Pyroelectric response of inhomogeneous ferroelectric-semiconductor films
We have modified Landau-Khalatnikov approach and shown that the pyroelectric response of inhomogeneous ferroelectric-semiconductor films can be described by using six coupled equations for the average displacement, its mean-square fluctuation and correlation with charge defects density fluctuation...
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| Цитувати: | Pyroelectric response of inhomogeneous ferroelectric-semiconductor films / A.N. Morozovska // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 85-89. — Бібліогр.: 16 назв. — англ. |
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Morozovska, A.N. 2017-05-28T15:42:09Z 2017-05-28T15:42:09Z 2007 Pyroelectric response of inhomogeneous ferroelectric-semiconductor films / A.N. Morozovska // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 85-89. — Бібліогр.: 16 назв. — англ. 1607-324X PACS: 77.80.-e, 77.84.Dy, 68.03.Cd, 68.35.Gy DOI:10.5488/CMP.10.1.85 https://nasplib.isofts.kiev.ua/handle/123456789/118062 We have modified Landau-Khalatnikov approach and shown that the pyroelectric response of inhomogeneous ferroelectric-semiconductor films can be described by using six coupled equations for the average displacement, its mean-square fluctuation and correlation with charge defects density fluctuations, average pyroelectric coefficient, its fluctuation and correlation with density fluctuations of charged defects. Coupled equations demonstrate the inhomogeneous reversal of pyroelectric response in contrast to the equations of Landau-Khalatnikov type, which describe the homogeneous reversal with sharp pyroelectric coefficient peaks near the thermodynamic coercive field values. Our approach explains pyroelectric loops observed in Pb(Zr,Ti)O₃ film. Модифiковано пiдхiд Ландау-Халатнiкова та показано, що пiроелектричний вiдгук неоднорiдної сегнетоелектрично-напiвпровiдникової плiвки з зарядженими дефектами може бути описаний за допомогою шести зв’язаних рiвнянь для шести параметрiв порядку: середня електрична iндукцiя, її середньоквадратичне вiдхилення, корелятор флуктуацiй iндукцiї та густини заряду дефектiв, пiроелектричний коефiцiєнт, його середньоквадратичне вiдхилення та корелятор з густиною заряду дефектiв. Зв’язанi рiвняння описують неоднорiдне переключення пiроелектричного вiдгуку на вiдмiну вiд рiвнянь типу Ландау-Халатнiкова, якi вiдповiдають випадку однорiдного переключення з рiзким максимумом пiроелектричного вiдгуку поблизу коерцитивного поля. Запропонована модель пояснює типовi петлi пiроелектричного гiстерезису у Pb(Zr,Ti)O₃ плiвках. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Pyroelectric response of inhomogeneous ferroelectric-semiconductor films Пiроелектричний вiдгук неоднорiдних сегнетоелектрично-напiвпровiдникових плiвок Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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| title |
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films |
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Pyroelectric response of inhomogeneous ferroelectric-semiconductor films Morozovska, A.N. |
| title_short |
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films |
| title_full |
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films |
| title_fullStr |
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films |
| title_full_unstemmed |
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films |
| title_sort |
pyroelectric response of inhomogeneous ferroelectric-semiconductor films |
| author |
Morozovska, A.N. |
| author_facet |
Morozovska, A.N. |
| publishDate |
2007 |
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English |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
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Пiроелектричний вiдгук неоднорiдних сегнетоелектрично-напiвпровiдникових плiвок |
| description |
We have modified Landau-Khalatnikov approach and shown that the pyroelectric response of inhomogeneous
ferroelectric-semiconductor films can be described by using six coupled equations for the average displacement,
its mean-square fluctuation and correlation with charge defects density fluctuations, average pyroelectric
coefficient, its fluctuation and correlation with density fluctuations of charged defects.
Coupled equations demonstrate the inhomogeneous reversal of pyroelectric response in contrast to the equations
of Landau-Khalatnikov type, which describe the homogeneous reversal with sharp pyroelectric coefficient
peaks near the thermodynamic coercive field values. Our approach explains pyroelectric loops observed in
Pb(Zr,Ti)O₃ film.
Модифiковано пiдхiд Ландау-Халатнiкова та показано, що пiроелектричний вiдгук неоднорiдної сегнетоелектрично-напiвпровiдникової плiвки з зарядженими дефектами може бути описаний за допомогою шести зв’язаних рiвнянь для шести параметрiв порядку: середня електрична iндукцiя, її середньоквадратичне вiдхилення, корелятор флуктуацiй iндукцiї та густини заряду дефектiв, пiроелектричний коефiцiєнт, його середньоквадратичне вiдхилення та корелятор з густиною заряду дефектiв.
