The magnetic field generated by a charge in a uniaxial magnetoelectric material

0.4 MB·

by  Michael Fechner

Themagneticfieldgeneratedbyachargeinauniaxialmagnetoelectricmaterial
M. Fechner and N. A. Spaldin
Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zu¨rich, Switzerland
I. E. Dzyaloshinskii
School of Physical Sciences, University of California Irvine, Irvine, CA 92697, USA
(Dated:January13,2014)
We revisit the description of the magnetic field around antiferromagnetic magnetoelectrics in the context
of recent developments regarding magnetoelectric monopoles. Using Maxwell’s equations, we calculate the
magneticandelectricfieldsassociatedwithafreechargeinabulkuniaxialmagnetoelectric,aswellasinafinite
sphereofmagnetoelectricmaterial. WeshowthatachargeintheprototypicalmagnetoelectricCr O ,which
2 3
isuniaxialwithadiagonalmagnetoelectricresponse,inducesaninternalmagneticfieldwithbothmonopolar
and quadrupolar components, but that only the quadrupolar contribution extends beyond the sample surface.
Wediscussthebehavioroftheexternalquadrupolarfieldandcompareitsmagnitudetothoseofmagneticfields
fromothersources.
4
1
0 A linear magnetoelectric is a material inwhich an applied magnetoelectric response, it emerges that the first isotropic
2 magneticfieldinducesanelectricpolarizationandanapplied part corresponds to a “magnetoelectric monopolar” compo-
electricfieldinducesamagnetization,withthesizeofthere- nent, and the second traceless part to a “magnetoelectric
n
sponseproportionaltothestrengthofthefield:1 quadrupolar” component2,5. (The antisymmetric part which
a
J is is zero in this case results from a magnetoelectric toroidal
0 Pi = αijHj moment6.) A free electric charge in such a magnetoelectric
1 M = α E . (1) should in turn generate a magnetic field that reflects these
i ji j
monopolarandquadrupolarcontributions.

] There has been increasing research interest in magneto- In this work we use Maxwell’s equations to calculate the
i
c electrics over the last fifteen years, motivated in part by the magneticandelectricfieldsassociatedwithafreechargeina
s technological appeal of electric-field-controlled magnetism, magnetoelectric material. We begin with the mathematically
-
l aswellastheintriguingmechanismsthatallowaferroicprop- straightforwardcaseofanisotropicmagnetoelectricmaterial
r
t erty to be modified other than by its conjugate field. In ad- and confirm that, as required by symmetry, it has a purely
m
dition, it has been pointed out in the last months that cer- monopolar response. While isotropic magnetoelectrics have
t. tain symmetry classes of magnetoelectrics exhibit behaviors been discussed in the literature7, none has been identified to
a that can be described in terms of so-called magnetoelectric date, as the requirement that time- and space-inversion sym-
m monopoles2leadingtopotentiallynewphysicssuchashidden metrybebrokenwithinacubicsymmetryisrestrictive. Inter-
- ordersandnoveltransportproperties3. estinthemhasbeenrenewed recentlyinthecontextoftheir
d
n Themagnetoelectrictensor,αij,isanaxialsecondrankten- relationship to strong Z2 topological insulators8, as well as
o sor,whichisantisymmetricunderbothspaceandtimeinver- theirpotentiallynoveltransportproperties3. Wethenproceed
c sion.Asforanysquaretensoritcanbedecomposedintosym- to the physically more abundant and mathematically more
[ metricandantisymmetricparts: complexuniaxialcaseandevaluatetherelativecontributions

ofthemonopolarandquadrupolartermstothemagneticand
1
α=α +α (2)
v S AS electricfields. Inbothcasesweobtainthesolutionforafree
8 charge in an infinite magnetoelectric medium, as well as for
withtheanti-symmetriccomponentsindicatingresponsesper-
8 thecaseofafinitesphericalsample. Finally,basedonourre-
3 pendicular to the external field. The form of α, of course sultswerevisitliteraturemagnetometrymeasurementsofthe
2 reflects the symmetry of the system, and there exists a large magnetic field around Cr O 9,10. These were discussed pre-
. numberofuniaxialmagnetoelectrics,includingtheprototype 2 3
1 viously in terms of the intrinsic quadrupolar response of an
Cr O 4,inwhichtheresponseispurelydiagonalinthebasis
0 2 3 uncharged magnetoelectric11 as well as in terms of the sur-
4 ofthecrystallographicaxes. ForsuchmaterialsαASvanishes, face magnetization caused by the antiferromagnetism12 and
1 and it is useful to reduce the symmetric part further into its
itscouplingtothemagnetoelectricity13. Weevaluatetherela-
: isotropicdiagonalandtrace-freesymmetriccontributions:
v tivemagnitudesofallthreecontributionstotheexternalmag-
Xi 1 0 0  neticfieldanddiscusstheimplications.
ar α=αS= 13(2α⊥+α(cid:107))I+α⊥−3 α(cid:107)0 1 0  . (3)
0 0 −2
I. ANISOTROPICMAGNETOELECTRIC
Here α and α are the responses parallel and perpendicu-
(cid:107) ⊥
lar to the high symmetry axis respectively, and I is the unit We provide a source of electric field within an isotropic
matrix. Whenacorrespondenceismadebetweenthesecond- magnetoelectric by introducing a point charge ρ = qδ(r).
