Very large dielectric response of thin ferroelectric films with the dead layers

5 Pages·0.14 MB·English

by  A. M. Bratkovsky

Very large dielectric response of thin ferroelectric films with the dead layers
A.M. Bratkovsky1 and A.P. Levanyuk1,2
1Hewlett-Packard Laboratories, 1501 Page Mill Road, Palo Alto, California 94304
2Departamento de F´isica de la Materia Condensada, CIII, Universidad
Aut´onoma de Madrid, 28049 Madrid, Spain
1
0 (October 31, 2000)
0
2
We study the dielectric response of ferroelectric (FE) thin films with “dead” dielectric layer at the

n interface with electrodes. The domain structure inevitably forms in the FE film in presence of the
a deadlayer. Asaresult,theeffectivedielectricconstantofthecapacitorǫ increasesabruptlywhen
eff
J
the dead layer is thin and, consequently, the pattern of 180-degree domains becomes “soft”. We
1 compare the exact results for this problem with the description in terms of a popular “capacitor”
2
model,whichisshowntogivequalitativelyincorrectresults. Werelatethepresentresultstofatigue

observed in thin ferroelectric films.
]
h
c 77.22.Ch, 77.80.Dj, 84.32.Tt, 85.50.+k
e
m
-
at We have shown recently that the dead layer forming stant ǫeff of the capacitor. It is easy to show that in
at the interface between ferroelectric (FE) thin film and the present case (180o domains) the linear response is
t
s
electrodes has drastic effect on the electric response of a not changed by electromechanical effect, the change in
.
t capacitor [1]. It has direct bearing on fatigue observed ǫ only appearing in quadratic terms in external field,
a eff
m inFE capacitorssince inmanycasesthe deteriorationof buthereweareinterestedinzero-fieldvalueofǫ only.
eff
the switching behavior,like the loss of the coercive force We haveassumeda quadraticcoupling betweenthe elas-
-
d and of the squareness of hysteresis loop, were attributed tic strains and the polarization (as in perovskites). In
n to the growth of a “passive layer” at the ferroelectric- this case the linear response of 1800 domain structure
o
electrode interface [2–5]. It is of principal importance without pinning is not affected by intimate contact be-
c
[ that the presence of a dead layer, no matter how thin tween the ferroelectric and the dead layer (excluding

in comparison with thickness of the ferroelectric layer, mere renormalization of materials constants by homo-
3
triggers a formation of the domain structure in FE film geneous stresses). Note that this is invalid in the case of
v
5 [1]. We have shown that when the thickness of the dead 90o domains [8] or a linear coupling between the strain
0 layerd is not verysmall, the apparent(net) polarization and the polarization [9].
0 P ofthe ferroelectricwith180 degreedomainwallsfol- Here we addressthe anomalousbehaviorof the dielec-
a
1 −
lows an approximate relation dP /dE ǫ /d, which is tricconstantǫ indetail. Wealsodiscusstheimportant
1 a ∝ g eff
ingoodcorrespondencewithavailableexperimentaldata issue regarding the relation of the present results to so-
0
0 (see e.g. [6,7] and references therein). Importantly, the called “capacitor” model [3,4,6]. The “capacitor” model
/ response of this structure to an externalbias voltagebe- (incorrectly)presumesthatthe dielectricresponseofthe
t
a comes more rigid when d increases, i.e. when the dead dielectriclayerisnotaffectedbythepresenceofthedead
m layer grows, even in the absence of pinning by defects. layer,so the system lookslike capacitancesin series. We
- The implication for real systems is that with the show that the effective “dielectric constant” of the FE
d
growth of the passive layer the hysteresis loop very layer,ǫ ,asfoundinthe “capacitor”model[3,4,6],isac-
n f
o quickly deteriorates and looses its squareness, as ob- tuallynegative. Inspiteofthisthesystemremainsstable
c served. Theapproximate1/ddependenceoftheresponse (stiffness of the domain pattern is positive).
