Local-field corrected van der Waals potentials in magnetodielectric multilayer systems

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by  Agnes Sambale

Local-field corrected van der Waals potentials in magnetodielectric multilayer systems
Agnes Sambale and Dirk-Gunnar Welsch
Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany
Ho Trung Dung
Institute of Physics, Academy of Sciences and Technology,
1 Mac Dinh Chi Street, District 1, Ho Chi Minh city, Vietnam
Stefan Yoshi Buhmann
Quantum Optics and Laser Science, Blackett Laboratory, Imperial College London,
Prince Consort Road, London SW7 2AZ, United Kingdom
(Dated: January 6, 2009)
9
0 Withintheframeworkofmacroscopicquantumelectrodynamicsinlinear,causalmedia,westudy
0 the van der Waals potentials of ground-state atoms in planar magnetodielectric host media. Our
2 investigationextendsearlieronesintwoaspects: Itallowsfortheatomtobeembeddedinamedium,
thuscoversmanymorerealisticsystems;andittakesaccountofthelocal-fieldcorrection. Two-and
n
three-layer configurations are treated in detail both analytically and numerically. It is shown that
a
J an interplay of electric and magnetic properties in neighbouring media may give rise to potential
wells or walls. Local-field corrections as high as 80% are found. By calculating the full potential
6
including the translationally invariant and variant parts, we propose a way to estimate the (finite)

value of the dispersion potential at the surface between two media. Connection with earlier work
]
h intendedfor biological applications is established.
p
- PACSnumbers: 34.35.+a,12.20.-m,42.50.Nn42.50.Wk,
t
n
a
u I. INTRODUCTION potential in an arbitrary geometry has been derived in
q theformofasumofatranslationallyinvarianttermand
[ and a term containing the uncorrected scattering Green

The dispersive interaction between small neutral, un- tensor modified by a local-field correction factor [9, 10].
1
v polarised particles (atoms, molecules etc.) and an un- Thisgeneralformulaisapplicableformeta-materials[12]
4 charged macroscopic object is a manifestation of the andwillserveasastartingpointforourtreatmentofthe
3 quantum nature of the electromagnetic field and related local-fieldcorrectedvdW interactionin a stratifiedmag-
6 tothezero-pointenergy[1]. Inthefirstquantumelectro- netodielectric. Note that a generalisation of the formal-
0 dynamical treatment of this van der Waals (vdW) type ism has recently been used to study local-field corrected
.
1 force [2], andin many other relatedworks,the atomwas interactions of an excited atom with a ground-state one
0 assumed to be located in free space. In reality, the atom across an interface between two media [13].
9 may be submerged in or be a part of a gas, liquid, or
0 even solid — a situation typical in diverse fields such The interaction between a neutral atom and a mate-
:
v as colloid science [3, 4, 5], surface engineering [6], and rial surface is customarily split into two parts: a short-
i biology [7]. As a particular example from biology, one rangerepulsivepart,significantwhentheatomic valence
X
can cite the transfer of a small particle diluted in cell electrons overlap with the surface, and a longer-range
r plasma through a cell membrane caused by vdW forces dispersive vdW part [14, 15, 16, 17, 18, 19, 20]. The-
a
[8]. A question that may arise when the atom is embed- ories that focus on the vdW interaction commonly ne-
dedinamediumishowthevdWinteractionistobecor- glect the first, thus are incapable of correctly predicting
rected due to the difference between the local fields act- the behaviour of the interaction potential at very short
ingontheatomandthemacroscopicfieldsaveragedover distances. They typically yield a power law of z−3 (z ,
A A
a region which contains a great number of the medium atom–surfacedistance)formaterialswithdominantelec-
constituents, thus ignoring the gaps between them. In tric properties, and a power law of z−1 for purely mag-
A
Ref. [9], the vdW interaction between two ground-state netic materials, which lead to divergent values for the
atomsembeddedinadjacentsemi-infinitemagnetodielec- vdW potential right on the surface [2, 21, 22]. Due to
tricmediahasbeenconsidered. Comparisonofthisresult the importance of phenomena such as physisorptionand
withthatdeducedfromthe Casimirforceonathincom- transportofparticlesthroughinterfacesandmembranes,
posite slab in front of a composite semi-infinite medium, muchefforthasbeendevotedtoabettertreatmentofthe
both obeying the Clausius-Mossotti relation, suggests a atom–surface interaction when the two are at very short
hintonhowtoaccountforthelocal-fieldcorrection. This distances. With respect to physisorption, this has been
confirmed shortly after in Ref. [10] on the basis on a done via introducing a reference plane [23]; via charac-
macroscopic quantum electrodynamics theory and the terising the materialsurface by a more realistic response
real-cavity model [11]. A general formula for the vdW function which includes spatial dispersion [24], smooth
2
variationofthedielectricpropertiesattheinterface,and where U (r ) is constant throughout any homogenous
1 A
the contribution from d electrons to the screening [25]; region and accounts for all scattering processes within
and via using an atomic polarisability going beyond the the cavity,
dipole approximation [17, 25]. These studies typically
~µ ∞
produce finite values for the interaction potential at the U (r )= 0 dξξ3α(iξ)C (iξ), (2)
interface [23]. 1 A −4π2cZ0 A
In the presentpaper, the local-fieldcorrectedvdW in-
where
teraction of a ground-state atom embedded in a planar,
dispersing and absorbing magnetoelectric host medium
C (ω)=
isstudiedwithintheframeworkofmacroscopicquantum A
′ ′
electrodynamics and the real-cavity model. We propose h(1)(z ) zh(1)(z) ε (ω)h(1)(z) z h(1)(z )
aprocedurethatwouldallowforobtaininganestimateof 1 0 1 − A 1 0 1 0 (3)
h i h ′i
the finite vdW energy of an atom right on the interface ε (ω)h(1)(z)[z j (z )]′ j (z ) zh(1)(z)
even within our predominantly macroscopic framework A 1 0 1 0 − 1 0 1
h i
treating the atom within electric dipole approximation.