ordertermsinthemagneticmultipoleexpansionandthelinear While in practice such a charge is likely to be an electron,
2
a) b)
herewetreatthecaseofaspinlesscharge,anddonotinclude
0.4
inouranalysisthereciprocalmagnetoelectricresponsearising
fromanelectronicspinmagneticmoment. Sincethematerial 15
isisotropic,theαtensorisgivenby EME(r) 0.2
α 0 0 V/m) 10 E(r) (m) 0.0
n y
α=0 α 0. (
E
0 0 α 5 -0.2
Themagnetoelectricmediumaugmentsthedisplacementfield -0.4
and magnetic induction from their usual (cid:15)E and µH to in- 0.2 0.4 0.6 0.8 -0.4 -0.2 0.0 0.2 0.4
cludethemagnetoelectriccrossterms: r(m) x(m)
D = (cid:15)E+αH
FIG.1: (a)Electricfieldaroundapointchargeinanisotropicdi-
B = µH+αE. (4) electric(bluesolidline)andinamagnetoelectricwithα increased
byafactorof20,000(reddottedline). Theparametersusedarethe
Here(cid:15)andµaretheusualdielectricpermittivityandmagnetic measured values for Cr2O3, averaged to give isotropic values. (b)
permeability tensors, which have the same isotropic symme- Calculatedmagneticfield,H(r),generatedbyapointchargeinan
try as the magnetoelectric tensor. Insertion of these field ex- isotropicmagnetoelectricinacutthroughtheaplanecontainingthe
charge.Thearrowsindicatethefieldorientation.
pressions into Maxwell’s equations yields a system of cou-
pleddifferentialequationswhichwesolvebyintroductionof
amagneticandelectricpotential14 (seeappendixA)toyield
component α(ps/m) εr(ε0) µr(µ0)
thefields:
⊥ 0.734 10.3 1.0014
µ q
E(r) = e (cid:107) -0.233 10.9 1.0001
4π(εµ−α2)r2 r
α q
H(r) =− 4π(εµ−α2)r2er , (5) TatAivBeLpEerIm:EeaxbpielirtiymµenrtfaolrvaClru2eOs3o1f9–α2,1.reεlartiivsempeearmsuirtetidviatytrεoroamndtermel--
wheree istheunitvectorintheradialdirection.Firstwedis- perature,whereasµrandα arethelowtemperature(4K)values.SI
r unitsareusedthroughout,soαhasunitsofinversevelocity.
cusstheelectricfield. Weseethatitsformisidenticaltothat
induced by a free charge in an ordinary dielectric – 1 q –
4πε r2
exceptthatthemagnitudeoftheinducedfieldismodifiedwith
aneffective(lowered)dielectricconstantε =(ε−α2/µ). Themagneticfield,H(r),showsasimilarpurelydivergent
eff
radial dependence as the electric field. We illustrate this in
To estimate the size of the lowering we use literature values
Fig. 1 right panel, where the arrows represent the magnetic
for Cr O (Table I) averaged as in Eqn. (3) to extract the
2 3
fieldlines.Note,however,thatwhileH(r)diverges,M(r)=
isotropic part, and find that the change in relative permittiv-
ityissmall,with∆ε =α2/(µε )∼1.5×10−8. InFig.1(a) αE alsodivergesandexactlycompensatesthedivergencein
r 0
H. As a result, ∇·B(r)=0, Maxwell’s equations are not
wecomparetheelectricfieldofapuredielectricwiththatof
violatedandno“true”magneticmonopoleisgenerated.
anisotropicmagnetoelectric,withα 20,000timeslargerthan
that of Cr O , for the true α the curves would be indistin-
2 3
guishablefromeachother.
Eqns.5indicatethatifα2→εµ,boththeelectricandmag-
A. Afinitesphereofanisotropicmagnetoelectric
neticfieldsinsidethemediumwoulddivergecausinga“mag-
netoelectriccatastrophe”. Thisisconsistentwiththeinequal-
We now extend our discussion to the case of a sphere of
ity
an isotropic magnetoelectric with a charge at its center in a
α ≤(cid:112)µ ε vacuum.Thefieldsinsideandoutsideofthespherearesubject
ij ii jj
totheboundaryconditions:
required for thermodynamic stability15 and the stricter con-
straint,derivedusingthermodynamicperturbationtheoryand φi (R) = φo (R)
e,m e,m
requiring positive definiteness of the free energy with re-
Di(R)·n = Do(R)·n
spect to external fields16, that α2 < χ χ . Even close
m e
to a phase transition where α may diverge, either ε or Bi(R)·n = Bo(R)·n (6)
µ will diverge simultaneously avoiding a magnetoelectric
catastrophe. This scenario was discussed recently for a whereφ aretheelectricandmagneticpotentials,theoandi
e,m
magnetoelectric-multiferroic phase transition using Landau superscriptsindicateoutsideandinsideofthesphere,respec-
mean-field theory17 and demonstrated numerically for the tively,Risthesphereradius,andnisaunitvectoralongthe
case of a strain-induced multiferroic phase transition18 in surface normal. Di and Bi are given by the expressions of
CaMnO . Eqns.4includingthemagnetoelectricresponses,whereasfor
3
3
thevacuumregiontheusualrelations
a) b)
D = ε E
0 -3
B = µ H
0 V) -2 A) 0.3
µ p
anpopmlyia.lsW,eanedxpsainncdetthheepsoytsetnetmialissirnadaiablalysissyomfmLeegtreincdarlelpcoolmy-- r()(e -1 r()(m 00..12
φ φ
ponentsapartfroml=0vanish. Thefullderivationisgiven
0 0.0
intheappendixB,andoursolutionsare: 0 R/3 2R/3 R 0 R/3 2R/3 R
c) d)
φi(r) = µ q+(α2+µ(ε−ε0)) q -3
e 4π(εµ−α2)r (α2−ε µ) 4πεR
0 m) -2 m) 0.4
φo(r) = 1 q V/ A/
e 4πε r m -1 n 0.2
0 ( (
φmi(r) = −4π(εµα−α2)rq+4π(εµα−α2)Rq Er() 00 R/3 2R/3 R Hr() 0.00 R/3 2R/3 R
φo(r) = 0 . (7)
m
FIG.2: (a)Electricand(b)magneticpotentialasafunctionofdis-
tance from the center of the sphere. (c) Electric and (d) magnetic
We plot our calculated potentials and fields, obtained using
field.Thesphereradius,R=1mm.