: suggests that the effective dielectric constant of the ca- The reasonfor this apparentlyunusualbehavior(ǫ <
v f
i pacitor 0) is that the “dielectric constant” of the FE layer ǫf
X
is the non-local quantity which characterizes the whole
ar ǫeff = 4πLC (1) system. This non-local behavior is due to long range
Coulomb interaction, which makes the response rigid
A
may become very large when the layer is thin. Here C evenwhentheFEitselfwouldhaveanegative“dielectric
constant” (cf. Ref. [10], Fig. 1). The “capacitor”model
is the capacitance of the electroded FE film of area ,
A neglectsthisessentiallynonlocalbehaviorand,therefore,
with L the separation between electrodes. Indeed, when
is hardlyapplicable to the problemofdielectric response
the dead layer is thin, the domain width a becomes very
large,it growsexponentially with 1/d2 [1]. The response of thin ferroelectric films, and to the problem of fatigue
forthatmatter. The presentconsiderationsremainvalid
of this domain structure is very soft, and this should
until the period of the domain structure remains much
translate into very abrupt increase of the dielectric con-
1
smallerthatthelateralsizeofthefilm(orthegrainsize). where σ = P is the density of the bound charge given
s
±
Those will set some cutoff for the effects considered be- by only the spontaneous polarization at the interfaces
low. between the FE and the dead layer, and the integration
We shallfind the responseof a ferroelectricfilm under goes over these interfaces. The periodic pattern consists
a bias voltage U with thickness l separatedfrom the top of c domains with widths a1 and a2, and the period
−
and bottom electrodes by passive layers with thickness T = a1 +a2. The solution of the equations (4) is then
d/2 (Fig. 1, inset) with the use of the thermodynamic readily found by Fourier transformation
potential [1]:
σ(x)= σ eikx, ϕ (x,z)= ϕ (z)eikx, (7)
F˜ =Fs+F˜es, (2) k α kα
k k
X X
electrodes 2iP
F˜es = dVE2/(8π)− eaϕa. (3) σk = kTs [1−exp(ika1)], k 6=0, (8)
Z Xa a1 a2
Here Fs is the surface energy of the domain walls, F˜es σk=0 ≡σ0 =Psa1+−a2 ≡Psδ, (9)
is the electrostatic energy, E~ is the electric field, while
ea (ϕa) is the charge (potential) on the electrode a. The wherek =2πn/T =πn/a,n=0,±1,...,a≡(a1+a2)/2
and the index α = f, g marks the quantities for the FE
lastterminEq.(3)accountsfortheworkoftheexternal
voltagesource(s). Notethattheresultswouldnotchange layerandthedeadlayer. NotethatbydefinitionPa ≡σ0
is the net spontaneous polarization of the FE layer. We
qualitatively is there were only one dead layer, since the
obtain
accompanyingdepolarizingfieldwouldhavethe sameef-
fect. F˜ =F˜ +F˜ , (10)
es h inh
We are interested in a case when the ferroelectric has
the spontaneous polarization P~s z (Fig. 1, inset), i.e. F˜h = 4πσ0d−ǫgU lσ0 + 1σelU, (11)
we are considering the case of 180k degreedomain walls. ǫgl+ǫcd 2 2 0
− A
The uniaxialferroelectricfilmhasthe dielectricconstant F˜ 4π σ 2
inh k
ǫ (ǫ ) in the z-direction (in the xy-plane), and the di- = | | , (12)
c a kD
electric constant of the passive layer is ǫ . We select the A k6=0 k
g X
x-axis perpendicular to the domain walls. The poten-
where
tial ϕ, which is related to the electric field in usual way,
E~ = ϕ, satisfies the following equations of electro- ǫ 1/2 kl kd
