limit
F =2Im | (3)
z
∂k ∂k
(cid:28) y x(cid:29) In many recent applications the Berry curvature F
here |usi is the Bloch state in the absence of the electric strongly depends on the momentum of the wave packet
field and impurities and s is the index of the band. k.5,19,30 However,itisinstructivetoconsiderfirsttheone
3
Bloch band 2D system with a constant Berry curvature changes of the momentum matters. However when the
F . Berry curvature is nonzero the additional (anomalous)
z
For a constant F the anomalous velocity (4) leads to shift does not disappear after the averaging. To see this,
z
the following contribution to the Hall current from the suppose that the term with the Berry curvature F in
z
unperturbed part of the distribution function in a single Eq. (2) is small in comparison with the first one and
band at zero temperature: calculate the corresponding correctionto the shift of the
particle during the scattering on an impurity. Integrat-
J(clean) =e d2kf (k)v(a) = ing(2)overthetimeintervalatwhichaparticlefeelsthe
yx 0 y
Z impurity potential during a single scattering and treat-
= e kF kdk 2π dφ eE F = e2ExFzkF2 (5) ing the second term in (2) as a small perturbation one
(2π)2 x z 4π can find that, after the scattering, a particle makes an
Z0 Z0
additional shift
where f (k) is the equilibrium distribution function and
0
kF is the Fermi momentum. The upper index (clean) ∆r(k,k′)=z×(k′ −k)F + t2 ∂ǫdt (6)
means that the quantity is calculated in the absence of z ∂k
disorderandthedistributionfunctionofthewavepackets Zt1
coincides with the one in equilibrium at E =0. here k′ and k are momentums respectively after and be-
x
The case of a constant Fz is realized in the conduct- fore the scattering, t1 and t2 are times of entering and
ing bands whose states are weakly hybridized with the leaving the impurity in a semiclassical picture. The sec-
states in the valent hole bands21 where the spin orbit ond term in the rhs. of (6) is just the shift due to the
coupling is allowed. If the kinetic energies of electrons normal velocity. To first order in Fz and ∇V it is not
in the conducting band are much smaller than the gap affectedbytheBerryphaseandhenceisaveragedtozero
between conducting and valent zones, the Berry curva- after many scatterings; therefore we will disregard it in
ture due to such an induced spin orbit coupling can be our future discussion.
considered as a constant for all conducting electrons. In Theparticle’sdisplacementduetothefirsttermin(6)
modern applications it can be realized in some limiting is due to the anomalous velocity. This shift does not
situations,likeinthecaseofR2DEGatastrongZeeman dependexplicitlyonthedetailsoftheimpuritypotential
field (see following sections). In this section we show and does not have the chiral symmetry. There are two
that for smooth impurity potentials the wave packet ap- mainratherdistincteffectsduetothe appearanceofthis
proacheasilyreproducesthesignreversaloftheHallcur- anomalousshift. Thefirsteffect,thesocalledside-jump,
rentwhen F =const. In comparisonto previous works, isthatthey-componentofthisshiftdoesnotcancelafter
z
however,ourapproachkeepsthederivationgaugeinvari- theaveragingovermanyscatteringsandthuscontributes
ant and hence is physically clear. to the drift velocity perpendicular to the electric field,
i.e. to the Hall current. We will focus on this effect in
the following subsection. The second effect is that when
B. Effects of disorder a scattering takes place in the presence of an external
electric field there is a change in the potential energy
Accordingto(5)ifthedistributionfunctionisthesame upon a scattering given by
as in the equilibrium at zero external field a Hall cur-
∆U =−eE ∆x (7)
rent appears in the external field due to the anomalous x
velocity. However, in the steady state the distribution where ∆x is the shift along the external electric
function is no longer f0(k). In the electric field and on field.11,12,13 Both of these effects, as shown below and
timescalesmuchlargerthanthescatteringtimeparticles inthe nextsubsection,givethe samecontributionasEq.
diffuse witha constantvelocityratherthanaccelerateas (5) but with an opposite sign in the particular case of a
intheabsolutelycleancase. Theanomalousvelocityand momentum-independent Berry curvature. Because both
hence the cleancontribution to the Hall currentare pro- contributions are linear in E we are able to consider
x
portionaltotheaccelerationk˙x. Sinceinthesteadystate them separetly when considering the linear response of
the average acceleration is zero up to the first order in the system.