the parameters for Cr O and a sphere radius, R=1mm, in
2 3
Fig.2.
Again we begin by discussing the electric field case. In
We obtain the fields in the same manner as in the isotropic
Figs. 2 (a) and (c) we show the radial components of the
case, working in this case in elliptical coordinates to more
electric potential and the electric field inside and outside of
readily treat the different responses along the perpendicular
thesphere. Weseethatthebehaviorisanalogoustothatofa
and parallel axes. We note that, by symmetry, the µ and (cid:15)
chargeatthecenterofadielectricsphere,withthefieldfalling
tensors have the same form as α, but the ratios of their par-
off radially within the sphere, then more sharply in the vac-
allelandperpendicularcomponentsarenotnecessarilyequal.
uumregion. Thereisarenormalizationofthedielectriccon-
Asaresultfullyanalyticalsolutionsarenotaccessibleandthe
stant,however,fromthemagnetoelectricresponse. InFigs.2
solutions–giveninappendixC–mustbeevaluatednumeri-
(b) and (d) we show the corresponding magnetic quantities.
cally.
As in the case of the infinite medium, the H field diverges
We begin by discussing the differences we expect from
withradialdistancefromthecharge,andM alsodivergesso
the isotropic case as a result of the anisotropy in each re-
that ∇·B is zero. Consequently the magnetic field vanishes
sponse tensor. In the isotropic case (Eqns. 5) we found
attheboundaryduetothezeromagnetoelectriceffectoutside
that the electric field is described by an effective dielectric
the sphere. As a result, all monopolar effects of the charge
constant ε = ε−α2/µ. For Cr O , ε ∼10−12F/m and
areconstrainedtowithinthesphere,andnomagneticfieldis eff 2 3
α2/µ ∼10−19F/m. Therefore we expect that the anisotropy
detectableoutside.
of the electric field will be dominated by the anisotropy in
Our main finding from this section, therefore, is that,
the dielectric constant. For the magnetic field, the ε−α2/µ
while a charge in an isotropic magnetoelectric induces a
term in the denominator is again dominated by ε. However,
monopole-likemagnetizationthroughthemagnetoelectricef-
thereisalsoaprefactorofα/µ,andinthecaseofCr O the
fect, the magnetic field associated with this magnetization 2 3
stronglyanisotropicmagnetoelectrictensorshoulddetermine
drops to zero at the boundary of a spherical sample. Thus,
theanisotropyinthemagneticfield. Finally,wenotethatthe
while the induced field might have a profound effect on the
magneticfieldcanbecome“accidentallyisotropic”evenwhen
magnetotransport3, it cannot be directly detected outside of
all tensors are anisotropic, if α /(µ ε )=α /(µ ε(cid:107)). We
thesample. Wepointout, however, thatanydeviationsfrom ⊥ ⊥ ⊥ (cid:107) (cid:107)
willdiscussanexampleofthisbehaviorlater.
ideal sphericity in the sample will allow a non-zero, non-
Now we look at our explicitly calculated azimuthal de-
monopolarexternalfield.
pendenciesoftheelectricandmagneticpotentials,whichwe
showaspolarplotsinFigs.3(a)and(b)forfourdifferentsets
ofparameters:
II. AUNIAXIALMAGNETOELECTRIC
• The isotropic case obtained using averaged values for
allthreetensors(blacklines),
Wenowdiscussthecaseofananisotropicuniaxialmagne-
toelectricwithamagnetoelectrictensorofthefollowingform:
• TheactualmeasuredparametersforCr O (reddashed
2 3
  lines),
α 0 0
⊥
α= 0 α⊥ 0. • The measured Cr2O3 ε and µ values, but with only
0 0 α thetracefreequadrupolarpartoftheα tensorincluded
(cid:107)
4
(greendottedline),and a) |φe(θ)|(arb.u.) b) φm(θ)(arb.u.)