Зв’язанi рiвняння описують неоднорiдне переключення пiроелектричного вiдгуку на вiдмiну вiд рiвнянь типу Ландау-Халатнiкова, якi вiдповiдають випадку однорiдного переключення з рiзким максимумом пiроелектричного вiдгуку поблизу коерцитивного поля. Запропонована модель пояснює типовi петлi пiроелектричного гiстерезису у Pb(Zr,Ti)O₃ плiвках.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/118062 |
| citation_txt |
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films / A.N. Morozovska // Condensed Matter Physics. — 2007. — Т. 10, № 1(49). — С. 85-89. — Бібліогр.: 16 назв. — англ. |
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AT morozovskaan pyroelectricresponseofinhomogeneousferroelectricsemiconductorfilms AT morozovskaan piroelektričniividgukneodnoridnihsegnetoelektričnonapivprovidnikovihplivok |
| first_indexed |
2025-11-27T09:23:21Z |
| last_indexed |
2025-11-27T09:23:21Z |
| _version_ |
1850809052410413056 |
| fulltext |
Condensed Matter Physics 2007, Vol. 10, No 1(49), pp. 85–89
Pyroelectric response of inhomogeneous
ferroelectric-semiconductor films
A.N.Morozovska∗
V.Lashkaryov Institute of Semiconductor Physics of the National Academy of Sciences of Ukraine,
Ave. Nauki 41, 03028 Kiev, Ukraine
Received September 4, 2006, in final form November 17, 2006
We have modified Landau-Khalatnikov approach and shown that the pyroelectric response of inhomogeneous
ferroelectric-semiconductor films can be described by using six coupled equations for the average displace-
ment, its mean-square fluctuation and correlation with charge defects density fluctuations, average pyroelectric
coefficient, its fluctuation and correlation with density fluctuations of charged defects.
Coupled equations demonstrate the inhomogeneous reversal of pyroelectric response in contrast to the equa-
tions of Landau-Khalatnikov type, which describe the homogeneous reversal with sharp pyroelectric coefficient
peaks near the thermodynamic coercive field values. Our approach explains pyroelectric loops observed in
Pb(Zr,Ti)O3 film.
Key words: ferroelectric-semiconductor film, pyroelectric response, charged defects
PACS: 77.80.-e, 77.84.Dy, 68.03.Cd, 68.35.Gy
1. Introduction
The main peculiarity of ferroelectric materials is hysteresis dependence of their dielectric per-
mittivity ε, spontaneous displacement D and pyroelectric coefficient γ on electric field E0 applied
to the sample [1]. The pyroelectric hysteresis loops of inhomogeneous ferroelectric-semiconductor
films have several characteristic features depicted in figure 1a.
Figure 1. Pyroelectric γ (E0) hysteresis loops. Different plots correspond to the data obtained for
a semiconductor-ferroelectric film (a), Landau-Khalatnikov model (b) and our coupled equations
(c) for a bulk sample (ω1 � ω2 are two frequencies of the applied electric field).
Such typical ferroelectric-semiconductors as slightly doped Pb(Zr,Ti)O3 solid solutions, their
films, multilayers and heterostructures are widely used in actuators, electro-optic, piezoelectric,
pyroelectric sensors and memory elements [2–4]. However, the task of creating the ferroelectric
∗E-mail: morozo@i.com.ua
c© A.N.Morozovska 85
A.N.Morozovska
material with pre-determined dielectric and/or pyroelectric properties is solved mainly empirically.
The correct theoretical consideration could answer fundamental questions as well as help to tailor
new ferroelectric-semiconductor materials, save time and expenses.
Conventional phenomenological approaches with material parameters obtained from first-prin-
ciple calculations give significantly incomplete picture of the pyroelectric hysteresis in the doped
or inhomogeneous ferroelectrics-semiconductors (compare figure 1b with figure 1a). In particu-
lar, Landau-Khalatnikov approach, evolved for the single domain perfect ferroelectrics-dielectrics,
describes homogeneous polarization reversal but does not describe the domain nucleation and
movement [5,6]. Therefore, its modification for inhomogeneous ferroelectrics-semiconductors seems
necessary [7,8].
In our recent papers [9–12] we have considered the displacement fluctuations caused by charged
defects and modified the Landau-Khalatnikov approach for the inhomogeneous ferroelectrics-semi-
conductors. In this paper we develop the proposed model for pyroelectric response (see figure 1c).