[z =ωR /c, z=n(ω)z , n(ω)= ε (ω)µ (ω), ε (ω)=
0 c 0 A A A
The paper is organised as follows. In Sec. II the main
ε(r ,ω), µ (ω) = µ(r ,ω)], with j (x) and h(1)(x), re-
results concerning the local-field corrected vdW interac- A A A p 1 1
spectively, being the first spherical Bessel function and
tion of a ground-state atom and an arbitrary absorbing
the first spherical Hankel function of the first kind,
and dispersing macroscopicbody are reviewed,and then
appliedto planarmultilayersystems. The two-layercase
sin(x) cos(x)
is considered and a procedure for an estimation of the j1(x)= x2 − x , (4)
vdW potential at the interface is given in Sec. III while
1 i
the three-layer case is discussed in Sec. IV. Connection h(1)(x)= + eix, (5)
1 − x x2
with earlier work which focus on biological systems and (cid:18) (cid:19)
concluding remarks are given in Sec. V.
andα(ω)denotingtheisotropicpolarisabilityoftheguest
atom in lowest non-vanishing order of perturbation the-
ory [26],
II. THE MODEL
2 ω d0k 2
k0
α(ω)= lim (6)
theWleocsaelt-fitehldecsotrargeectbedyvbdrWieflpyorteecnatilalilnogftahgeroreusnudlt-sstafoter η→0+3~Xk ωk20−ω(cid:12)(cid:12)2−(cid:12)(cid:12)iηω
atom within an arbitrary geometry and applying them with ω =(E E )/~ being the (unperturbed) atomic
k0 k 0
to a planar multilayer system. transitionfreque−nciesanddlk k dˆ l beingtheatomic
≡h | | i
electric-dipole transition matrix elements.
The term U (r ) involves all interactions associated
2 A
A. Local-field corrected vdW potentials in
with the particular shape and size of the magnetodielec-
arbitrary geometries
tric host medium,
We consider a ground-state guest atom A located at U (r )=~µ0 ∞dξξ2D2(iξ)α(iξ)TrG(1)(r ,r ,iξ)
r in an absorbing and dispersing host medium of arbi- 2 A 2π A A A
A Z0
trarysizeandshapecharacterisedbymacroscopicε(r,ω) (7)
and µ(r,ω). To account for the local-field correction we where
employ the real-cavity model by assuming the atom to
be surrounded by a small spherical free-space cavity of DA(ω)=
radius Rc. The radius of the cavity is a measure of the j (z ) z h(1)(z ) ′ [z j (z )]′h(1)(z )
distance between the guest atom and the surrounding 1 0 0 1 0 − 0 1 0 1 0
.
hostatoms[10]. By construction,the macroscopicquan- h i′
µ (ω) j (z ) zh(1)(z) ε (ω)[z j (z )]′h(1)(z)
tities ε(r,ω) and µ(r,ω) do not vary appreciably on the A 1 0 1 − A 0 1 0 1
microscopic length scale R . To apply the real cavity h h i i(8)
c
modeloneshouldkeepinmindthat ε(0)µ(0)R should
c
be small compared to the maximum of all characteristic The Green tensor G(1)(r,r′,ω) of the electromagnetic
p
atomicandmediumwavelengthsaswellastothesepara- field accounts for scattering at the inhomogeneities of
tion from any surface of the host medium. In particular, the (unperturbed) magnetoelectric host medium while
the application of model to metals is very problematic. the factor DA(ω) comes from the local-field correction.
Usingsecond-orderperturbationtheory,the vdWpoten- Within the real-cavity model, the potentials U1(rA)
tial for such an atom can be written in the form [10] and U2(rA) are well approximated by their asymptotic
limit of small cavity radii, i.e., we keep only the lead-
U(r )=U (r )+U (r ), (1) ing non-vanishing order in ε(0)µ(0)ω R /c where
A 1 A 2 A max c
| |
p
3
ω represents the maximum of characteristic atomic [j >0, for j =0 set d =0] with the abbreviation
max 0
and medium frequencies (for details, see Ref. [10]),
Dσ =1 rσ rσ e−2βjdj. (14)
j − j− j+
U (r )= ~ ∞dξα 3 εA−1 1 In Eq. (13), p(s) denotes p(s) polarisations. The reflec-
1 A −4π2ε 2ε +1R3 tion coefficients obey the recursion relations
0 Z0 (cid:20) A c
+9ξ2 ε2A[1−5µA]+3εA+1 1 (9) rσ = rlσl±1+e−2βl±1dl±1rlσ±1± , (15)
c2 5[2εA+1]2 Rc# l± 1+rlσl±1rlσ±1±e−2βl±1dl±1
and
ε β ε β µ β µβ
rp = l+1 l− l l+1, rs = l+1 l− l l+1 (16)
U (r )= ~µ0 ∞dξξ2α 3εA 2 ll+1 µl+1βl+εlβl+1 ll+1 µl+1βl+µlβl+1
2 A
2π Z0 (cid:18)2εA+1(cid:19) [r1σ−=rnσ+=0], where the modulus of wave vector in the
TrG(1)(r ,r ,iξ). (10) z direction is given by
A A
×
Here and in the following the dependence of ε, µ and α β = k2+q2 (17)
l l
on the iξ is suppressed for brevity. Note that in leading q
order the local-field factor [3εA/(2εA +1)]2 depends on with kl, which always appears in the form of kl2 in the
dielectric properties only. The associated (conservative) Green tensor, Eq. (13), being
vdW force is given by
ξ2
k2 = ε µ . (18)
~µ ∞ l c2 l l
F(r )= ∇U (r )= 0 dξξ2α
A − 2 A − 2π The s- and p-polarisation unit vectors are defined as
Z0
×(cid:18)2ε3Aε+A 1(cid:19)2∇TrG(1)(rA,rA,iξ). (11) e±s =eq×ez, e±p = ik1j (qez ∓iβjeq) (19)
The cavity-induced part of the potential, U1 according [eq = q/q,q = |q|]. Substitution of Eqs. (12) and (13)
to Eq. (2) does not leadto a force but to an energy shift into Eq. (10) leads to
whose influence on the overall potential will be studied
in Secs. III and IV. U (z )= ~µ0 ∞dξξ2αµ 3εj 2
2 A 8π2 j 2ε +1
Z0 (cid:20) j (cid:21)
∞ q rs q2c2 rp