1.5 1.5
isotropic
• The “accidentally isotropic” case, enforced by setting
quad
(ε,α)⊥= 21(ε,α)(cid:107)andµ(cid:107)=µ⊥(bluedashedline). 1.0 Cr2O3 1.0
0.5 acc.iso. 0.5
We see in Fig. 3 (a) that the electric potentials for the
θ
true, isotropic, andquadrupolar-onlycaseslieexactlyontop 0.0
of each other, and are almost symmetrical, confirming our 0.5 1.0 1.5 -1.0-0.5 0.5 1.0
-0.5 -0.5
reasoning above that the asymmetry in ε (which is small in
Cr2O3)determinesthatintheelectricpotential. Theacciden- -1.0 -1.0
tallyisotropiccasehasanelliptical-shapedelectricpotential,
sinceweartificallysetε = 1ε . -1.5 -1.5
⊥ 2 (cid:107)
In contrast there are large differences in the magnetic po- c) d)
tential for the four different cases. As expected there is no 0.4 0.4
angulardependencefortheisotropicoraccidentallyisotropic
cases, which overlie each other. The quadrupolar-only case 0.2 0.2
(greenline)changessignwithangle,consistentwithitstrace- m) m)
less magnetoelectric tensor. The true Cr2O3 potential (red z(0.0 z(0.0
line)isthesumoftheisotropicandquadrupolarcontributions
-0.2 -0.2
andshowsnosignchangeastheisotropicmonopolarcontri-
butiondominates. InFigs.3(c)and(d)weshowvectorplots
-0.4 -0.4
of the magnetic fields for the true Cr O parameters and the
2 3 -0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4
purely quadrupolar case respectively. (d) shows clearly the x(m) x(m)
fieldlinespointingoutwardsvertically,andinwardshorizon-
e) f)
tally,inthemannerofaquadrupole,whereas,becauseofthe
0 0
additional, larger, monopolar component in true Cr O , (c)
showsoutwardpointingfieldlinesinalldirections. 2 3 u.) u.) Cm
To quantify this division into monopolar and quadrupolar arb. -5 Cm arb. -5 C0m2
( 0 (
contributions,weexpandthecalculatedelectricandmagnetic Cn-10 Cm2 Cn-10
potentials in linear combinations of Legendre polynomials,
-2 -1 0 1 2 0 1 2
wherebysymmetryonlyeventermsarenon-zero:
α(cid:107)/α⊥ ε(cid:107)/ε⊥
q ∞
φe,m(r,θ)= 4π2r ∑(4n+1)C2e,nmP2n(cos(θ)) . (8) FIG. 3: (a) Electric and (b) magnetic potentials for four example
n=0 setsofparemeters(fordetailsseetext.) (c)and(d): Magneticfields
H(r)inacutthroughaplanecontainingthehighsymmetryaxis,for
Theneachcoefficient,C ,intheexpansionindicatesthecon-
2n Cr O (c)anditsquadrupolar-onlycontribution(d).Thearrowsindi-
2 3
tributionofthetermwiththecorrespondingazimuthaldepen-
catethefieldorientation.(e)and(f)Evolutionoftheexpansioncoef-
dence, withC0 representing the monopolar contribution, and ficentsCmandCmcorrespondingtothemonopolarandquadrupolar
0 2
C2 thequadrupolar. Notethatallcontributionshavethesame componentsofthemagneticpotential,asafunctionoftheratio α(cid:107)
r−1 dependence,incontrasttotheusualmultipoleexpansion α⊥
and ε(cid:107).
inwhichamultipoleoforderndecaysasr−(n+1),becauseof ε⊥
theinfinitesizeofthesystem.
The relative contribution of eachC is determined by the
2n
relative ratios of α(cid:107), ε(cid:107) and µ(cid:107). In Table I we already saw TbPO4 which has one of the largest known magnetoelectric
α⊥ ε⊥ µ⊥ coefficients22 – charges in such materials do not generate
that the highest anisotropy occurs in the α tensor, with the
purely quadrupolar electric and magnetic fields. This is be-
dielectric constant and permeability being almost isotropic.
cause, while the magnetoelectric tensor can be traceless, the
(Note however that the magnetic susceptibility is strongly
permittivityandpermeabilitytensorscannot,sincethediag-
anisotropic with χ /χ ∼15 in Cr O 19). In Fig. 3 (e) we
⊥ (cid:107) 2 3 onal components of ε and µ are required for stability to be
plottheCmandCmexpansioncoefficientsasafunctionofthe
0 2 greaterthan1.
ration α(cid:107), with the ε and µ tensors constrained to be purely
α⊥
isotropic.Wefind,asexpected,thatfor α(cid:107) =−2theresponse
α⊥ A. Afinitesphereofauniaxialmagnetoelectric
ispurelyquadrupolar,whereasat α(cid:107) =1itispurelymonopo-
α⊥
larwithalinearconnectionbetweenthetwolimits. Forcom-
Finally,weinvestigatethecaseofasphereofuniaxialmag-
parison we show in (f) the same coefficients as a function of
netoelectricsurroundedbyvacuumwithapointchargeatits
the corresponding asymmetry in ε and with an isotropic α.