external electric field Ex one can expect that disorder Let us focus first on the second effect. Here only the
should strongly influence the Hall current. anomalouspartofthisshiftisimportantforHallcurrent
A natural way to study the effect of disorder on the calculations
transport in our case is the classical Boltzmann equa-
∆x=−F (k′ −k ) (8)
tion. Thereishoweveracomplicationwhenbothnonzero z y y
Berry curvature and finite sizes of impurities must be
and the effect of the normal part of the shift only renor-
considered. At a scattering on an impurity potential not
malizes the diagonal current response to E . Since the
x
onlythe momentumbutalsothe coordinateofaparticle
totalenergyremainsthe sameafteranelasticscattering,
changes. Usually such a coordinate shift at the scatter-
the kinetic energy must change by the amount
ing is discarded since after averaging over many scatter-
ings such random shifts cancel each other and only the ∆ǫ(k,k′)=ǫ(k′)−ǫ(k)=eE ∆x (9)
x
4
This effect leads to an instability of the initial equilib- shifts do not compensate each other after many scatter-
rium distribution function f (k) = f (ǫ(k)) due to scat- ings, they should contribute to the total Hall current.
0 0
terings. The Boltzmann equation is given by This phenomenonis knowninthe theoryofthe extrinsic
Hall effect as the side jump. If scatterings happen with
∂f =− ω(k,k′)[f(ǫ(k))−f(ǫ(k′))] (10) the rate ω(k,k′) in averagethe particle moving with the
∂t momentum k also acquires the anomalous drift velocity
k′
X
perpendicular to the electric field
where ω(k,k′) is the scattering rate. Kinetic energies
before and after a scattering do not coincide but the dif- v(sj)(k) = ω(k,k′)∆y(k,k′)=−F k /τ
y z x ||
ference between them is small. Since we are seeking a imp
k′
contributionlinearinEx wecansubstitutef byf0which D E X (16)
dependsonlyonthekineticenergyǫ(k)sowecanexpand where
1/τ = ω(k,k′)(1−cos(k,k′)) (17)
f (ǫ(k))−f (ǫ(k′))= ||
0 0 k′
(11) X
= −df0(ǫ(k′)−ǫ(k))=−df0eE F (k −k′) and”(sj)”marksthesidejumpcontributiontothephys-
dǫ dǫ x z y y
ical quantity.
This yields In the equilibrium the side jump does not lead to the
Hallcurrentbecausetheanomalousvelocity(16)changes
∂f −df
∂t =− ω(k,k′) dǫ0eEFz(ky −ky′)6=0 (12) the sign under the transformation kx → −kx. and the
k′ distributionfunctionintheequilibriumisinvariantunder
X
this transformation. However, when the electric field is
The distribution function will relax until a contribution
appliedthenon-equilibriumcorrectiontothedistribution
to it compensates this relaxationin the stationary limit.
function appears which has no such a symmetry under
To find the new equilibrium distribution one should
the momentum reflection.
substitute f(k)=f (ǫ(k))+g (k) into the rhs. ofequa-
0 E This correctionto the distributionfunction canbe de-
tion (10). To make ∂f/∂t = 0 in equation (12) the cor-
rived by means of the standard approach to the Boltz-
rection g (k) must be the following
E mann equation. Up to the first order in the electric field
the standardBoltzmannequationfor one band, ignoring
−df
gE(k)=− 0eExkyFz (13) the asymmetric contribution considered in the previous
dǫ
subsection, is
This contribution is not symmetric in the y-direction.
This means that already the normal (i.e. usual ∂ǫ/∂k) −eE v(normal)(−∂f0)=− ω(k,k′)(f(k)−f(k′))
velocitycancontributetotheHallcurrentinthestation- x x ∂ǫ
k′
ary state: X (18)
where f (k) is the equilibrium distribution function.
0
d2k Equation (18) has a solution f(k)=f0(k)+g(k) where
Jynxormal = e (2π)2gE(k)(vy(normal)) to the first order in the electric field
Z
= −e2(E4πxF)zkF2 (14) g(k)=eExτ||(−∂∂fǫ0)vx(normal) (19)
which has the opposite sign and is exactly the same in vx(normal) is the normal velocity along the electric field,
the absolute magnitude as the anomalous velocity con- which in our case is
tribution in the clean limit given by Eq.(5).
