Table Of ContentLyddane-Sachs-Teller relationship in linear magnetoelectrics
Raffaele Resta
Dipartimento di Fisica, Universita` di Trieste, Italy,
and DEMOCRITOS National Simulation Center,
IOM-CNR, Trieste, Italy
(Dated:)
In a linear magnetoelectric the lattice is coupled to electric and magnetic fields: both affect the
longitudinal-transversesplittingofzone-centeropticalphononsonequalfooting. Aresponsematrix
relatesthemacroscopicfields(D,B)to(E,H)atinfraredfrequencies. Itisshownthattheresponse
matrices at frequencies 0 and ∞ fulfill a generalized Lyddane-Sachs-Teller relationship. The rhs
member of such relationship is expressed in terms of weighted averages over the longitudinal and
transverse excitations of themedium, and assumes a simple form for an harmonic crystal.
PACSnumbers: 75.85.+t,77.22.-d
1
1
0
The original Lyddane-Sachs-Teller (LST) relation- to macroscopic electric fields, in linear MEs it is due to
2
ship[1]appliestothesimplecaseofacubicbinarycrystal thecouplingofboth(electricandmagnetic)fieldsonthe
n
in the harmonic regime. It relates four macroscopically same footing: this is perspicuous in the lhs of Eq. (3).
a
J measurable constants as The simple form of Eqs. (1) and (3) requires a
4 ε(0) ω2 crystalline system with only a single IR-active mode at
= L. (1) the zone center. The additional requirement of cubic
] ε(∞) ωT2 symmetry can be relaxed; it is nonetheless convenientto
i
c consideronlycrystalswhosesymmetryisorthorombicor
s Here ε(0)is the static dielectric constant,whichincludes
higher;thenallcrystallinetensorscanbesimultaneously
- thelatticecontribution,ε(∞)isthesocalled“statichigh
rl frequency” (a.k.a “clamped-ion”) dielectric constant, diagonalized (as e.g. in Cr2O3). This allows us to adopt
t simple scalar-like notations, as in Eqs. (1) and (3) with
m which accounts for the electronic response only, and ωL
the proviso that we deal separately with each principal
. (ωT) is the zone-center longitudinal (transverse) optical
t direction.
a frequency [2]. It is remarkable that all microscopic
m parameters (force constants, masses, Born effective OvertheyearstheLSTrelationshiphasbeenextended
in several ways, to cover cases where more than one IR-
- charges, cell volume) disappear from Eq. (1).
d active mode per direction exists [12, 13], the crystal is
Magnetoelectrics (MEs) are insulators where electric
n in a low-symmetry class [14, 15], or even the material is
fields control magnetization, and conversely magnetic
o
noncrystalline and/or anharmonic [16–19]. The general
c fields control polarization; they attracted considerable
case can be written as
[ theoretical and technological interest in recent times [3–
3 11]. The simplest and most studied single-crystal linear ε(0) hω2iL
v ME is antiferromagnetic Cr2O3 [3, 5–7]. In any linear ε(∞) = hω2iT. (4)
6 ME the role of the dielectric function ε(ω) is played
8 by the 2×2 response matrix—called R(ω) here—which Thequantitiesintherhsareweightedaverages,obtained
1
yields the macroscopic fields (D,B) in terms of (E,H) frommomentsoftheappropriatespectralfunctions. The
4
. at frequency ω: derivation is based on general principles of statistical
7
mechanics and does not require an Hamiltonian, even
0 D E ε(ω) α(ω) E
0 =R(ω) ≡ , (2) less an harmonic one [16, 18]; in the special case of a
B H α(ω) µ(ω) H
1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19) single IR-active harmonic mode per direction Eq. (4) is
: equivalent to Eq. (1). In this Letter we generalize the
v where ε, µ, and α are permittivity, magnetic permeabil-
Xi ity, and ME coupling, respectively. In this Letter we are viewpoint of Ref. [18] to the ME case, showing that
going to show that a generalized LST relationship holds
r tr {R−1(∞)R(0)}−2 hω2iL
a in the form = , (5)
2−tr {R−1(0)R(∞)} hω2iT
tr {R−1(∞)R(0)}−2 ω2
L
= . (3)
2−tr {R−1(0)R(∞)} ω2 where the rhs is defined below, Eqs. (10) and (11).