center, and solve the corresponding equations subject to the
For ε(cid:107) =1apurelymonopolarstatefound. sameboundaryconditionsasinSectionIA.Thesolutionsin-
ε⊥
Finally, we note that, in spite of the fact that materials side the sphere are identical to the previous solution for the
with traceless magnetoelectric tensors exist – an example is infinite uniaxial case. Each coefficient Ce,m then couples to
2n
5
a) b)
a corresponding outside solution, with the exception of Cm, (cid:107) (cid:107)
0 Cr O quadrupolar
whichvanishesoutsideofthesphereasintheisotropiccase, 2 3 ⊥ ⊥
0.2 0.2
ensuring that ∇·B is always zero and Maxwell’s equations
arenotviolated. HigherordercoefficientsofH(r)andE(r) m)0.1 0.1
A/ T)
are, however, non-zero in the outside region, and in particu- (n0.0 (f0.0
larafinitequadrupolarmagneticfieldpropagatesbeyondthe H0.1 B0.1
sphere. Thefullderivationofthesolutionispresentedinthe
0.2 0.2
appendix. 0 R/2 R 3R/2 0 R/2 R 3R/2
Asinthecasefortheinfinitesystem,wefindthattheelec- r r
tricfieldforrealisticparametersisclosetoisotropictherefore c) d)
we move directly to a discussion of the magnetic field. We 2R
consider two sets of parameters: The genuine Cr O values,
2 3
andtakingonlythetracefreequadrupolarpartoftheαtensor. R
InFig.4,(a)weshowthenormalcomponentofthemagnetic
field in both cases (dotted blue line: quadrupolar only, red 0
z
line: full response) along the two high symmetry directions
(⊥ and (cid:107)). In contrast to the isotropic monopolar case, the -R
fielddoesnotvanishattheinterfacebutextendsintothevac-
uum area. Note that, since only the quadruoplar component -2R
ofthemagneticfieldextendsoutofthesphere, thefieldsare
-2R -R 0 R 2R -2R -R 0 R 2R
identicaloutsideofthesphereinthetwocases,inspiteofthe x x
factthattheydifferconsiderablywithinthespherewherethe
monopolar contribution can manifest. In Fig. 4 (b), we plot FIG.4:(a)Magneticfieldand(b)magneticinductionasafunctionof
thenormalcomponentofthemagneticinduction,B,whichis radialdistancefromthecenterofthespherealongthehighsymmetry
identicalinthetwocases. ⊥and(cid:107)directions. Thebluelinescorrespondstothequadrupolar-
onlyresponse(dashed⊥,dot-dashed(cid:107)),andtheredlineshowsthe
In Figs. 4 (c) and (d) we show the magnetic field lines
caseofCr O (solid⊥,long-dashed(cid:107)). (c)and(d)showthemag-
in a cut through a plane containing the high symmetry axis 2 3
neticfieldsinaslicethroughaplanecontainingthehigh-symmetry
forCr O andthepurelyquadrupolarcaserespectively. The
2 3 axisforCr O andthequadrupolar-onlymaterialrespectively. The
2 3
redcircleindicatesthesphereboundaryinbothcases. Inside
arrowsshowtheorientationandmagnitudeofthefield,withthemag-
the sphere the solutions are identical to the infinite solutions nitudeweightedbythesquareofthedistancefromthecharge.
showninFig.3(c)and(d). Againweseethatintheoutside
regionagainbothfieldsareidentical,asonlythequadrupolar
field extends out of the sphere. This indicates that magneto- even without an internal charge. The order of magnitude of
electricswiththesametracefreepartofthetheαtensorarein- thisfieldatthesurfaceis µB ,withaanatomicdistanceand
distinguishablefrommeasurementsoftheirexternalmagnetic aV1/3
V thevolumeofthesample. Forthesphericalsamplewithra-
field,providedthattheε andµ tensorsareclosetoisotropic.
dius 1mm that we have considered here, explicit calculation
ofthesurfacemagneticinductionfromthiscontributiongives
B =7×10−7T.Thecontributiontotheexternalquadruopo-
int
B. Comparisonwithothersourcesofmagneticfieldandwith lar field from the charge-induced effect derived in this work
experiment thus becomes equal to the intrinsic contribution at an elec-
tronconcentrationof∼107 per1mmradiussphere,or∼1010
Finally,wecompareourcalculatedvaluesforthemagnetic cm−3. Thisconcentrationislikelyachievablewithfieldeffect
fieldgeneratedbyachargeinamagnetoelectricwiththataris- doping (although this might not be convenient in the spher-
ing from two other sources: the intrinsic magnetoelectric re- ical geometry). For the larger sphere used in the measure-
sponse of an uncharged material, and the surface moments ments of Ref. 9 the predicted value from the intrinsic mech-
from the antiferromagnetism. We then discuss the relevance anism is correspondingly smaller because of the 1 depen-
V1/3
ofthesepossiblecontributionstotheearliermeasurementsof dence, and is indeed consistent with the measured value of
Astrovandcoworkers9. B =1×10−9T9. The contribution from the charge effect
exp
First we point out that, in Fig. 4 (b) we see that an elec- becomes comparable at even smaller charging levels, and a
tron inside a magnetoelectric sphere of radius 1mm gener- superposition of the two contributions might be responsible
atesamagneticinductionofB ∼10−16Tatthesurfaceof forthecomplicatedmeasuredradialdependence.