B. Atom in a magnetoelectric multi-layer system dq e−2βjzA j− 1+2 j−
×Z0 βj( "Djs −(cid:18) ξ2εjµj(cid:19) Djp#
So far we have not been specific about the geometry rs q2c2 rp
of the macroscopic body. Consider a stack of n layers +e−2βj(dj−zA) j+ 1+2 j+
labeled by l (l = 1,...,n) of thicknesses dl with planar "Djs −(cid:18) ξ2εjµj(cid:19) Djp#)
µpa(rr,aωlle)l=bouµnl(dωa)ryinsularfyaecresl,. wThhereecεo(orr,dωi)na=te εsly(sωt)emanids + ~4πµ20 ∞dξξ2αµj 2ε3ε+j 1 2
chosen such that the layers are perpendicular to the z Z0 (cid:20) j (cid:21)
axis and extend from z = 0 to z = dl for l 6= 1,n and ∞dq q rjσ−rjσ+e−2βjdj, (20)
from z =0 to z = ( ) for l =1 (n). The scattering × β Dσ
−∞ ∞ Z0 j σ=s,p j
part of the Green tensor at imaginary frequencies for r X
and r′ in layer j is given by (see, e.g., Ref. [22]) where we have used the relations
e± e± =e± e∓ =1, (21)
G(1)(r,r′,iξ)= d2qeiq·(r−r′)G(1)(q,z,z′,iξ) (12) s · s s · s
q2c2
Z e± e± =1, e± e∓ = 1 2 (22)
p · p p · p − − ξ2ε µ
[q e ] where j j
z
⊥
tocalculatethetrace. Itisworthnotingthatthetermin
µ 1 curlybracketsinEq.(20)describesprocessesthatinvolve
G(1)(q,z,z′,iξ)= j
8π2β Dσ an odd number of reflections at the interfaces while the
jσX=s,p( j second term accounts for an even number of reflections,
e+e−rσ e−βj(z+z′)+e−e+rσ e−2βjdjeβj(z+z′) as can be seen from
× σ σ j− σ σ j+
+rjσh−rjσ+Dejσ−2βjdj e+σe+σe−βj(z−z′)+e−σe−σeβj(z−z′))i 1−rrjσjσ−−er−jσ+2βej−zA2βjdj =e−βjzArjσ−e−βjzA
h i
(13) +e−βjzArσ e−βjdjrσ e−βjdjrσ e−βjzA +..., (23)
j− j+ j−
4
rσ e−2βj(dj−zA) A. Position-dependent part
j+ =e−βj(dj−zA)rσ e−βj(dj−zA)
1 rσ rσ e−2βjdj j+
− j− j+
Foraplanarmagnetoelectrictwo-layersystemwiththe
+e−βj(dj−zA)rσ e−βjdjrσ e−βj(dj−zA)+... (24)
j+ j− guestatomplacedin,say,layer2,substitutionofrσ =0
n+
andrσ inaccordancewithEqs.(16)inEq.(20)leadsto
n−
and the following expression for the position-dependent part
of the vdW potential
1−rjσ−rjσr−jσ+rjσe+−e2−βj2dβjjdj =e−βjzArjσ−e−βjdjrjσ+ U2(zA)= ~8πµ20 Z0∞dξξ2α(cid:18)2ε32ε+2 1(cid:19)2
×e−βj(dj−zA)+e−βjzArjσ−e−βjdjrjσ+e−βjdj µ ∞dq q µ1β2−µ2β1
×rjσ−e−βjdjrjσ+e−βj(dj−zA)+.... (25) × 2Z0 β2 (cid:20)µ1β2+µ2β1
ε β ε β q2c2
1 2− 2 1 1+2 e−2β2zA, (26)
Obviously, the expression presented in Eq. (25), which − ε1β2+ε2β1 (cid:18) ε2µ2ξ2(cid:19)(cid:21)
corresponds to even numbers of reflections, is indepen-
which differs from its atom-in-free-space counterpart by
dent of the position of the atom and rapidly decreases
2
with increasing distance between two plates. Hence, it the local-field correction factor 2ε32ε+21 and by ε2 6= 1
cannot lead to a force but only to a layer-dependent en- and µ2 =1 [22] (cid:16) (cid:17)
ergyshiftwhichissmallcomparedtothe otherterms. It 6
iscompletelyabsentinthetwo-layercasesinceonlysingle
reflectionsatthe interfaceoccur. Italsovanishesforsuf- 1. Retarded limit
ficiently dilute media,anexpansionforsmallχ =ε 1
j j
−
and ζ = µ 1 showing no contribution to linear or-
j j It is instructive to study the potential in the retarded
−
der, which reflects the fact that at least three medium-
limit
assisted reflection processes are involved. On the con-
c c
trary, Eqs. (23) and (24) depend on the position of the z >> and z >> , (27)
A ω− A ω−
atom, where waves propagating to the left and to the A M
right of the atom have to be distinguished. where ω− and ω− being the minima of all relevant
A M
It is worth noting that all formulas in this section are atomic transition and medium resonance frequencies,
valid for arbitrary (passive) magnetoelectric media, in- respectively. Introducing a new integration variable
cluding metamaterials and in particular, lefthanded ma- v =cβ /ξ and replacing α(iξ) α(0), ε (iξ) ε (0),
2 1,2 1,2
≃ ≃
terialswithsimultaneouslynegativerealpartsofεandµ, andµ (iξ) µ (0),the integrationoverξ canbe per-
1,2 1,2
≃
duetothefactthat ε (ω)µ (ω)hasnobranchingpoints formed and we obtain for the potential
j j
in the upper half of the complex frequency plane. How-
ever, left-handed mpaterial properties being only realized U (z )= C4, (28)
2 A z4
in finite frequency windows, they can not be expected A
tohaveastronginfluenceonground-statedispersionpo-
where
tentialswhichdependonthe mediumresponseatallfre-
quencies;alternatively,thiscanbeseenfromthefactthat 3~c 3ε (0) 2 ∞
2
C = α(0) µ (0) dv
tpheermpeoatebniltiitaielsisteaxkpenresastibimleaignintearrmysfroefqpueernmciietstiwvihtiiecshaanrde 4 64ε0π2 (cid:18)2ε2(0)+1(cid:19) 2 Z√ε2(0)µ2(0)
always positive [22, 27]. The situation can change when 1 µ (0)v µ (0) v2 ε (0)µ (0)+ε (0)µ (0)
1 2 2 2 1 1
− −
theatomisinanexcitedstateandtheatom-fieldinterac- × v4 "µ1(0)v+µ2(0)pv2 ε2(0)µ2(0)+ε1(0)µ1(0)
tion resonantly depends on the medium response at the −
atomic transition frequencies [28, 29]. ε1(0)v ε2(0) vp2 ε2(0)µ2(0)+ε1(0)µ1(0)
+ − −
ε (0)v+ε (0) v2 ε (0)µ (0)+ε (0)µ (0)
1 2 p 2 2 1 1
−
v2
p
(1 2 ) . (29)
× − ε (0)µ (0)
III. VDW POTENTIAL NEAR AN INTERFACE 2 2 (cid:21)
It can be proven that ∂C /∂ε (0)<0, ∂C /∂µ (0)> 0,
4 1 4 1
and ∂C /∂µ (0)<0.