T
The presentation proceeds as follows. We start at a
In the simple case of a magnetically inert material (i.e. very general level without any microscopic assumption
µ ≡ constant, α ≡ 0) the lhs of Eq. (3) equals indeed about the ME medium, and using only very general
ε(0)/ε(∞). While in ordinary dielectrics the LO-TO principles in order to arrive at Eq. (5). We will then
splittingisduetothecouplingoftheionicdisplacements apply the general results to a crystalline system in the
2
harmonic regime, and finally we will show that Eq. (5) Next we take the trace of Eqs. (13) and (14); permuting
reduces to Eq. (3) in the single-mode case. the matrices in the product we get the identity
We write explicitly the linear response matrix of the
∞
ME medium as the sum of its real and imaginary dω ω tr {R(∞)−1R′′(ω)}
part: R(ω) = R′(ω) + iR′′(ω), and analogously for 0
Z
its inverse; both R(ω) and R−1(ω) obey the Kramers- = − ∞dω ω tr {R(ω)−1′′R(∞)}. (15)
Kronig relationships in the form
0
Z
2 ∞ ω′R′′(ω′) In order to arrive at Eq. (5) it is enough to put together
R′(ω)−R(∞) = dω′ , (6)
π 0 ω′2−ω2 Eqs. (8), (9), (10), (11), and (15).
Z
R−1′(ω)−R−1(∞) = 2 ∞dω′ ω′R−1′′(ω′). (7) ideWnteitys,trEeqss.(t1h5a)t,iastthteheassruomotptoifonthtehastuωpe=rc∞onvaecrtgueanlclye
π 0 ω′2−ω2
Z means ω much higher than all the frequencies of ionic
From these, it follows immediately that the numerator motions, yet lower than the frequencies of electronic
and denominator in the lhs of Eqs. (3) and (5) are excitations [2]. Therefore the clamped-ion response
R(∞) is a real matrix.
2 ∞ dω Next we address an harmonic crystal. A zone-center
tr{R−1(∞)R(0)}−2= tr{R−1(∞)R′′(ω)},
π 0 ω optical mode is lattice-periodical; it is then expedient
Z
(8) to consider the energy per cell of a (macroscopically
homogeneous) solid with a frozen-in phonon distortion.
2−tr{R−1(0)R(∞)}=−2 ∞ dωtr{R−1′′(ω)R(∞)}. This energy is well defined only when a prescription for
π 0 ω taking the thermodynamic limit is given. We cut a
Z
(9) sample in the shape of a slab parallel to the principal
The integrands in the rhs of Eqs. (8) and (9) are axes; we remind our assumption that all crystal tensors
interpreted here as the transverse and longitudinal are diagonal on them. If the slab is free-standing in
spectral weights, respectively, by means of which we vacuo, all fields vanish outside (E=D=H=B=0), while
define the second moments the value of the fields inside depend on the polarization
of the mode. Simple electrostatics and magnetostatics
hω2iT = 0∞∞dωωdωωt2rtr{R{R−−1(1∞(∞)R)R′′(′′ω(ω)})} (10) ismlapbly(“tthraatnsivfetrhsee”p),hobnootnh pEolaarnidzatHionvaisnipsha,rawllehlilteoifthiet
R 0 ω
hω2iL = R0R∞0∞dωωdωωωt2rtr{R{R−−1′1′(′′ω(ω)R)R(∞(∞)})}. (11) Tgisohipenegroprtdeonerdiinocfifunltaihrtey()“lilmiosnitgesist(sufiednristntiaalal,”)salaDnbd,atnthhdeenBtiwtvsoanthiesnihcekr[ng2ei2es]ss.
(longitudinal and transverse) are indeed different in the
R
The reason for the semantics (transverse and longitudi-
thermodynamic limit. Similar reasonings apply if the
nal) may appear obscure at this point; it will become
lattice-periodical mode is regardedas the k→0 limit of
clearwhenspecializingEqs.(10)and(11)toanharmonic
a finite-k optical phonon [2].
crystal—see also Eq. (26) below.