ME
thesphereviathemagnetoelectriceffect. Wenotethatasin- Finally we point out that there is an additional contribu-
gleelectronspininanon-magnetoelectricsphereofthesame tion to the magnetic field around a sphere of antiferromag-
dimension and permeability would generate a much smaller netic material, resulting from truncation of the discrete spin
fieldofonly10−23T,andsointhiscontext,themagnetoelec- arrangementofanantiferromagnet12. These“Andreevfields”
tricresponseshouldnotberegardedassmall. areinprincipledistinguishablefromtheintrinsicandcharge-
InRef.11weshowedthattheintrinsicstructureandmag- induced magnetoelectric fields as they are genuine surface
neticorderinCr O giverisetoanexternalquadrupolefield fields which decay exponentially with the distance from the
2 3
6
surface. In addition, their form is sensitive to the details of ming from the isotropic or tracefree parts of the ME ten-
the surface termination and is not required to be quadrupo- sor respectively. Consistent with Maxwell’s equations, how-
lar. Recently it was argued that the Andreev field is funda- ever,themonopolarmagneticcontributiondoesnotpropagate
mentallydifferentinamagnetoelectricmaterialthaninanon- outside of a finite sample, and so magnetoelectrics with the
magnetoelectricantiferromagnet13. same tracefree part of the α tensor are indistinguishable by
magnetic field measurements. Interestingly, even a sphere
of purely monopolar isotropic magnetoelectric material will
C. Conclusion have an external quadrupolar magnetic field because of the
interactions between dielectric and magnetic permeabilities
In summary, we have derived the static electric and mag- and the magnetoelectric tensor. The magnitude of the ex-
netic fields induced by a free charge in a diagonal magne- ternal magnetic field induced by charges can be comparable
toelectric. We found that the electric fields are analogous with the intrinsic quadrupolar field as well as that from the
to those generated by a charge in a simple dielectric, with surfacemagnetizationforrealisticgeometries,andallcompo-
a renormalization of the dielectric permittivity by the mag- nents should be considered in interpreting measurements of
netoelectric response. A charge generates both monopolar fieldsaroundmagnetoelectrics.
and quadrupolar magnetic fields inside the material, stem-
1 M.Fiebig,J.Phys.D38,R123(2005). AppendixA:Detailedderivationforapointchargeinan
2 N. A. Spaldin, M. Fechner, E. Bousquet, A. Balatsky, and isotropicmagnetoelectric
L.Nordstro¨m,Phys.Rev.B88,094429(2013).
3 D.I.Khomskii,arXivp.1307.2327v1(2013).
Herewepresentthedetailsofthederivationforthecaseof
4 I.Dzyaloshinskii,SovietPhys.Jetp-USSR10,628(1960).
5 N. A. Spaldin, M. Fiebig, and M. Mostovoy, J. Phys. Condens. a point charge in an isotropic magnetoelectric material. We
follow Ref. 14 to calculate the scalar electric and magnetic
Matter20,434203(2008).
6 C.EdererandN.A.Spaldin,Phys.Rev.B76,214404(2007). potentials,φe(r)andφm(r),usingthematerialsequations
7 F.W.Hehl,Y.N.Obukhov,J.P.Rivera,andH.Schmid,Eur.Phys.
J.B71,321(2009). B = µH+αE
8 S.CohandD.Vanderbilt,Phys.Rev.B88,121106(2013).
D = (cid:15)E+αH ,
9 D.N.Astrov,N.B.Ermakov,A.S.BorovikRomanov,E.G.Kol-
evatov,andV.I.Nizhankovskii,PismaZh.Eksp.Teor.Fiz.[JETP
Lett.]63,745(1996). wherethelasttermsineachexpressionresultfromthemagne-
10 D. N. Astrov and N. B. Ermakov, Pisma Zh. Eksp. Teor. Fiz. toelectriceffect,andthepermittivityandpermeabilitytensors
[JETPLett.]59,297(1994). µ and (cid:15) have the same symmetry as α. Using ∇×H =0
11 I.Dzyaloshinskii,SolidStateCommun.82,579(1992). and thus H =−∇φ (r) (as there is no free charge current),
m
12 A.F.Andreev, PismaZh.Eksp.Teor.Fiz.[JETPLett.]63, 758 ∇·B=0and∇·D=ρ,weobtain
(1996).
13 K.D.Belashchenko,Phys.Rev.Let.105,249902(2010). µ∇2φ (r) + α∇2φ (r)=0
14 Jackson,J.D.,ClassicalElectrodynamics3rdEdition(JohnWi- m e
ley&Sons,2007). (cid:15)∇2φe(r) + α∇2φm(r)=qδ(r) . (A1)
15 T.H.O’Dell,Philos.Mag.8,411(1963).
16 W.Brown, R.M.Hornreich, andS.Shtrikman, Phys.Rev.168,
SolvingEqs.A1simultaneouslyweobtaintrivially:
574(1968).
17 I.Dzyaloshinskii,Europhys.Lett.96,17001(2011).
qµ 1
18 E.BousquetandN.A.Spaldin,Phys.Rev.Lett.107(2011). φ (r) =
19 S.Foner,Phys.Rev.130,183(1963). e 4π(εµ−α2)r
20 H.Wiegelmann,A.Jansen,P.Wyder,J.P.Rivera,andH.Schmid, α qα 1
Ferroelectrics162,141(1994). φm(r) = −µφe(r)=−4π(εµ−α2)r .
21 H.B.Lal,R.Srivasta,andK.G.Srivastava,Phys.Rev.154,505
(1967).
22 G. T. Rado and J. M. Ferrari, AIP Conference Proceedings
Thefieldsarethenobtainedstraightforwardlyfromthegradi-
10, 1417 (1973), URL http://scitation.aip.org/
entsofthepotentials:
content/aip/proceeding/aipcp/10.1063/1.
2946809.