−∇ a
statics in the ferroelectric and the passive layer [1], Dk =√ǫaǫccoth +ǫgcoth , (13)
ǫ 2 2
"(cid:18) c(cid:19) # (cid:18) (cid:19)
∂2ϕ ∂2ϕ ∂2ϕ ∂2ϕ
f f g g
ǫ +ǫ =0, + =0, (4)
a ∂x2 c ∂z2 ∂x2 ∂z2 and
+wi(th)(tlh+e db)o/u2n,daanrdy conditions ϕ = −(+)U/2, z = σ0el =−4ǫπg 4πǫlσl0++ǫǫdcU (14)
− g c
∂ϕ ∂ϕ
ǫ f ǫ g =4πσ(x), ϕ =ϕ , at z =l/2, (5) is the netchargedensityinduced onthe electrodeatz =
c g f g
∂z − ∂z L/2, with the area of the film. One can calculate the
A
where the subscript f (g) denotes the ferroelectric (pas- apparent dielectric constant of the whole capacitor from
sive layer, or a vacuum gap). Here σ is the density of
the boundchargedue to spontaneouspolarizationatthe ǫeff =4πLσ0el/U. (15)
ferroelectric-passive layer interface, σ = P = P , de-
ns ± s The term F˜ inEq.(10) is due to the net polarizationof
pending on the normal direction of the polarization at h
the FE film induced by the bias voltageU,with the first
the interface, alternating from domain to domain, Fig. 1
terminF˜ (11)correspondingtothe termk =0,singled
(inset). Thus, we assume that the absolute value of the h
out in the stray field energy F˜ (12). With the use of
spontaneous partof the polarizationP is constantin all inh
s
σ from (8) we obtain
domainsandonlyitsdirectionisalternatingfromdomain k
to domain. We have also assumed a usual separation of
F˜ 32a ∞ 1
linearandspontaneouspolarization,sothatthedisplace- inh =
ment vector is Di = ǫikEk +4πPsi, where i,k = x,y,z, Ps2A π2 n=0(2n+1)3D2n+1
X
and the dielectric response ǫik in uniaxial, Fig. 1(inset). 16a ∞ ( 1)n1 cosπnδ
Within this approximationwe obtain + − − , (16)
π2 n3 D
n=1 n
electrodes X
1 1
F˜es = 2ZFEdAσϕ− 2 a eaϕa, (6) where Dn =Dkn, kn =πn/a.
X
2
4πdl 16ln2
The total free energy of the domain pattern per unit
S = a . (25)
K
area is ǫgl+ǫcd − ǫ˜
F˜ γl F˜ +F˜ When d > a , one can neglect the second (stray) term
= + h inh, (17) K
a in this ex∼pression for stiffness, thus recovering our ap-
A A
proximate expression for the net spontaneous polariza-
whereγ =P2∆is thesurfaceenergyofthe domainwall,
s tion Pa σ0 of the ferroelectric film from (18) [1]
with ∆ the characteristic microscopic length [1]. This ≡
freeenergyallowsonetodeterminetheequilibriumprop- P R ǫ
a g
= . (26)
erties and response of the domain pattern to external U S ≈ 4πd
field.
This approximationbreaks down for thinner films but it
We can determine a linear response of the system to
is obvious that the response becomes softer for thinner
bias voltage from the total energy (17), which, for small
dead layers. The breakdown of this approximate behav-
biasU,canbeexpandeduptotermsquadraticinU and
iorhasalsobeennoticedbyKopaletal. forsimilarmodel
σ0,
[11], but they have not analyzed what actually happens
F˜ F˜ (a) 1 to the response at very small thicknesses d of the dead
= stray + 2Sσ02−RUσ0, (18) layer. Note that Sinh < 0, Eq. (22), and, consequently,
A A the following inequality always holds: P /U >ǫ /4πd.
a g
Fw˜shterarey(aσ)0=≡F˜iPnshδ(,a;Fσ˜s0tr=ay(δa=) i0s),thgeiveunsubayl tshtreafiyrsetneterrgmy, ofTcohtihnbdyeaudnliatyyeirs(naot≫aldlo)w.−eIdn. tFhoirsaca<seltahnedre√pǫlaacǫecm=enǫgt
in Eq. (16). This expression results from expanding the we have Dn =ǫg(1+cothkd/2) and∼
Eqs.(12),(16)inpowersofU andδ (note thatσ0 Psδ).