4 2
Let us first apply the theory to the simplest case of a
To deduce some physics from Eq. (29), we consider
single interface between two homogeneous semi-infinite
some limiting cases. Assuming that the atom is located
magnetoelectric half spaces, where we first concentrate
infree spaceµ (0)=ε (0)=1,itcanbe shownthat, for
2 2
on the position-dependent part of the potential respon-
a purely electric halfspace 1,
sible for the vdW force and then interpret the (layer-
dependent) constant part. C [µ (0)=1,µ (0)=1,ε (0)=1]<0 (30)
4 1 2 2
5
and, for a purely magnetic halfspace 1 with µ (0)>1, 2. Non-retarded limit
1
C [ε (0)=1,µ (0)=1,ε (0)=1]>0. (31)
4 1 2 2
The non-retarded limit corresponds to atom–surface
The inequality (30) means the atom is attracted toward
distances z small compared with the typical wave-
A
the electric halfspace — the case which is commonly
lengths of the medium and the atomic system,
treatedin earlierliterature. On the other hand, the pos-
itivity ofC inEq. (31)means the atomis repelledfrom c
4 z << and/or (39)
the magnetic half space. This, coupled with the signs of A ω+[n (0)+n (0)]
A 1 2
the derivatives given below Eq. (29), imply that electric c
z << (40)
properties tend to make the potential attractive while A ω+[n (0)+n (0)]
magnetic ones tend to make the potential repulsive. M 1 2
Now if the atom is embedded in a material halfspace, [n (0)= ε (0)µ (0); ω+ and ω+, maxima of the
while the opposite halfspace is empty µ (0)=ε (0)=1, 1,2 1,2 1,2 A M
1 1 relevant atomic transition and medium resonance fre-
it can be shown that, for a purely electric material, p
quencies]. The conditions (39) and (40) imply
C [µ (0)=1,ε (0)=1,µ (0)=1]>0, (32)
4 1 1 2
ξz
A
theatomisrepulsedfromtheinterface,whileforapurely ε1µ1 ε2µ2 <<1, (41)
c | − |
magnetic material with µ (0)>1,
2 ξz p
A
√ε µ <<1, (42)
C4[µ1(0)=1,ε1(0)=1,ε2(0)=1]<0, (33) c 2 2
the atom is attracted towards the interface. Since
where we have used that the ξ-integration is practically
∂C4/∂µ2(0)<0, if one starts from a purely electric ma- limited to a region where ξ .ω+ . We substitute
A,M
terialwhichisaccompaniedbyarepulsivepotential,then
enhances the magnetic properties of the material by in-
creasing µ2(0), one would obtain with an attractive po- q = β22−ε2µ2ξ2/c2, (43)
tential in accordance with Eq. (33). qdq =qβ dβ , (44)
2 2
Another particular case is when the magnetodielectric
contrast between the contacting media is small β = ξ2/c2(ε µ ε µ )+β2 (45)
1 1 1− 2 2 2
ε (0)=ε (0)+χ(0), (34) q
1 2
inEq.(26)and,onrecallingEq.(41),performaleading-
µ (0)=µ (0)+ζ(0) (35)
1 2 order Taylor expansion in ξ2/(c2β2)(ε µ ε µ ). After
2 1 1− 2 2
[χ(0) ε (0), ζ(0) µ (0)], one can further treat the carrying out the β -integral, we arrive at
2 2 2
≪ ≪
integrals analytically by keeping only terms linear in χ
and ζ, C3 C1
U (z )= + , (46)
2 A −z3 z
µ (0)v µ (0) v2 ε (0)µ (0)+ε (0)µ (0) A A
1 2 2 2 1 1
− −
µ (0)v+µ (0) v2 ε (0)µ (0)+ε (0)µ (0) where
1 2 p 2 2 1 1
−
1p ε2(0) ζ(0) µ2(0)χ(0), (36) ~ ∞ 9ε2 ε1 ε2
≃(cid:18)2µ2(0) − 4v2 (cid:19) − 4v2 C3 = 16π2ε0 Z0 dξα(2ε2+1)2ε1+−ε2, (47)
ε (0)v ε (0) v2 ε (0)µ (0)+ε (0)µ (0)
1 2 2 2 1 1
− −
ε (0)v+ε (0) v2 ε (0)µ (0)+ε (0)µ (0)
1 2 p − 2 2 1 1 ~µ ∞ 3ε 2
ε2(p0)ζ(0)+ 1 µ2(0) χ(0). (37) C1 = 16π02 dξξ2αµ2 2ε +2 1
≃− 4v2 (cid:18)2ε2(0) − 4v2 (cid:19) Z0 (cid:18) 2 (cid:19)
µ µ ε ε 2ε (ε µ ε µ )
1 2 1 2 1 1 1 2 2
The v-integration is straightforwardand we arrive at − + − + − , (48)
× µ +µ ε +ε µ (ε +ε )2
(cid:20) 1 2 1 2 2 1 2 (cid:21)
9~c 23µ (0)χ(0)+7ε (0)ζ(0)
2 2
C4 = 640π2ε0α(0) −ε2(0)µ2(0)µ2(0)[2ε2(0)+1]2. aeqnudaEltqo.o(n42e.)Bhayspbuetetinngusεed=toµse=t 1e,xpE(q−s.2(√4ε72)µa2nξdzA(4/8c))
(38) 2 2
p reduce to those for an atom in free space [22].