∂ǫ(k)
v(normal) = =bk (20)
x ∂k x
x
C. The side jump here we assumed that the conducting band has a trivial
dispersion ǫ(k)=bk2/2 where b=1/m . The side-jump
e
Theanomalouschangeofenergyafterthescatteringis contributionofimpuritiestotheanomalousvelocityleads
not the only important effect of the anomalous shift (7). to the Hall current
Theanomalousshiftduringthe scatteringshasgenerally
kdkdφ
acomponentperpendiculartothedirectionoftheelectric J(sj) = e v(sj)(k) g(k)
field. yx Z D y Eimp (2π)2
e2E F 2π
∆y(k,k′)=Fz(kx′ −kx) (15) = − (2πx)2z dφcos2(φ) dǫkk2δ(ǫF −ǫk)
Z0 Z
Itdoesnotcontributetothechangeofenergyduringthe = −e2ExFzk2 (21)
scatteringbutitshiftsaparticlealongthey-axes. Ifsuch (4π) F
5
As in the previous impurity contribution, the Hall cur-
rent due to the side jump has the opposite sign and the
same magnitude as the one in the clean limit. The side
jump current appears because the anomalous velocity is
proportionaltotheaccelerationoftheparticlebutinthe
stationary state the average acceleration should be zero
(up to the terms of the first order in the electric field),
namely, while the external electric field accelerates the
wave packet during its motion between impurities, the
impuritypotentialgenerallydeceleratesit. Duringsucha
deceleration the wave packethas the anomalous velocity
with the opposite sign and hence the side jump contri-
bution should compensate or at least decrease the pure
limit result. The described picture is of course oversim-
plified, which will be clear from the discussion of a mo-
mentum dependent F . In reality, the external electric
z
fieldequallyacceleratesallelectronsintheFermiseabut
effective deceleration due to impurity scatterings is, in
average, seen only by the electrons near the Fermi level
FIG. 1: Scattering of a particle on impurity. The impurity
because there the distribution function acquires a cor-
is assumed to have the shape of a long stripe with linearly
rection required to compensate the acceleration by the
growing potential inside.
external field. Hence the side jump contribution is the
propertyofthe Fermisurfacewhile the cleanlimit Berry
phase contribution appears from all electrons even deep fromanothersideandscattersfromk′ intok. Thusboth
in the Fermi sea. processeshave equal probability and ω(k,k′)=ω(k′,k).
Fig.1demonstratesascatteringonsuchanimpurity. We
would like to point to the analogy of the scattering in
D. Total current and numerical check Fig.1 with Goos-H¨anchen’s and with Fedorov’s shifts in
geometrical optics29.
The total Hall current (not including effects of skew One can consider the effect of the wall on a wave
scatteringsduetoasymmetryofthecollisiontermkernel) packetas the one due to an electric field E acting on it
⊥
is the sum of the clean limit contribution, of the side only inside the wallin the directionperpendicular to the
jump and of the normal velocity contributions. For the wall. Althoughthe potentialin the wallcanbe strongin
constant F we find comparison with the external electric field, we suppose
z
that the motion inside the wall is still governed by the
J(total) =J(clean)+J(sj)+J(normal) =−e2ExFzk2 equations of the wave packet dynamics. Such a scatter-
yx yx yx yx (4π) F ing problem can be solved exactly. This solution shows
(22) that for the given type of scatterings the theory of the
Asitwaspredictedintheformerliteraturewhichfocused anomalous shift is valid even if we do not assume that
on models with a constant F , the total current has the the anomalous velocity term in wave packet equations is
z
samemagnitude butwith the oppositesignas the onein small. This is useful because the bigger anomalous ve-
the absolutely clean system. Our derivation however is locity is needed to accelerate the numerical calculations.