µ q
E(r) = e
4π(εµ−α2)r2 r
α q
H(r) = − e ,
4π(εµ−α2)r2 r
withe theradialunitvector.
r
7
AppendixB:Detailedderivationforapointchargeinasphere where n is a unit vector in the radial direction. Inserting D
ofisotropicmagnetoelectricinvacuum and B explicitly in the last two equations and changing to
potentialsratherthenfieldsyields
For the boundary problem we proceed by expanding the
potentialsφm(r)andφe(r)intermsofLegendrePolynomials, (εEi+αHi)·n = (ε Eo)·n
P(cos(θ)): o
l ε∇ φe(r)+α∇ φi(r) = ε ∇ φo(r)
r i r m o r e
φi(r,θ)=∑(2l+1)(cid:104)Alrl+Blr−(l+1)(cid:105)Pl(cos(θ)) (B1) (µHi+αEi)·n = (µoHo)·n
l µ∇ φm(r)+α∇ φi(r) = µ ∇ φo(r) , (B4)
r i r e o r m
withexpansioncoefficientsA andB.Inthelimitr→∞both
l l
potentialstendtozero,andsoAl=0fortheoutsidepotentials. leadingtothesolution
Forr→0thepotentialsshouldnotdivergethusB iszerofor
l
theinsidepotentials. Therefore,usingourresultsforthepoint
(εµ)−µε
chargeinaninfinitemedium,wewritetheAnsatz: A = 0
0 4π(εµ−α2)R
qµ 1 q
φi(r) = +∑(2l+1)ArlP(cos(θ)) B =
e 4π(εµ−α2)r l l 0 4πε
l 0
α
φeo(r) = ∑(2l+1)Blr−(l+1)Pl(cos(θ)) C0 = 4π(εµ−α2)R
l
qα 1 D0 = 0 , (B5)
φi(r) = − +∑(2l+1)CrlP(cos(θ))
m 4π(εµ−α2)r l l
l
φo(r) = ∑(2l+1)Dr−(l+1)P(cos(θ)) , (B2) whereRisthesphereradiusandallothercoefficientsarezero.
m l l AppendixC:Detailedderivationforapointchargeina
l
diagonaluniaxialmagnetoelectric
where the i and o superscripts indicate the solutions for in-
side and outside of the sphere. (Since the Legendre expan-
Nowweconsiderthecaseofauniaxialmaterialwithmag-
sion solves only the homogenous equation, the source terms
netoelectrictensoroftheform:
must be added to account for the charge in the center.) The
coefficients are found from the boundary conditions for the
 
correspondingfields: α⊥ 0 0
α= 0 α⊥ 0
φi (R) = φo (R) 0 0 α
e,m e,m (cid:107)
Di(R)·n = Do(R)·n
Bi(R)·n = Bo(R)·n , (B3) TheninsteadofequationsA1wehave:
µ ∇2φ (r)+µ ∇2φ (r)+α ∇2φ (r)+α ∇2φ (r) = 0
⊥ ⊥ m (cid:107) (cid:107) m ⊥ ⊥ e (cid:107) (cid:107) e
ε ∇2φ (r)+ε ∇2φ (r)+α ∇2φ (r)+α ∇2φ (r) = qδ(r) , (C1)
⊥ ⊥ e (cid:107) (cid:107) e ⊥ ⊥ m (cid:107) (cid:107) m
Fouriertransformationoftheaboveexpressionyields
µ k2φ (k)+µ k2φ (k)+α k2φ (k)+α k2φ (k) = 0
⊥ ⊥ m (cid:107) (cid:107) m ⊥ ⊥ e (cid:107) (cid:107) e
ε k2φ (k)+ε k2φ (k)+α k2φ (k)+α k2φ (k) = q . (C2)
⊥ ⊥ e (cid:107) (cid:107) e ⊥ ⊥ m (cid:107) (cid:107) m
Eqns.C2canthenbesolvedtoyieldthefollowingexpressionsforthescalarpotentialsintheuniaxialcase:
µ k2 +µ k2
⊥ ⊥ (cid:107) (cid:107)
φ (k) = q
e (ε k2 +ε k2)(µ k2 +µ k2)−(α k2 +α k2)2
⊥ ⊥ (cid:107) (cid:107) ⊥ ⊥ (cid:107) (cid:107) ⊥ ⊥ (cid:107) (cid:107)
α k2 +α k2
⊥ ⊥ (cid:107) (cid:107)
φ (k) = −1 φ (k) . (C3)
m µ k2 +µ k2 e
⊥ ⊥ (cid:107) (cid:107)
8
Transformingtosphericalcoordinatesk =ksin(θ)andk =kcos(θ)yields
⊥ (cid:107)
µ⊥sin(θ)2+µ(cid:107)cos(θ)2 q q
φ (k)= = f (θ) . (C4)
e (ε sin(θ)2+ε cos(θ)2)(µ sin(θ)2+µ cos(θ)2)−(α sin(θ)2+α cos(θ)2)2k2 e k2
⊥ (cid:107) ⊥ (cid:107) ⊥ (cid:107)
Therealspacesolutionisthenobtainedfromtheinversetrans- Settingφ−φ(cid:48)=φ,theintegrationoverφ gives
formation
φ (r)=(cid:90) 2π(cid:90) π(cid:90) ∞ f (θ) q eik·rsin(θ)k2dφdθdk, 1(cid:90) 2πeiksin(θ)sin(θ(cid:48))cos(φ)dφ =2πJ (ksin(θ)sin(θ(cid:48))),
e e k2 r 0
0 0 0 0
wherebysymmetry
withJ (x)theBesselfunction,andthatoverkisthestandard
0
k·r=cos(θ)cos(θ(cid:48))+sin(θ)sin(θ(cid:48))cos(φ−φ(cid:48)) . tabulatedintegral
(cid:40)