ThisresultgeneralisestheoneobtainedinRef.[22]tothe AscanbeseenfromEqs.(46)–(47),onecandistinguish
case of an atom embedded in a medium, with local-field tworegimeshavingdifferentpowerlaws. Thefirstregime
correctionincluded. Itisricherincontentthanitsatom- is where the two contacting media have unequal electric
in-free-space counterpart. For example, when χ(0) = properties. Then the first term in Eq. (46) dominates
ζ(0) and the atom in free space ε (0) = µ (0) = 1, the and the power law is z−3. The atom is pulled toward
2 2 A
potential is attractive, whereas when the atom is in a (repelled from) the interface if the medium it is located
medium of ε (0)/µ (0)>23/7,a repulsive potential can in has stronger (weaker) electric properties than that on
2 2
be realized. the other side of the interface.
6
0.06
In the case of equal electric properties C3 = 0 and )
U (z )=C /z with 2| (1)
2 A 1 A 0
d1 (2)
C = ~µ0 ∞dξξ2α 3ε2 2(µ µ ) 3ω|10 (3)
1 16π2 Z0 µ2(cid:18)2ε2+11(cid:19) 1− 2 3c/(0 0
+ . (49) ε
×(cid:18)µ1+µ2 2(cid:19) 22π
1
The atom experiences a force which points away from )
A (a)
(toward) the interface if the magnetic properties of the (z
2
medium the atom is situated in are weaker (stronger) U -0.06
than those of the medium on the other side of the inter- -1 0 1
)
face. 2| 0.02
Notethatthedependenceofthedirectionsoftheforces 10
d
on the difference in strength of the medium responses |
0
31
is opposite in the two cases of dominantly electric and ω
(
purely magnetic media. In both cases the strength of /
3
c
theforceincreaseswithincreasingdifferencebetweenthe 0
ε 0
electric and magnetic parameters of the contacting me- 2
π
dia. Theseresultsareconsistentwithearlierones[8,30]. 12
)
Amoredetailedcomparisonofourresultswiththosepre- A
z
sented in Ref. [8] will be made in the last section. Since (
2
U (b)
the coefficient C of the leading-order term depends on
3 ∆
electric properties only, the influence of electric prop- -0.02
erties on the behaviour of the potential tends to dom- -1 0 1
inatethatofthemagneticproperties,exceptforthecase
zAω10/c
whentheelectricpropertiesoftheneighboringmediaare
similar. At more moderate distances, competing effects
of electric and magnetic properties may create potential
FIG.1: (a)Position-dependentpartofthevdWpotentialex-
walls or wells, as is also evident from the numerical re- periencedbyaground-statetwo-levelatominamagnetoelec-
sults below. tric two-layer system as a function of atom–surface distance
forfixedε1,µ1,µ2,andforωPe2/ω10 =1(1),0.4(2),and0.2
(3). Solid lines denotethe potentials with the local-field cor-
3. Numerical results rection,whiledashedlinesrepresentthosewithout. Otherpa-
rameters are ωTe1/ω10 =ωTe2/ω10 =1.03, ωPe1/ω10 =0.75,
Tostudythelocal-fieldcorrectedpotentialatinterme- ωTm1/ω10 = ωTm2/ω10 = 1, ωPm1/ω10 = 2.3, ωPm2/ω10 =
0.4, γm1,2/ω10 = γe1,2/ω10 = 0.001, and the cavity radius
diate distances and to elucidate the combined influence
is Rcω10/c = 0.01. (b) Difference between local-field cor-
of electric and magnetic properties of the media, we cal-
rected and uncorrected (position-dependent) vdW potential
culate the positiondependent partU2(zA) inaccordance ∆U2 versus atom–surface distance where the solid, dashed,
with Eq. (26) numerically. A two-level atom of transi- and dotted lines refer to the curves (1), (2), and (3), respec-
tion frequency ω is assumed and material electric and tively.
10
magnetic properties are described using single-resonance
Drude-Lorentz-type permittivities and permeabilities
(1) ε > ε , it can be seen from the figure that the po-
2 1
ω2
ε (ω)=1+ Pei , (50) tential at very short atom–surface distances is repulsive
i ωT2ei−ω2−iωγei in medium 2 and attractive in medium 1, in consistency
ω2 with the analytical result (46) and (47). Similarly, in
µi(ω)=1+ ω2 ωP2mi iωγ , i=1,2. (51) cases (2) and (3) ε2 < ε1, the potential is attractive in
Tmi− − mi medium 2 while repulsive in medium 1. As the atom–
Figure 1 (a) illustrates the atom–surface-distance de- surfacedistanceincreases,the secondterm C1 inthe po-
zA
pendence of the U (z ) potential. The interface is at tential(46)graduallycomesintoplayandifthemagnetic
2 A
z = 0, the medium 1 is on the left, while the medium properties are strong enough, may switch the signof the
A
2 is on the right. The parameters ε , µ , µ are fixed potential and create potential walls or wells in the pro-
1 1 2
whereas ε takes on three different values of the plasma cess, as is clearly visible in case (2).
2
frequency ω . In all three cases ε and ε have same To show how the net effect of the local-field correc-
Pe2 1 2
(transverse)resonancefrequenciesω anddampingcon- tion depends on the distance and on the properties of
Te
stants γ , but different plasma frequencies (i.e., ε =ε ) the media surrounding the guest atom, we plot the un-
e 1 2
and the z−1 term in Eq. (46) is negligibly small. In6 case correctedpotential by dashedlines inFig. 1 (a), and the
A
7
)
2
|10 0.0012 108
3dω|10105 2)|10
3(/ d|0 106
c 31
0 ω
ε (
2 0.0008 /
)12π 0 3εc0 104
A 2
(z2 12π
U )
zA 102
-105 0.0004 (1
1 2 3 U
∓
ε2(0) 100
0 0.05 0.1
FIG. 2: Position-dependent part of the vdW potential as Rcω10/c
a function of the static permittivity ε2(0) (more specifi-
cally ωPe2/ω10) for two values of the atom–surface distance
FIG. 3: The exact layer-dependent constant part of the po-
zAω10/c = 0.01 (scale to the left) and zAω10/c = 3 (scale to
tential (solid line), Eq. (2), and approximate results (dashed
theright). Solidlinesarewiththelocal-fieldcorrectionwhile
line),Eq.(9),areshownasfunctionsofthereal-cavityradius.
dashedlinesarewithoutone. Otherparametersarethesame
asinFig.1. Notethearrowswhichindicatetheordinatescale Theupperpairofcurvesshows−U1(zA)12π2ε0c3/(ω130|d10|2)
(thesignhasbeenreversedsothatalogarithmicscalecanbe
to beused.
used) for a pure electric material with ωPe2/ω10 =0.4, while
thelowerpairofcurvesshowsU1(zA)12π2ε0c3/(ω130|d10|2)for
apuremagneticmaterialwithωPm2/ω10 =0.4. Allotherpa-
difference between the corrected and uncorrected results
rameters are the same as in Fig. 1. The radius of the cavity
in Fig. 1 (b). The ratio between the corrected and un- Rcω10/c starts from 0.001.
correctedresultsis notalwaysagoodmeasureofthedif-
ferencebetweenthe twobecause oneofthemcanvanish.