considerablysimpler andis straightforwardto generalize Let components of the momentum of the particle in-
to a case with a more complicated band structure such cident on the wall be k and k parallel and perpen-
|| ⊥
as Rashba coupled 2DEG. dicular to the wall respectively. In the presence of an
To confirm the analytical result (22) numerically we additional external electric field the total force has gen-
simulate the motion of particles according to equations erally also a nonzero component parallelto the wall eE
||
of motions (1) and (2). Initially all particles were pre- where E is the projection of the external electric field
||
pared uniformly distributed over the momentum space on the direction along the wall. Expressions for the fi-
having the absolute value of the momentum less than a nalmomentumandcoordinatesrightafterthescattering
specified Fermi momentum. The action of an impurity are strongly simplified in the limit of large E⊥, so that
was simulatedby the potential ofa wallof a finite thick- all terms O(1/E⊥) can be dropped. In this limit the
nesswithaveryfastlinearlygrowingpotentialinsidethe scattering time is vanishing and in (2) one can disregard
wall. The wallis sufficiently long to disregardthe effects the first term in comparison with the second one since
on its edges. This type of impurity does not allow the k˙⊥ ∼ E⊥ is large. Dropping the term with ∂ǫ/∂k⊥ the
skew scattering mechanism to appear because for every equation(2)canbereadilyintegratedoverthetimeofthe
scatteringonthewallthatchangesmomentumfromkto scattering leading to the relations ∆r|| = eFz(k⊥ −k⊥′ )
k′ there is a process when a particle hits the same wall and k′ = k . Solving them together with the energy
|| ||
6
conservation equation ǫ(k)+eE r =ǫ(k′) we arrive at
|| ||
following expressions for the momentum of the outgoing
particle
k′ =k
|| || (23)
k′ =−k −E F
⊥ ⊥ || z
In addition, the scattered particle appears in a point of
the interface shifted in comparison to the incident point
by the amount.
∆r =2k F +E F2 (24)
|| ⊥ z || z
Theshiftr isexactlytheanomalousshiftintheimpurity
||
potential. Note also that, as it follows from (23), when
FIG. 2: Hall conductivity vs strength of the Berry curvature
a scattering happens in the additional external electric
fieldwithnonzeroprojectiontothedirectionofthisshift in the model with momentum independent Fz. Solid line is
the theoretical prediction that includes impurity scatterings
E , the absolute magnitude of the momentum changes,
|| and dotted line is the prediction of the clean limit contri-
whichleadstothe instabilityoftheequilibriumdistribu-
bution. Triangles show results of the numerical simulations.
tion function, discussed in previous sections. The scat- Simulations were performed in units e = ~ = m = 1 and
teringtimeinthislimitissmallenoughtobedisregarded kF =1.
so numerically such a scattering can be easily simulated
bychoosingthecoordinatesystemrelatedtothewallori-
entation and updating the momentums and coordinates currentcanbe derivedas the sum oftotal displacements
accordingto(k ,k ,r ,r )→(k′,k′ ,r +∆r ,r ),r of all particles divided by the evolution time.
|| ⊥ || ⊥ || ⊥ || || ⊥ ⊥
isthesamefortheincidentandtheoutgoingparticle. In In Fig.2 we compare the analytical result (22) with
between scatterings equations of free motion in the ex- ournumericalsimulations forthe fixedFermienergybut
ternalelectricfieldarealsotriviallysolvable. Inoursim- different strengths of the spin orbit coupling, and hence
ulations we assumed the ”noncrossing approximation”, F . Bothanalyticalandnumericalresultsareinexcellent
z
namely we suppose that every scattering happens on a agreement with each other. This result survives at an
different impurity. The algorithm consists of two parts. arbitrary magnitude of the Berry curvature.
First we generate randomly the distance L to the next
impurity. We suppose that between impurities particles
move only under the action of the electric field. The in-
tegrationofwavepacketequations leads to the following
trajectories
k′ =k +E t
x x x
k′ =k
y y (25)
x′ =x+k t+ Ext2
x 2
y′ =y+(k +F E )t
y z x
The time t of the motion to the next impurity was esti-
mated from the equation((x′−x)2+(y′−y)2))1/2 =L.
Solving this relation with the solution (25) we find the
momentumandcoordinatesoftheparticlebeforethenew
scattering. In the second step we randomly generate the
orientationangle of the impurity, switch into the related
coordinatesystem andupdate particle’s coordinatesand FIG. 3: Distribution of particles over the momentum space
the momentum according to (23) and (24). Then we re- in the stationary state with electric field applied along the
turn to the initial coordinate system and repeat the cir- y−axes at zero Berry curvature. Parameters are as follows.
cle. Werepeatthiscirclesufficientlymanytimestoallow kF = 1, Ey = 0.003, Ex = 0, Fz = 0. Average distance
the distribution to relax to the stationary one (usually a between impurities is l∼0.5 in unitsm=~=e=1.