2. Coupled equations
Figure 2. Spatial distribution of displacement
D(r, t1,2), free carriers charge density n(r, t1,2)
and sluggish defects density ρS in an inhomoge-
neous ferroelectric-semiconductor film.
Let us consider n-type ferroelectric-semi-
conductor with sluggish randomly distributed
defects. The charge density of defects ρs(r)
is characterized by the positive average value
ρ̄S and random spatial fluctuations δρS(r), i.e.
ρS(r) = ρ̄S + δρS(r) (the dash designates av-
erage values). The average distance between
quasi-homogeneously distributed defects is d.
Screening clouds δn(r,t) with Debye screening
radius RD surround each charged center, so
the free carriers charge density is n(r, t) =
n̄ + δn(r, t). Screening clouds are deformed in
the external field E0, and the system “defect
center δρS + screening cloud δn” causes dis-
placement fluctuations δD(r, t) in accordance
with Maxwell’s equations divD = 4π (n + ρS),
div (∂D/∂ t + 4π jc) = 0 (see figure 2).
In this way we obtained six coupled equations for the average displacement D, its mean-square
fluctuation δD2 and correlation δD δρS, pyroelectric coefficient γ̄ = ∂D̄
/
∂ T , its deviation δγ =
∂ δD2
/
∂ T and correlation with charge defects density fluctuations δγ δρS = ∂ δD δρS
/
∂ T (see
[13]):
Γ
∂D̄
∂ t
+
�
α + 3 β δD2� D̄ + βD̄3 = E0(t) + Ei(l, t), (1)
ΓR
2
∂ δD2
∂ t
+ �αR + 3βD̄2� δD2 + β
�
δD2�2
= �E0(t) �δD δρS
n̄
− δEi�+
4πkBT
n̄e
δρS (δρS + δn)� , (2)
ΓR
∂ δD δρS
∂ t
+
�
αR + 3 βD̄2 + β δD2� δD δρS = −E0(t)
δρSδn
n̄
. (3)
Γ
∂ γ̄
∂ t
+
�
α + 3 β δD2 + 3βD̄2� γ̄ = −
�
αT + 3 β δγ2� D̄, (4)
ΓR
∂δγ δD
∂ t
+ 2
�
αR + 2 β δD2 + 3 β D̄2� δγ δD = �E0(t)
δγ δρS
n̄
− �αRT + 6 β D̄ γ̄� δD2
+
4πkB
n̄e
δρS (δρS + δn)� , (5)
ΓR
∂ δγ δρS
∂ t
+
�
αR + β δD2 + 3 β D̄2� δγ δρS = − �αRT + 2β δγ δD + 6 β D̄ γ̄� δD δρS . (6)
86
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films
The built-in electric field
Ei (l, t) =
4πγ
l
(δn (t) + δρS)
x,y ���
+l/2
−l/2
in (1) and its deviation
δEi =
2π
l n̄ �� z
z0
dz (δn + δρS)
�2
x,y
�������
+l/2
−l/2
in (2) are inversely proportional to the film thickness l, thus it vanishes in the bulk material. For a finite
film it is induced by the space charge layers accommodated near the non-equivalent boundaries z = ±l/2
of the examined heterostructure/multilayer (e.g. near the substrate with bottom electrode and free surface
with top electrode depicted in figure 2). Such layers are created by the screening carriers [5–7]. In general
case the field Ei (l, t) can be time-dependent, its amplitude is proportional to the space charge fluctuations
|δn + δρS|. Also Ei diffuses paraelectric-ferroelectric phase transition. In particular, it shifts and smears
dielectric permittivity temperature maximum.
Bratkovsky and Levanyuk [8] predicted the existence of built-in field in a finite ferroelectric film within
the framework of phenomenological consideration. Our approach confirms their assumption and gives the
expression of the field existing in the inhomogeneous ferroelectric-semiconductor film.
Renormalization ΓR ≡ Γ + τm of Khalatnikov kinetic coefficient is connected with the contribution
of free carrier Maxwellian relaxation. The renormalization of coefficients αR ≡ α + (γ + kBT/4π n̄e)�d2
and αRT ≡ αT + kB�4π n̄ed2 is connected with the contribution of correlation and screening effects [9,13].
Coefficient α = −αT(TC−T ) is negative in the perfect ferroelectric phase without random defects (δρ2
S = 0).