If the crystalhas N IR-active modes in each principal
Inordertoarriveatourmainresult,Eq.(5),weexploit
direction, we denote with ω the zone-center TO
n
the “superconvergence” theorem [20]. For large ω (i.e.
frequencies (i.e those with E=0 and H=0); equivalently,
forω muchlargerthanalltheresonancesofthemedium) ω2 are the eigenvalues of the analytical part of the
R′′(ω) vanishes; Eqs. (6) and (7) yield, to leading order n
dynamical matrix at k = 0. The free energy per cell in
in 1/ω2,
function of the normal coordinates u , E, and H (taken
n
as independent variables) is expanded to second order
2 ∞
R(∞)−1R(ω)≃I− dω′ ω′R(∞)−1R′′(ω′), as [3, 5]
πω2 0
Z
(12) 1
2 2
F(E,H,{un})=F0+ 2 ωnun
R(ω)−1R(∞)≃I− 2 ∞dω′ ω′R−1′′(ω′)R(∞), Xn
πω2 0 − Ω [ε(∞)E2+2α(∞)EH +µ(∞)H2]
Z
(13) 8π
whereI is the 2×2identity. InversionofEq.(12)to the − (u Z∗E+u ζ∗H), (16)
n n n n
same order gives the alternative expression
n
X
R(ω)−1R(∞)≃I+ 2 ∞dω′ ω′R(∞)−1R′′(ω′). where we adoptatomic Gaussianunits [21], and Ω is the
πω2 0 cell volume. Zn∗ are the mode-effective charges and ζn∗
Z
(14) their magnetic analogues; notice that in ordinary units
3
thenormal-modecoordinateswouldincludeafactorwith Eliminationof the u ’s fromEqs.(21) and(23) provides
n
the dimensions of (mass)1/2, while Z∗ and ζ∗ would R−1(ω) in the form
n n
include a factor with the dimensions of (mass)−1/2.
The derivatives of F provide the equations of motion R′−1(ω) = R−1(∞)− 4π Z˜nZ˜n† (24)
in the form Ω ω˜2 −ω2
n n
X
D = −4Ωπ∂∂EF =ε(∞)E+α(∞)H + 4Ωπ Zn∗un R′′−1(ω) = −4Ωπ2 Z˜nZ˜n†δ(ω˜n2 −ω2). (25)
Xn Xn
4π ∂F 4π
B = − Ω ∂H =α(∞)E+µ(∞)H + Ω ζn∗un Usingtheaboveresults,wearriveataverytransparent
n expression for the rhs member of Eq. (5):
X
∂F
fn = −∂un =−ωn2un+Zn∗E+ζn∗H. (17) hω2iL = n ω1n2Zn†R−1(∞)Zn, (26)
Wf e=the−nωc2ouns.ideErlfiomrcineadtioosncilolaftitohnesuat’sfrefqroumencEyqω. ,(1i.7e). hω2iT P n ω˜1n2Z˜n†R(∞)Z˜n
n n n
provides the linear ME response, including the lattice i.e. hω2iT is the weighPted harmonic mean of the ωn2’s,
contribution; we cast it in compact form as with weights Zn†R−1(∞)Zn, and hω2iL is the weighted
harmonic mean of the ω˜2’s, with weights Z˜†R(∞)Z˜ .
4π Z Z† n n n
R′(ω) = R(∞)+ n n (18) The two sets of weights are in general different, except
Ω ω2 −ω2
n n whenthetransverseandlongitudinaleigenmodeshappen
X
4π2 to be the same. However, “superconvergence”,Eq. (15),
R′′(ω) = Z Z†δ(ω2 −ω2), (19)
Ω n n n implies the same normalization in any case:
n
X
where the ME lattice coupling vectors are Zn†R−1(∞)Zn = Z˜n†R(∞)Z˜n. (27)
n n
Z∗ X X
Z = n , Z† = Z∗ ζ∗ . (20)
n ζ∗ n n n In the special case where a single IR-active mode exists
(cid:18) n (cid:19) (cid:0) (cid:1) hω2iT =ωT2 and hω2iL =ωL2: this concludes the proof of
Eq. (18) is the elegant result obtained in 2008 by J.
Eq. (3).
`In˜iguez[5];theTOfrequenciesω areclearlytheN poles
n A well known bound requires the matrix R to be
of R(ω).
positive definite [3]. Since this is based on stability
Actual computations performed for the paradigmatic
arguments,it only concernsR(0) and R(∞): the former
material Cr2O3 and based on Eq. (18) show that the is a genuine static property of the real system, while the
magnetoelectric coupling α(0) is significantly enhanced
latter canbe regardedas a static propertyin the infinite
by the lattice contribution [5]. Clearly, a large coupling
nuclear mass limit. At any other frequency the matrix
isthe keypropertytobe exploitedindeviceapplications
R(ω) accounts for the forced oscillations of the system,
[3].
whichisclearlyoutofequilbrium. ThereforeR(ω)isnot
It is now expedient to express Eq. (17) in terms of D
required, in general, to be positive definite; because of
and B.
the same reason, ε(ω) is not a positive real function in
E = R−1(∞) D − 4π Z u (21) ordinary dielectrics [2].