few scatterings is enough), however the total evolution
time was small enough in order to avoid strong heating In Fig.3 and Fig.4 we visualized the stationary dis-
of the system. In our simulations, every particle makes tribution function of particles in the momentum space,
about 102 −103 scatterings during the whole time. To calculated numerically for the system placed in the uni-
prevent strong heating the electric field is chosen suffi- form electric field. Brighter areas represent higher den-
ciently small E l/E ∼10−4 where l is the typical scat- sities. To increase the contrast,we subtractedthe initial
x F
tering length and E is the Fermi energy. The total equilibrium distribution (i.e. without electric field) from
F
7
equations (1), (2). The new system (26), (27) already
corresponds to the classical Hamiltonian evolution with
the following Hamiltonian
H(k,r)=ǫ(k)−F(k×(eE−∇V(r))+V(r)−eEr (28)
This Hamiltonian has the same structure as in typical
models ofthe extrinsicAHE11,12,13 andthe Berrycurva-
turenowplaystheroleofaneffectivespinorbitcoupling
constant. As we mentioned, the main contributions to
the extrinsic Hall current have been proved to be the
side jump and the skew scattering. In our previous cal-
culations, including numerics, we ignored the skew scat-
tering mechanism, nevertheless the skew scattering can
be calculated by same weave packet techniques because
FIG. 4: Distribution of particles over the momentum space the second term in the rhs. of (28) is responsible for
in the stationary state with electric field applied along the
both skew scattering and the side jump contributions in
y−axes at the non-zero Berry curvature. Parameters are as
the extrinsic AHE. Note also that the skew scattering
dfoilsltoawnsc.e bkeFtw=ee1n,iEmypu=rit0ie.0s0i3s,lE∼x0=.5.0, Fz = −1. Average currentwilldepend onthe Berrycurvatureandthus will
also have a geometric interpretation. The fact that the
side jump is the only other surviving contribution in the
the calculated one. In Fig.3 the Berry curvature is set extrinsicAHEisinagreementwithourpreviousfindings.
to zero. In this case the correction to the equilibrium
distribution is symmetric along the electric field. Fig.4
shows that when the Berry curvature becomes nonzero, IV. RASHBA 2DEG WITH EXCHANGE
the distribution function acquires the additional asym- COUPLING
metriccontribution,whichobviouslydepositstotheHall
current. Recently studied systems with AHE suchas DMS and
Rashba 2DEG have Berry curvature that strongly de-
pends on momentum of the wave packet30. When F
z
E. Analogy between extrinsic and intrinsic AHE is no longer a constant the simple arguments leading to
canceling of some terms may not work. One can notice
As we showed above, in the AHE induced by Berry thatwhenanelectrondeceleratesithasingeneraldiffer-
curvature there is a possibility of disorder effects similar ent momentum than when it accelerates, in other words
to previously studied extrinsic contributions to Hall cur- the uniform electric field accelerates all electrons down
rent due to gradient terms of the disorder potential. In to the bottom of the Fermi sea but impurities produce
this subsection we discuss how this relation is encoded nonzero deceleration in average only for electrons near
in the wave packet equations (1), (2). We consider here the Fermi surface. The Berry phase acquired by acceler-
only the case of a constant Berry curvature. The wave atingelectronsdependsnownotonlyonthe acceleration
packet dynamic equations do not arise from a particular butonthemomentumitselfthereforecontributionsfrom
Hamiltonianinasensethatcoordinatesandmomentums impurities andfromuniformelectricfieldnotnecessarily
of a wave packet are not canonical variables. Neverthe- cancel each other.
less,itispossibletomakethemarisefromaHamiltonian The Hamiltonian of R2DEG with the electric field
byaddingtermsofhigherorderinthepotentialgradient. along the x-direction and the exchange field in the z-
Weremindthereaderthatequations(1)and(2)werede- direction is
rivedin the approximationofa smoothpotential sothat
itshighergradientscouldbediscarded. Thereforesucha
H =H +H +H −eE x+V (29)
procedure should not change the physics at least to the 0 SO exch x imp
orderoftheapproximationofthewholetheory. Consider
the following system
bk2
d H = σ , b=1/m (30)
k=eE−∇V(r)−∇(k×∇V(r))F (26) 0 2 0
dt
d ∂ǫ(k) H =λ(k σ −k σ ) (31)
r= −(eE−∇V(r))×F (27) SO y x x y
dt ∂k
Except for one term of the second order in gradients of
the potentialthe systemisequivalentto the wavepacket H =hσ (32)
exch z
8
where V is the impurity potential. note that here k is the Fermi momentum of the mi-
imp F+
DiagonalizingtheunperturbedpartoftheHamiltonian nor band. The above formulas are valid when there are
we find the energy dispersion electronsinbothbands. Ata sufficiently lowFermilevel
(E < h,λk ) the minor band becomes depleted and
F F−
the transportis only in the major band. In this case the
ǫ (k)=bk2± (λk)2+h2 (33)
± clean limit Hall current is
Therearetwobands,theminporbandhastheplussignin
e2E 1
the above expression. Fermi momentums can be derived j(clean) = x 1− (39)
by inversion of the equations (33). We denote them as yx 2(2π) 1+(kF−λ/h)2!