The local-field correction factor [3ε /(2ε +1)]2 is posi-
j j correctedanduncorrectedcurvestendstothestaticvalue
tive, larger than one, and increases with εj [j indicative of the local-field correction factor [3ε (0)/(2ε (0)+1)]2
2 2
ofthelayercontainingtheguestatom]. Itapproachesthe
(which lies between 1 and 9/4), in agreement with the
maximum value of 9/4 as ε . Note that a larger-
j analytical analysis given in Sec. IIIA1. For the smaller
→ ∞
than-unitylocal-fieldcorrectionfactordoesnotnecessar-
value of the atom–surface distance z ω /c = 0.01, a
A 10
ily lead to an enhancement of the potential because the
crossing point between the corrected and uncorrected
uncorrected factor in the integrand can change sign as
curves is observed, where the local-field correction pro-
ξ varies. The local-field correction has a clear-cut ef-
duces no net change.
fect of increasing or decreasing the potential only when
the uncorrected integrand is purely repulsive or attrac-
tive, for which cases (1) and (3) can serve as examples.
In the middle case (2), the two curves with and with- B. Layer-dependent constant part
out local-field correction cross, that is, there exists an
atom–surface distance at which the effect of the local- Fig.3showsthedependenceoftheconstantpartU of
1
field correction is canceled out due to the ξ-integration. the potential on the real-cavity radius R . To gain some
c
In addition, one can notice that the local-field correc- insight, we consider the two limiting cases of a purely
tion leads a small shift of the position of the peak. Fig- electric and a purely magnetic material. According to
ure 1 (b) which shows the difference between the local- the analytic result (9), in the first case the leading term
field corrected and uncorrected potentials, reveals quite is proportional to [R ω /c]−3, while in the second case
c 10
significant corrections of up to 30% of the uncorrected theleadingtermisproportionalto[R ω /c]−1. Thusfor
c 10
values. smallenoughR ω /c, U forapureelectricmaterialis
c 10 1
| |
The behaviour of the local-field correctedvdW poten- generally larger than that for a pure magnetic material
tialwithrespecttothestaticpermittivityofthemedium - a fact that is confirmed by the figure. It can be seen
theatomisembeddedinisshowninFig.2fortwodiffer- that U1 < 0 in the first case and U1 > 0 in the second
entvaluesoftheatom–surfacedistance. Withinthescale case, again in agreement with Eq. (9). The figure also
of the figure, the curves for the larger distance from the indicatesthatthe magnitudeofU1,whichisentirelydue
interfacepeakatcertainvaluesofε (0). Thepositionsof to the local-fieldcorrection,decreaseswith anincreasing
2
thepeaksaredifferentduetotheeffectsofthelocalfield. real-cavity radius, that is, the effects of the local field
Asω /ω andasaconsequenceε (0)increases,anin- becomes weaker as the medium becomes more dilute.
Pe2 10 2
spection of the figure reveals that the ratio between the For comparison, we also plot the potential U accord-
1
8
ing to the approximate result (9) (dashed curves). It 0
can be seen that for the parameters used in the figure,
the approximate result reproduces quite well the exact )
2
one, especially in the case of a pure electric material. |10
d
The agreement in the case of a pure magnetic material |0
31
is good for very small R ω /c, but worsens as R ω /c ω
c 10 c 10 (/ -5x105
increases. 3
c
0
ε
2
π
2
1
)
A
z
(
U
6
-10
C. Total vdW potential and its value at the
-0.05 0 0.05
interface
zAω10/c
We have separately investigated the position-depen-
dent part U (z ) of the potential, which determines FIG.4: Local-fieldcorrectedtotalvdWpotentialofaground-
2 A
the force acting on the atom, and the constant, layer- state two-level atom in a magnetodielectric two-layer sys-
dependentpartU ,whichisrelatedtothelocal-fieldcor- tem as a function of the atom–surface distance. Different
1
rection. For problems such as the transfer of an atom or curvesarefordifferent(equal)electricandmagneticcoupling
a small particle through an interface, it is of relevance strengths of themedium 2: ωPm2/ω10 =ωPe2/ω10 =0 (solid
line),0.4(dashedline)and1(dottedline). Otherparameters
to evaluate the potential right at the interface. For this
are thesame as in Fig. 1.
purpose, the total value of the potential is needed. In
Fig. 4, we have calculated U +U (z ) on both sides of
1 2 A
the interface with the medium 1 fixed while the medium the potential right at the interface, we suggest that
2varyingfromvacuumtoamoredensemediumwithbal-
anced electric and magnetic properties. The case repre- 1
U(z =0)= [U(R )+U( R )]
sentedbydashedlineisnothingelseratherthancase(2) A 2 c − c
inFig.1. Only veryshortdistances |εiµi|zAω10/c≪1 ~ ∞ ε1 1 ε2 1
are presented. Since ε = ε in general, the position = dξα 12 − + −
dependence of the poten1ti6al at2shortpdistances is mostly −32π2ε0Rc3 Z0 ( (cid:18)2ε1+1 2ε2+1(cid:19)
dtheeteirnmteingreadndb.yNthuemeCzrA33icatlerrmesuwlthsiicnhtchoenfitagiunrseεa1re−coεn2siisn- ε1−ε2 1 3ε1 2 1 3ε2 2 (52)
tent with this and show that the potential is attractive − ε1+ε2 "ε1 (cid:18)2ε1+1(cid:19) − ε2 (cid:18)2ε2+1(cid:19) #)
(repulsive) if the atom is located in a medium which is
[recallEqs.(9),(46),and(47)]. Visually,thismeansfirst
electricallymoredilute (dense)thanthatinthe opposite
plottingthepotentialasafunctionoftheatomicposition
side ofthe interface. Additionalstructures inU like po-
2 up to distances z = R (R being the radius of the
A c c
tential wells or walls are typically overwhelmed by the | |
cavityinthe real-cavitymodel),thenconnectingthe two
magnitude of U . Baring the visual suppression of the
1 loose ends on the two sides of the interface to find the
potential wells or walls in U due to large magnitude of
2 valueofthepotentialatz =0. Ourresultisremarkably
A
U ,afullpotentialismorestraightforwardthanU alone
1 2 similar to what has been obtainedby calculating the on-
in predicting the movement of an atom across a surface.