sHpeercetStoisexthteernstailffnbeiasss ovfoltthaegedoUma[wineshtaruvectuormeitw≡teitdhtrhee- PF˜si2nAh = π126ǫag (cid:20)87ζ(3)−Li3 e−b + 18Li3 e−2b (cid:21), (27)
constant term U2 in Eq. (18)]. We have 8a 2 (cid:0) (cid:1) (cid:0) (cid:1)
∝ S = ln , (28)
inh −ǫ 1+e−b
ǫ l g
g
R= , (19)
ǫgl+ǫcd where b = πd/a, Lin(z) = ∞k=1zk/kn [12]. By using
S =S +S (20) the asymptotic expansion of Li (z) we obtain for d a
h inh, n
P ≪
4πdl our earlier result [1]
S = , (21)
h
ǫ l+ǫ d
g c πd ǫ ∆l
g
a= exp , (29)
where S is the (negative) contribution of the inhomo- 2e1/2 4d2
inh (cid:18) (cid:19)
geneous(stray)energy,correspondingtothesecondterm
whichcorrespondsto veryabruptincreaseofthe domain
in (16),
width to values a a , with a the Kittel width (24),
K K
≫
∞ ( 1)n when the dead layer is very thin, Fig. 1. This approx-
S =16a − . (22) imation, however, is not sufficient to make an estimate
inh
nD
n=1 n of the stiffness for very thin dead layers. Indeed, for-
X
mally the stiffness there becomes small and negative,
We can now consider the following limiting cases:
= 4πd[1/(ǫ +ǫ d/l) 1/ǫ ] < 0. This fact simply in-
Thick dead layer (a d). Therewecanreplaceboth g c − g
≪ − dicates that for thin dead layers the stiffness diminishes,
coth in (13) by unity, with the result
S 0 and has to be calculated more accurately. The
→
F˜(δ =0) ∆l 28ζ(3)a exact results, illustrating the abrupt softening of the di-
= + , (23) electric response for small d, are shown in Fig. 1.
P2 a π2˜ǫ
sA One may wish to interpret the approximate result for
slope of the net polarizationP 1/d,givenby Eq.(26),
with ǫ˜= √ǫ ǫ +ǫ . The domain structure has a mini- a
a c g ∝
in terms of the “capacitors in series” model [3,4,6]. In-
mum energy for the equilibrium (Kittel) domain width
deed, a similar result follows if one were to assume that
π2ǫ˜ 1/2 the capacitance of the dead layer, 1/d, dominates at
a=aK ∆l √l, (24) small thicknesses of the layer d. ∝
≡ 28ζ(3) ∝
(cid:18) (cid:19) An opinion has even been voiced that the effective di-
so that a √∆l l, a usual relation. Then inho- electric constant of the FE layer is infinite (ǫf = ∞?)
mogeneous∼contribu≪tion to stiffness S 16ln2a , since the domain walls in our model are not pinned [13].
inh ≈ − ǫ˜ K However,suchaninterpretationwouldbeincorrectsince
Eq.(22), and we obtain from Eqs.(20),(21)
ǫ ,asfoundinthe“capacitor”model,isnotinfinite, but
f
3
create a net electric field in the capacitor, see Eq. (11)
andsecondterminEq.(18),andits energyis the source
ε =ε L/d L/W=100 of finite stiffness of the domain pattern, even when the
1 g wallsarenotpinned,asisverywellknownsince1960[14].
The reasonfor this apparentlyunusualbehavior(ǫ <
100 f
0) is that the “dielectric constant” of the FE layer ǫ
d/2 f
z ε in (31) is the non-local quantity which characterizes the
L a a l c ε whole system: it depends on the properties of the dead
1 2 a
x layer. This non-local behavior is due to long range
ε Coulomb interaction, which makes the response rigid
10 g
even when the FE itself would have a negative “dielec-
ε /ε -ε/ε tric constant” (for analogous situation in FE films with
eff 1 f 1
a/a
K depletion charge see Ref. [10], Fig. 1). Thus, the “ca-
pacitor”modelactuallyoperateswithobscurequantities
without much physical meaning.