k andk respectivelyforthemajorandminorbands.
F− F+ Weaddressnexttheeffects opfimpurityscatteringwithin
The Berry curvature is different for different bands
the formalism that we have developed.
∂us ∂us As in the case of the constant Berry curvature, the
Fs =2Im | (34)
z ∂k ∂k non-equilibriumcontributiontothedistributionfunction
(cid:28) y x(cid:29) isthe sumofthe usualonegs(k), whichappearsto com-
here|usiistheBlochstateintheabsenceofelectricfield
pensate the electron acceleration
and impurities and s = ± is the index of the band: the
plus is for the minor and the minus is for the major one. ∂f
gs(k)=eE τ (− 0)v(normal) (40)
In the case of the Hamiltonian (29) the Berry curvature x || ∂ǫ s,x
is30, (see also Appendix for an alternative derivation).
and of the anomalous one gs(k) due to the shift of
E
hλ2 the kinetic energy of the scattered particle. We sup-
Fs(k)=−s (35)
z 2[(λk)2+h2]3/2 pose here that the impurity potential is weak, namely
V(r) << E ,λk and disregard the variation of the
F F
at λk << h this gives Fzs(k) ≈ −sλ2/(2h2) = const and Berry curvature near a wave packet during a scattering
we reduce the problem to the one consideredin previous because the absolute magnitude of the momentum does
sections. In the opposite limit at λk >>h we find notchangeappreciably. Thederivationoftheanomalous
sh correction is analogous to (10)-(13).
Fs(k)≈− (36)
z 2λk3
−df
gs(k)=− 0eE k Fs (41)
The anomalous velocity has the same magnitude but E dǫ x y z
differentsignsfordifferentbandssofilledstateswiththe
same k from different bands will compensate each other The normal velocity of the wave packet is
in calculations of the clean contribution and to find the
∂ǫ (k)
Hall current we should only integrate over uncompen- v(normal) = s (42)
s,x ∂k
satedstatesofthemajorband. TheHallcurrentfromthe x
unperturbed electron distribution function in the clean The expression for the drift velocity due to the side
case is jumps along the y-axes is the same as in (16). The side
jump current is
2π kF−
kdk
jy(cxlean) =e dφ (2π)2(eExFz−(k)) j(sj) = e hv(sj)(k)i gs(k)kdkdφ
Z0 kFZ+ yx s=± Z s,y dis (2π)2
X
2π kF− kdk eExhλ2 = dφcos2(ϕ) dǫ e2E Fs(k)k2δ(E −ǫ )=
= e dφ (2π)2 s,k x z F s,k
(2π)22[(λk)2+h2]3/2 s=±Z Z
Z Z X
0 kF+ he2E λ2 k2
e2Ex 1 = 2(4πx) [(λk )2F++h2]3/2
= (cid:18) F+
2(2π) 1+(kF+λ/h)2 k2
− F− (43)
− p1 (37) [(λkF−)2+h2]3/2(cid:19)
1+(kF−λ/h)2! At E >>λk >>h this expression can be simplified
F F
in the case EpF >> λkF± >> h the expression (37) is he2E
simplified j(sj) ≈ x (44)
yx 8πE
F
k2 e2E h(k −k )
j(clean) ≈ λk ≪E ≈ F = x F− F+ If E <h only the major band survives and
yx F 2m (2π)2λk2 F
(cid:26) (cid:27)
= {(k −k )≈2λm}= e2Exh (38) j(sj) = −he2Exλ2 kF2− (45)
F− F+ 4πEF yx 2(4π) [(λkF−)2+h2]3/2
9
V. DISCUSSION AND CONCLUSION
We calculated the anomalous Hall currentin the pres-
ence of a smooth disorder potential by combining the
wave packet dynamic equations with the classical Boltz-
mann equation. For the case of a constant Berry curva-
ture our results are in agreement with some known pre-
dictions of the very earliest works on AHE. The gauge
invarianceofourapproachclarifiesthe physicalmeaning
of various disorder contributions to the Hall current and
revealestherelationbetweentheextrinsic,relatedtothe
gradients of the disorder potential, and Berry curvature
effects in the presence of disorder.