surface potential of a molecule of finite size s [31],
Take,forexample,case(2)inFig.1. Anatomlocatedin
layer 2 close to the surface will be attracted to it, and if ~ ∞ 1 1 1
theatomcancrosstheinterface,itwillbepushedfurther U(zA =0)= 2π5/2ε s3 dξα 2 ε + ε
away from the surface into layer 1. Fig. 4 (dashed line) 0 Z0 (cid:20) (cid:18) 1 2(cid:19)
1ε ε 1 1
provides us with some additional information. It shows 1 2
+ − . (53)
explicitlythatthetotalpotentialinlayer2ishigherthat 3ε1+ε2 (cid:18)ε1 − ε2(cid:19)(cid:21)
that in layer 1, by giving the difference between the po-
The second terms in Eqs. (52) and (53), which represent
tentialsinthetwolayers;anditmayalsohelptoestimate
the interface contribution to the potential, agree when
the amount ofenergy requiredfor the atom to penetrate
setting s = (3 16/3/π−1/6)R 1.4R and neglecting
into layer 1. c ≈ c
the local-field correction in Eq. (52) which was not con-
p
In Fig. 4, we have not displayed the results for dis- sidered in Ref. [31]. The first terms, which represent
tances z < R εµ where the real-cavity model can bulk contributions from the two interfacing media, dif-
A c
| | | |
no longer be applied. This gives rise to a gap between fer in the two approaches, where Eq. (53) still contains
p
the potentials on the two sides of the interface. To ex- self-energycontributions whichdo not vanishin the vac-
tend our theory so that it can be employed to estimate uumcaseε =1whileEq.(52)does not. Ourresult(52)
i
9
0.04 0.04
thus represents animprovementof the previous one (53)
(a) (d)
in that local-field corrections are taken into account and
self-energycontributionshaveconsistentlybeenremoved. 0 0
-0.04 -0.04
IV. THREE-LAYER SYSTEM
-1 0 1 2 3 4 5 6 -1 0 1 2 3 4 5 6
0.04 0.04
(b) (e)
A systemconsisting ofanatomin a three-layerplanar
structure can serve as a prototype for the problem of a 0 0
small particle near or inside a membrane [8]. The trans-
lationally invariant part and position-dependent part of -0.04 -0.04
thevdWpotentialcanagainbedeterminedinaccordance
-1 0 1 2 3 4 5 6 -1 0 1 2 3 4 5 6
withEqs.(2)and(20),respectively. Iftheatomisinone
0.04 0.04
of the two outer layers, Eq. (20) simplifies greatly via
(c) (f)
either rσ = 0 or rσ = 0. If the atom is in the middle
j+ j−
layer, the overall form of the potential U (z ) remains 0 0
2 A
as in Eq. (20) and there exists a term which contains
the product rσ rσ and is position-independent. This -0.04 -0.04
j− j+
position-independent term in U (z ) is irrelevant when
2 A -1 0 1 2 3 4 5 6 -1 0 1 2 3 4 5 6
it is the force which is concerned, but has to be kept,
zAω10/c zAω10/c
together with U , if the potential at the surfacesis ofin-
1
terest. Since the formulas are complicated, we resort to
numerical computation. FIG. 5: Position-dependent part of the vdW potential as a
Figure5showsthe behaviourofthe vdWpotentialfor functionofzAω10/cforthreedifferentthicknessesofthemid-
an atom placed in an asymmetric (left column) and a dlelayer d2ω10/c=5[(a) and(d)],2[(b)and(e)],and1[(c)
symmetric (right column) three-layer magnetodielectric and (f)]. For the left column, the configuration is asymmet-
structure for different thicknesses of the middle layer. ric with ωTe1/ω10 = 1.03, ωPe1/ω10 = 0.75, ωTm1/ω10 =
Note that the parameters for layer 1 are the same as 1, ωPm1/ω10 = 2.3, ωTm2/ω10 = 1, ωPm2/ω10 = 0.4,
those for layer 1 in Fig. 1 and the parameters for layer ωTe2/ω10 = 1.03, ωPe2/ω10 = 0.4, γm2,1 = γe2,1 = 0.001ω10,
ε3 =µ3 =1, and Rcω10/c=0.01. For the right column, the
2 are the same as those for layer 2, case (2), in Fig. 1.
configuration is symmetric with the vacuum in layer 3 being
Thatis,theinterface1-2hereisthesameastheinterface
replaced by a medium of the same characteristics as those
1-2 in Fig. 1(2). In the asymmetric configuration, layer of layer 1. The curves without the local-field correction are
3 is vacuum while in the symmetric configuration, layer shown bydashed lines.
3 and 1 have the same characteristics.
Let us first analyse the asymmetric configuration and
seehowthepresenceofathirdlayer3ontheright,which First, it will be attracted to the 2-3 interface. If it can
isvacuum,affectsthebehaviourofthepotentialnearthe be transported through the interface and gather enough
1-2 interface. In case (a) where the middle-layer thick- momentum going down the slope, it can pass the finite
ness d ω /c = 5 is the largest among three cases, the potential wall and is then attracted to the 1-2 interface.
2 10
behaviour of the potential U (z ) in the boundary re- After this interface, and maybe some oscillations, it will
2 A
gions is similar to that in the two-layer systems. Near be suspended in the potential well near the surface.
the 1-2 interface, we find a potential well in layer 1 and Wenowturntothesymmetricconfiguration(rightcol-
a potential wall in layer 2, just as in Fig. 1(2). Near the umn of Fig. 5) where the middle layer 2 is sandwiched
2-3 interface the potential is repulsive in the more dense between two identical layers 1 and 3. The behaviour of
medium2andattractiveinthemoredilutemedium3. A the potential on the right is just a mirror image of that
newfeatureappearsaroundthecenterofthemiddlelayer on the left with the mirror plane being the one paral-
whereafinitepotentialwallonthe leftandanattractive lel to the surface and passing through the centre of the
one towards the right interface combine to a potential middle layer. When the thickness of layer 2 is largest
well. Clearly, if an atom at rest is put in the well, it will d ω /c=5[case(d)],weseeinlayer2acombinationof
2 10
remain there. When the thickness of the middle layer 2 two potential walls as found in Fig. 1, with a well in the
is reduced, the well becomes more shallow and eventu- middle. Thus even as the middle layer is in general less
ally disappears, as is visible in Figs. 5(b) and (c). For dense than the two surrounding layers in both electric
the parameters used in the figure, the potential well in and magnetic aspects, there exists a possibility that an
the middle layer occurs when d ω /c 1 and is over- atominitiallylocatedinthepotentialwellremainsthere.