1
0 5 10 15 20 We illustrate the difference between the exact results
d/W for ǫ and that following from the “capacitor” model
eff
ǫeff ǫ1 ǫgL/d in the assumption ǫf = [13] in
FIG.1. The dielectric constants of the ferroelectric with ≈ ≡ ∞
Fig. 1. The softness of the domain structure, charac-
thedead layers of thicknessd/2, as definedin thetext. W is
terized by P /U, increases very abruptly when the dead
thedomainwallthickness. Noteanabruptincreaseoftheef- a
layer is thin and the width of the domains is a a
fectivedielectricconstantofthecapacitorǫeff incomparison ≫ K
[Kittel width (24)]. According to (32), ǫ increases
withǫ1 =ǫgL/d,andthatthe“dielectricconstant”oftheFE eff
layerisnegative,ǫf <0. Therelationǫeff ≈ǫ1,whichfollows abruptly and becomes ≫ ǫ1 in this region, Fig. 1, in
fromthe“capacitor”model,isclearlyviolatedwhenthedead stark deviation from the prediction of the “capacitor”
layer is thin and the domain width a is large (a ≫ aK, the model ǫeff ǫ1 [13]. For thick “dead layers” the effec-
Kittelwidth,andǫeff becomes≫ǫ1). Insetshowsschematics tive dielectr≈ic constant ǫeff becomes comparable to ǫ1,
of the electroded ferroelectric film (capacitor) with the dead as we showed earlier [1].
layers. Wethinkthattheaboveclearlydemonstratesadanger
ofapplyinganaiveelectriccircuitanalysistoFEsystems
where the addition of one “circuit element” (dead layer)
finite andactuallynegative. To establishthis, onehas to
radically changes the electric response of the other, FE
findthe voltagedropsacrossthe deadlayerandFE film.
layer, by introducing a domain structure. It is not sur-
The homogeneous part of the electric fields inside the
prising,therefore,thatsuchanapproachcannotexplaina
FE (dead) layer E (E ) and the corresponding voltage
f d
fatigue observedin FE films (see e.g.[6,7]andreferences
drops are found from the standard equations
therein). Certainly, there might be various reasons for
E d+E l=U, thefatigueinFEcapacitors. Wesimplyobservethatthe
d f
growth of the dead layer at the interface with electrode
ǫ E =ǫ E +4πP , (30)
g d c f a
makesthe dielectric response ofthe film rigid evenwhen
where from one can, for instance, easily recover the ex- the domain walls are not pinned. The present mecha-
pression (14) for the net charge density on electrodes. nism gives a correct order of magnitude for the tilt of
One obtains the “capacitor” model by assuming that the hysteresis loops [1], therefore the growth of passive
the interface between the FE film and the dead layer layer might be the main source of fatigue. In the limit-
is equipotential and can be viewed as the metallic film ingcaseofthindeadlayerstheperiodoftheferroelectric
separating two areas. Simple calculation gives domains in the absence of pinning becomes much larger
than the standard Kittel width. As a consequence, the
ǫ 4πlσ0el =ǫ 1+4πPal/(ǫcU) <0 (!), (31) domainpatternbecomesvery“soft”intheabsenceofthe
f c
≡ Uf 1 4πPad/(ǫgU) pinning of the domain walls, and its contribution to the
−
4πLσel Lǫ ǫ dielectricresponsebecomesverylarge,sincethedomains
ǫ 0 = g c [1+4πP l/(ǫ U)], (32)
eff a c with polarization parallel to the external field can easily
≡ U ǫ d+ǫ l
c g
grow at the expense of the domains with the opposite
where U E l. Since always P /U >ǫ /4πd, Eq. (26), polarization.
f f a g
≡
we obtain ǫ < 0, Fig. 1. In spite of formally negative
f
“dielectricconstant”oftheferroelectriclayerǫ ,Eq.(31),
f
which is an artifact of the “capacitor” model [3,4,6,13],
the system remains stable (stiffness of the domain pat-
ternispositive). Indeed,shiftingthedomainwallswould
4
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electrics 223, 127 (1999).
[12] Lin(z) = zΦ(z,n,1), see I.S. Gradshteyn and I.M.
Ryzhik,Tableof Integrals, Series, and Products, 5thed.,
edited by A. Jeffrey (Academic, New York, 1994), Sec.
9.55.
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Thoseauthorsgaveatranscendentalequationformagne-
tization versus magnetic field, M =M(H), for a slab of
ferromagneticmaterialintermsofsomeseries,whichwas
evaluated numerically. Note that the effects of screening
of electric field by electrodes, which are responsible for
the present unusual behavior of the domain pattern in
ferroelectric capacitor, haveno analogy in magnetic sys-
tems.
5