We haveshownthatearlyresults,likethatofthe Lut-
FIG.5: (Color online) Hall conductivityvstheFermienergy tinger’s sign reversal prediction, cannot be used directly
in the R2DEG. Black line is the pure limit Berry phase pre- for new applications such as diluted magnetic semicon-
diction. The red curve shows the prediction of the theory ductors, where the Berry curvature is strongly momen-
that includes impurity scatterings (43), (47). Data are given
tum dependent.
in units e = ~ = m = 1 for the case h = 0.7, λ = 0.4.
Our prediction for the Hall current Fig.5 is distinct
Sharp change in the behavior above EF = 0.7 is due to the
fromthecleanlimit(orthepureBerryphase)one. Thus
appearance of electrons in theminor band.
the total Hall current changes the sign when the Fermi
level increases. In contrast, the clean limit prediction
The normal current contribution is for the models consideredremainsof the same sign. The
filledstatesinbothbandswiththesamemomentumcom-
j(normal) = e d2k v(normal)gs(k)= pensate each other, hence for the clean case the band
yx s=± (2π)2 s,y E
havingmorefilledstatesalwayswins. Incontrast,impu-
= dPφsin2(ϕ)R dǫ e2E Fs(k)k2δ(E −ǫ ) rity contributionsare sensitivenot only to the density of
s=± (2π)2 s,k x z F s,k states near the Fermi level, but also to the magnitude of
P R R (46) the Berry curvature near the Fermi level.
Comparing the last expression with the one for the side The change of sign of the Hall current as a function
jump current(43)we findthatthey aredifferentonly by
ofchemicalpotentialoriginatesfromthecompetitionbe-
anexchangeofcos(ϕ) andsin(ϕ) under the integralover tween two bands. The minor band has the smaller num-
the angle. Thus we arrive at the general result
ber of filled states than the major one but it also has
lower k and hence the stronger F (k ). According to
j(normal) =j(sj) (47) F z F
yx yx our calculations, such a change of sign should generally
happenwhenthechemicalpotentialisclosetothedeple-
Finally,
tion point of the minor band. When the minor band is
totallydepletedwealwaysobservethereversalofthesign
j(total) =j(clean)+j(sj)+j(normal) (48)
yx yx yx yx of the Hall current in comparison to the clean limit con-
tribution. We also note that such a competition should
Asitisseenfrom(44),(38)thesidejumpcontribution
be applicable to the skew scattering mechanism as well.
may not change the sign in comparison with the clean
This follows from the fact that the strength of the skew
contribution for sufficiently large Fermi energy. This is
scatteringshouldbe proportionalto the Berrycurvature
very distinct from the case of the constant Berry curva-
at the Fermi surface.
ture where such a contribution exactly cancels the pure
Our results can be generalized to DMS and although
one. Atthe limit E >>λk >>h wefind forthe total
F F
in this case the complicated behaviour of the bands, as
current
compared to the R2DEG, may change the quantitative
he2E behavior there is still a competition in DMS of Hall cur-
j(total) ≈ x (49)
yx 2πE rents from major and minor bands and Berry curvature
F
is also usually decreasingwhen momentum is increasing.
The totalHallcurrentnotonlyhasthe samesign,asthe HowevertheintricatedependenceofthecleanlimitAHE
clean limit prediction, but also increases due to scatter- observed in these systems versus the simple dependence
ings. Notealso,thatuptoacoefficientoftheorderunity, observed in the R2DEG will require a careful analysis.
in this limit, the total current has the same dependence Although we have verified most our predictions by
on various parameters. numerical simulations, the correspondence between the
In Fig.5 we compare the theoretical prediction of Eqs. wave packet approach and the evolution of a true quan-
(48) with the prediction of the pure Berry phase contri- tumsystemwithdisorderremainstobeinvestigated. An
bution (37). effect not taken in to account in our calculations is that
10
the spin density in the wave packet polarizes when the
wave packet is accelerated31. This effect is crucial for
k
the Spin Hall Effect and its importance for calculations y˙ = y +2λS (A3)
x
m
by the Boltzmann equation should be understood.