r1(cid:90)0∞J0(ksin(θ)sin(θ(cid:48)))eikcos(θ)cos(θ(cid:48))dk=ξ(θ,θ(cid:48))= (0(cid:112)cos(θ)2cos(θ(cid:48))2−sin(θ)2sin(θ(cid:48)))−1 efolsre cos(θ)2cos(θ(cid:48))2>sin(θ)2sin(θ(cid:48))2
sowearriveatthefinalexpressionforφ : with
e
2πq(cid:90) π
φ (r,θ(cid:48))= f (θ)ξ(θ,θ(cid:48))sin(θ)dθ . (C5)
e e
r
0
Usingthesameapproachweobtain
2πq(cid:90) π
φ (r,θ(cid:48))= f (θ)ξ(θ,θ(cid:48))sin(θ)dθ (C6)
m m
r
0
α sin(θ)2+α cos(θ)2
⊥ (cid:107)
f (θ)= . (C7)
m (ε sin(θ)2+ε cos(θ)2)(µ sin(θ)2+µ cos(θ)2)−(α sin(θ)2+α cos(θ)2)2
⊥ (cid:107) ⊥ (cid:107) ⊥ (cid:107)
Wecanrewritetheintegralequationsintermsofthefunction noms:
ξ(θ,θ(cid:48))as
φm(r,θ(cid:48))= 2πrq(cid:90)|ππ/−2−|πθ/(cid:48)2|−θ(cid:48)| fm(θ)ξ2(θ,θ(cid:48))sin(θ)dθ , φe(r,θ) = 1rn∑=∞1C2enP2n(cos(θ))
(C8) 1 ∞
with φm(r,θ) = r ∑C2mnP2n(cos(θ)), (D1)
n=1
(cid:18)(cid:113) (cid:19)−1
ξ (θ,θ(cid:48))= cos(θ)2cos(θ(cid:48))2−sin(θ)2sin(θ(cid:48)) (C9)
2
where the coefficientsCe andCm arise from the solutions of
thebulkuniaxialcase. Duetosymmetryonlyeventermsap-
pearinthisexpansion. Furthermoreforphysicallyreasonable
AppendixD:Detailedderivationforapointchargeina
diagonaluniaxialmagnetoelectricsphere parametersthecoefficientsconvergerapidlyandtermslarger
than2n=8arepracticallyzerosotheseriescanbetruncated
inthisorder.
Fortheuniaxialspherewestartfromthepreviousconsider-
ationsandexpandthetwopotentialsagaininLegendrePoly- Next we consider the boundary problem where we solve
9
againthefollowingsystemofequations: becomesinsphericalcoordinates
φi (R) = φo (R)  
e,m e,m α α 0
11 12
Di(R)·n = Do(R)·n αsph=α21 α22 0 , (D5)
Bi(R)·n = Bo(R)·n 0 0 α33
Ei(R)·t = Eo(R)·t
Hi(R)·t = Ho(R)·t , (D2) withthefourcoefficientsgivenby
wheretisaunitvectorinthetangentialdirection. Hereiand 1(cid:2) (cid:3)
α = (α +α )+(α −α )cos(2θ)
o superscript denote the solutions inside and outside of the 11 2 ⊥ (cid:107) (cid:107) ⊥
sphere. Thepotentialsaregivenbythefollowingequations: α = α =(cid:2)(α −α )(cid:3)sin(θ)cos(θ)
12 21 ⊥ (cid:107)
(cid:18)Ce (cid:19) 1(cid:2) (cid:3)
φei(r) = ∑ rl +(2l+1)Alrl Pl(cos(θ)) α22 = 2 (α⊥+α(cid:107))+((α⊥−α(cid:107))cos(2θ)
l α = α . (D6)
33 ⊥
B
φo(r) = ∑(2l+1) l P(cos(θ)) For the radial contributions at the sphere boundary only α11
e r−(l+1) l andα enterandthenormalcomponentofDibecomes:
l 12
(cid:18)Cm (cid:19)
φi(r) = ∑ l +(2l+1)Crl P(cos(θ))
m l r l l Di·n=ε ∂φei +ε 1∂φei +α ∂φmi +α 1∂φmi (D7)
11 12 11 12
D ∂r r ∂θ ∂r r ∂θ
φo(r) = ∑(2l+1) l P(cos(θ)). (D3)
m r−(l+1) l
l andDo:
Again we superimposed here our solutions with those of the
∂φo
homogenousLaplaceequations,takingintoaccountthelimits Do·n= e . (D8)
r →0 and r →∞, for inside and outside of the sphere, re- ∂r
spectively. Thefoursetsofcoefficients,A,B,C andD,are
l l l l
nonzeroonlyforl=2n. Asbefore,thedisplacementfieldDi Forthetangentialcomponentoftheelectricfieldwehavethe
andmagneticinductionBiinsidethespherearegivenby: followingequations
Di = εE+αH 1∂φi
Bi = µH+αE Ei·t= r ∂θe (D9)
andoutsidethespherebytheusualexpressionsforthevacuum
and
Do = E
1∂φo
Bo = H. Eo·t= e . (D10)
r ∂θ
Since we are working in spherical coordinates, we have to
transformourtensorstothesphericalcoordinatesystem.Thus Finally, toobtainthesystemofequationsforthecoefficients
α incartesaian: wemultiplybyPn(cos(θ))(fornormalizationoftheLegendre
Polynomials=andintegrateover 2n+1(cid:82)πdθ Weobtainthatall
  n 0
α 0 0 odd coefficients are zero, and that the coefficients B and D
⊥ l l
αcart = 0 α⊥ 0 (D4) decayasexpectedwithincreasingl.
0 0 α
(cid:107)