2 10
≫
whelmedwhend ω /c 1. Thecurvesforthepotential Withdecreasingthicknessd ,thebottomofthewellrises
2 10 2
∼
canhelponetopredictthemovementofanatomlocated andthetwowallseventuallymergeintoone[cases(e)and
nearorinside a membrane(layer2). Forexample,let an (f)]. Clearly, in the cases (e) and (f), any atom initially
atom be initially located in layer 3 (vacuum) of case (a). situated in the middle layer will be transported to the
10
neighboring layers. Numerical computation also shows an equal volume of solvent from its path (with associ-
that the magnitude of the position-independent term in ated pressure forces being present); the excess polaris-
U (z ), i.e., the last term in Eq. (20), is negligible com- ability and hence also the potential must vanish when a
2 A
paredtotheposition-independentterms,duetothesmall dissolvedparticle hasthe same propertiesas the solvent.
exponential factor. Equation (54) is the macroscopic counterpart of our
In Fig. 5, the uncorrected potentials are plotted by microscopic equations (46)–(48), where the position-
dashed lines. In the two outer layers, the effects of the dependent part of our potential for purely dielectric sol-
local field almost remain the same as in the two-layer vent media reads
configuration, which can be explained by that the pres-
ence of a third layeris screened by the middle layer. For ~ ∞ α 3ε2 2 ε1 ε2
U (z )= dξ − .
the middle layer,we havefound that typically,the local- 2 A −16π2ε z3 ε 2ε +1 ε +ε
0 A Z0 2 (cid:18) 2 (cid:19) 1 2
field correction is most significant around the centre of (56)
the layer, as is most manifest in case (f) where a cor- The results from a microscopic description of the parti-
rection[(U U )/U ]ofmore cle as a system of bound point charges, leading to our
2corrected 2uncorrected 2uncorrected
−
than 80% is observed. A variation of the middle layer Eq. (56), are thus formally very similar to those from a
thickness does not affect much the strength of the local macroscopic model of the particle as a dielectric sphere,
fieldcorrectionnearthesurfaces. Thiscanbeunderstood Eq. (54). The main difference is the fact that the micro-
as resulting from the fact that when an atom is located scopic polarisability (6) together with a local-field cor-
veryclosetoasurface,itwilltendtoseeonlythenearest rectionfactorappearsinEq.(56), whilethe macroscopic
neighboring layer. excess polarisability (55) which effectively accounts for
Ifonewishtoknowthepotentialsrightonthesurfaces, pressureforces enters Eq.(54). Depending on the size of
onewouldhavetoevaluatethefullpotentialU +U (z ) theimmersedparticle,oneofthetwomodelsmayprovide
1 2 A
for atom–surface distances large enough such that the a more realistic description: For very small particles like
macroscopic theory applies, then use it as an input to atomsorsmallmoleculeswhosesizeiscomparabletothe
the procedure proposed in the previous section. Since free interspaces between the atoms forming the solvent
theposition-independentpartU doesnotdependonthe medium, local-field effects are important while (macro-
1
layer thicknesses, at each surface in a more-than-two- scopic) pressure forces can not even be defined, so that
layersystem,theresultswillbecloselyanalogoustothose the microscopic result (56) should be used. For larger
of the two-layercase. molecules whose volume covers a region that would oth-
erwise be occupied by a large number of solvent atoms,
local-field effects can be neglected while pressure forces
V. DISCUSSION AND SUMMARY become relevant, so Eq. (54) should be given preference.
As an additional difference, note that our microscopic
Our results might be of interest in biological applica- calculation has also given layer-dependent constant con-
tions such as the transfer of a small molecule through a tributions to the potential, which are absentin Eq.(54).
membranefromonecelltoanother. Earliertheorieshave Similar considerations apply in the retarded limit,
beendevelopedto describe the vdWinteractionbetween where the macroscopic potential of the small sphere was
moleculesorsmallparticlesandasolventmedium[8]. In found to be [8]
particular,suchparticlesweremodeledbyasmalldielec-
23~c α (0) ε (0) ε (0)
tric sphere of radius R and (macroscopic) permittivity s 1 2
s U(z )= − ; (57)
εs. It was found that the nonretarded dispersion poten- s −320π2ε0zs4 ε32/2(0) ε1(0)+ε2(0)
tial of such a sphere near the interface of two dielectric
media (with the sphere being situated in a medium of whereasourmicroscopicresultasgivenbyEqs.(28)and
permittivity ε , and ε denoting the permittivity of the (29) reads
2 1
medium on the far side of the interface) is given by
3~c α(0) 3ε (0) 2
~ ∞ α ε ε U (z )= 2
U(zs)=−16π2ε0zs3 Z0 dξ ε2s ε11−+ε22. (54) 2 A 64π2ε0zA4 ε31/2(0)(cid:20)2ε2(0)+1(cid:21)
∞ 1 2 ay y2 1+a
Here, z is the distance from the sphere to the surface dy − −
and s ×Z1 "(cid:18)y4 − y2(cid:19)ay+py2−1+a
1 y py2 1+a
ε (ω) ε (ω)
αs(ω)=4πε0Rs3ε2(ω)εss(ω)+−2ε22(ω) (55) + y4y+−py2−−1+a# (58)
p
is the excess or effective polarisability [32, 33] of the dis- [a = ε (0)/ε (0)]. In addition to the observations made
1 2
solved particle in the medium. This potential already for the nonretarded limit, the interface-dependent pro-
accounts for the fact that the (macroscopic) particle has portionality constants [i.e., the last factors in Eqs. (57)
a finite volume and can hence only move by displacing and(58)]arenowalsodifferentingeneral,notethatthey