ThefinaltheoryofdisorderintheAHEcanbeachieved
only by purely quantum mechanical calculations for ex-
ample, based on the Keldysh technique. At present k˙x(t)=eEx (A4)
new approaches to systems with Berry curvature have
emerged32, as well as new mechanisms to generate the
anomalous Hall current33, but we believe that the wave k˙ (t)=0 (A5)
packet approach combined with the classical Boltzmann y
equation is worth studying because it provides the sim-
plest, to our knowledge, demonstration of the related
physics and the wave packet equations themselves have S˙ =−∆×S (A6)
been well justified by the quantum theory.
Note added. After completion of this work we became where∆istheeffectivemagneticfieldactingonthespin.
aware of the related effort by V. K. Dugaev et al.35 of a From the Hamiltonian (A1) we can read this effective
morequantummechanicalapproachtotheproblemwith magneticfieldactingontheparticlewiththemomentum
similarqualitative conclusions. Ourapproachhoweveris k=(kx,ky)
quite different and in combination with this work may
shed further light on the physical interpretation of the ∆=−2(λky,−λkx,h) (A7)
effect.
The absolute magnitude of this field is
Acknowledgments. Theauthorsaregreatfulforinsightful
and useful discussions with A. H. MacDonald and V. L.
|∆|= 4(k2λ2+h2) (A8)
Pokrovsky. This work was supported by Welch Founda-
tion, by DOE grant DE-FG03-02ER45958 and by NSF p
Assuming that the electric field E is weak, the varia-
x
under the grant DMR-0115947.
tionofthemomentumandhenceoftheeffectivemagnetic
fieldisveryslowand,ifattheinitialmomenttheclassical
spinisdirectedalongthemagneticfield,itwillfollowthe
APPENDIX A: CLASSICAL INTERPRETATION direction of this slowly varying field. This adiabatic ap-
OF THE WAVE PACKET DYNAMICS IN A proximation follows from the Landau-Lifshitz equations
RASHBA 2DEG and yields an approximate solution for (A6) given by
S(a)(t) = ±|S|∆(t), where ”+” and ”−” correspond to
|∆|
To make the semiclassical result more transparent we the initial direction along or opposite to the magnetic
show how the anomalous velocity arising from the Berry field respectively. Substituting this time dependence of
curvature can appear in a purely classical system. This the spin direction into equations for particle velocities
classical anomalous velocity originates from the non- one can find that in the adiabatic approximation
adiabaticcontributionstotheequationsofmotioninthe
linear response regime and are not present when a sim- x˙ = ∂ǫ0(k)
pler adiabatic approximation is considered. ∂kx (A9)
We consider the motion of a classical particle having y˙ = ∂ǫ0(k)
the electric charge e and the classicalspin S attached to ∂ky
it. Choose the Hamiltonian to be whereǫ (k)= k2 ∓ (kλ)2+h2,i.e. aparticlewithspin
0 2m
k2 along the field ∆ has energy dispersion as an electron
p
H = +2λ(kySx−kxSy)−eExx+2hSz (A1) in the major band of R2DEG. In the strict adiabatic
2m
approximation of this classical Hamiltonian particles do
not exhibit the anomalous velocity seen in the second
The analogy with the Rashba Hamiltonian (29) is obvi-
term of the group velocity of the quantum-mechanical
ous. In the classical problem we substitute operators by
wavepacketdynamics described by Eq. (2).
corresponding classical variables. We will suppose also
However,this classicalmodel does capture the correct
that|S|=1/2as fortrue electrons. The Hamiltonequa-
contribution to the anomalous Hall effect related to the
tions for the evolution of coordinates (x,y) and momen-
Berry curvature once we go beyond the strict adiabatic
tum (k ,k ) and Landau-Lifshitz equations for the evo-
x y
approximationandconsiderinsteadthemoregenerallin-
lution of spin components read
ear response. The nonadiabatic corrections describe de-
flection of the classical spin from the direction of the
k instant magnetic field. They are small but can still be
x
x˙ = −2λSy (A2) ofthe firstorderin the electric field andhence affect the
m