H B Ω n n All of the above results reduce to previously known
(cid:18) (cid:19) "(cid:18) (cid:19) Xn # ones for a magnetically inert material, where the 2×2
4π
−ω2un = −ωn2un− Ω Zn†R−1(∞)Xn′ Zn′un′ wmoarttrhixeRxa(mω)inhinagsEthqe. (u2n6i)qufoernζon∗n=tri0v:ial entry ε(ω). It is
+ Zn†R−1(∞) DB . (22) hω2iL = 1 n ω1n2(Zn∗)2. (28)
As explained above, in the L(cid:18)O mo(cid:19)des D=0 and B=0 by hω2iT ε2(∞)Pn ω˜1n2(Z˜n∗)2
definition [22]. The first line of Eq. (22) clearly shows The weights in the harmonicPmeans (Z∗)2 and (Z˜∗)2
thatthe LOeigenmodesareingeneraldifferentfromthe n n
are the (squared) transverse and longitudinal effective
TO ones; an explicit N ×N diagonalization is needed
charges, respectively. If (and only if) the transverse
in order to find the LO frequencies. We indicate with
and longitudinal eigenmodes are the same, then Z˜∗ =
ω˜ these frequencies, and we also transform the lattice n
n Z∗/ε(∞). Actually, this is the well known relationship
couplingvectorstothe LOeigenmodesasR−1(∞)Z → n
n between the transverse (a.k.a. Born) end longitudinal
Z˜ , in order to write Eq. (22) as
n (a.k.a. Callen) effective charges.
−ω2u =−ω˜2u +Z˜† D . (23) For an ordinary dielectric the N poles of ε(ω) are the
n n n n B TO frequencies ω , while the N poles of 1/ε(ω) are the
n
(cid:18) (cid:19)
4
LOonesω˜ : seeEqs.(18)and(24). Sincetheresponseis approximation,andfinallyisidenticaltotheoriginalLST
n
asingle-componentfunction,thepolesof1/ε(ω)coincide one when only one mode is IR active. The relationship
withthezerosofε(ω). Simpleconsiderationsaboutzeros showsthat the LO-TO splitting in a ME originatesfrom
andpolesofε(ω)eventuallyleadtothesimpleexpression the coupling of ionic displacements to both electric and
magnetic macroscopic fields.
2 N
hω iL ω˜n
= , (29)
hω2iT n=1ωn I thank J. `In˜iguez and D. Vanderbilt for illuminating
Y
discussions about magnetoelectrics. Work supported by
firstfoundin1961byKurosawa[12,19]. Thisresultdoes
the ONR grant N00014-07-1-1095.
not generalize to the ME case, for the good reason that
the responseis a2×2matrix: the presentformulationis
based on traces throughout, and the inverse of the trace
bears no simple relationship to the trace of the inverse,
at variance with the purely electrical case. [1] R. H Lyddane, R. G. Sachs, and E. Teller, Phys. Rev.
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Eq. (2) onwards. Other pairings are possible. In
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[4] W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature
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(2009).
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[10] A. M. Essin, A. M. Turner, J. E. Moore, and D.
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[14] M. Lax and D.F. Nelson, Phys.Rev.B 4, 3694 (1971).
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entries of the response matrices R to the same absolute
[19] A. J. Sievers and J. B. Page, Phys. Rev. B 41, 3455
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In conclusion, this Letter shows that the ratio [21] We adopt Gaussian units, where E and H have the
ε(0)/ε(∞) entering the LST relationship must be same dimensions, and we measure charges in atomic
replaced, for a linear ME, by the lhs of Eqs. (3) and (5), units. In such a way theparameters Zn∗ and ζn∗ are both
dimensionless before being transformed to normal-mode
whoseingredientsarethefullMEresponsesatfrequency coordinates. AsforZn∗,thisisstandardintheliterature.
0and∞,i.e. staticandclamped-ion. Inthemostgeneral [22] R. Resta, J. Phys.: Condens. Matter 22 123201 (2010).
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istheratiobetweenspectralmomentsofthelongitudinal may become much larger, while in the presence of soft
and transverse excitations of the medium. It assumes modes ε(0) may increase by some orders of magnitude.
a simple form for a crystalline ME in the harmonic [24] J. H.Lee et al.., Nature466, 954 (2010).