For the partially disordered ferroelectric with charged defects (δρ2
S > 0) coefficient αR is positive and
αR � |α|, αRT � αT. For example, for Pb(Zr,Ti)O3 solid solution α ∼ − (0.4 ÷ 2) · 10−2 [3], gradient
term λ ≈ 5 · 10−16 cm2, screening radius RD ∼ �10−6 ÷ 10−4� cm [5], average distance between defects
d ∼ �10−6 ÷ 10−4� cm and thus αR ∼ 1 ÷ 102. So the ratio ξ = −αR/α ≈ αRTT/αT(TC − T ) is greater
than 100.
The dimensionless amplitude g = 4πkBT · n̄��−αD2
Se�of displacement fluctuations δρS (δρS + δn)
varies in the range from 102 to104 for Pb(Zr,Ti)O3 (DS =
−α/β). The positive correlator R2 (t) =
− δρSδn (t)�n̄2 was calculated in [9] at small external field amplitude and low frequency. Under the con-
dition of prevailing extrinsic conductivity n̄ ≈ −ρ̄S the correlator R2 ( t) varies in the range (0; 1) because
its amplitude is proportional to the charged defects disordering δρ2
S�ρ̄2
S.
Hereinafter we discuss only the pyroelectric response near the equilibrium states. The system (1)–
(6) quasi-equilibrium behavior is described by the dimensionless built-in field amplitudeEm = Ei/EC ∼
(δn + δρS) and frequency w = −Γω/α as well as by the aforementioned parameters ξ, R2 (w),g and
temperature T/TC (EC = −α
−α/β). Under the conditions w < 1, g � 1 and ξ � 1 the scaling
parameter gR2�ξ determines the system behavior [9].
Figure 3. Hysteresis loops of pyroelectric coefficient γ (E) for different R2 values. Other param-
eters: g = 100, ξ = 100, T/TC = 0.45, w = 0.05 and Ei = 0 (plot (a) for the bulk sample) and
Ei = ±0.1 · R, (plot (b) for the film).
87
A.N.Morozovska
Figure 3 demonstrates the typical changes of pyroelectric hysteresis loop caused by the increase of
charged defects disordering (note, that gR2�ξ ∼ δρ2
S�ρ̄2
S). It is clear that the increase of gR2�ξ value leads
to the essential decrease and smearing of pyroelectric coefficient peaks near the coercive field as well as to
the decrease of the coercive field value (compare Landau-Khalatnikov loops (R2 = 0) with dashed curves
(R2
> 0.2)).
Let us underline that we do not know any experiment, in which pyroelectric coefficient peaks near the
coercive field have been observed. Moreover, usually pyroelectric hysteresis loops in doped ferroelectrics
have a typical “slim” shape with coercive field values much lower than the thermodynamic one [14,15].
The quantitative comparison of our results with typical PZT-pyroelectric loops is presented in the next
section.
2.1. Discussion
Dopants, as well as numerous unavoidable oxygen O−2 vacancies, can play a role of randomly distributed
charged defects in “soft” PZT. In this case ferroelectric and pyroelectric hysteresis loops have got relatively
high γ and D remnant values, but reveal low coercive fields [3]. Usually pyroelectric hysteresis loops of
PZT are rather slim and sloped even at low frequencies ω ∼ (0.1 ÷ 10)Hz [3], no pyroelectric coefficient
maximum near the coercive field is observed [14–16].
The pyroelectric response of the PZT films was registered by means of dynamic pyroelectric measure-
ments (see [14,16] for details). During the measurements, the quasi-static voltage Uvaried in the range
(–11V, +11V) at the low-frequency ω ∼ 0.01 Hz, the temperature T changes near the room temperature
with the frequency about 20 Hz. Pyroelectric hysteresis loops for Uπ 1 ∼ γ̄ and Uπ 2 ∼ γ̄/ε̄ are presented
in figures 4.
Figure 4. Pyroelectric hysteresis loops (Uπ1 ∼ γ̄ and Uπ2 ∼ γ̄/ε̄) of 1.9 µm-thick PZT(46/54):
Nb film. Squares are experimental data measured by Bravina et al. [14], solid curves are our
calculation with the fitting parameters w = 0.1, R2 = 0.5, g = 100, ξ = 100, Em = −0.03.
It is clear from the figures that our model both qualitatively and quantitatively describes pyroelectric
hysteresis loops in thick “soft” PZT films. The modelling of ferroelectric and dielectric hysteresis loops was
performed earlier (see e.g. [9]).
Earlier we proved [9–13] that the effect of random defect leads to the non-zero average values of δD2
even at D̄ = 0. This means that the sample is divided into polar regions with opposite polarization, i.e. the
domain structure originates from charged defects. In our model we neither consider the spatial distribution
of the emerged domain structure nor incorporate its initial distribution. We calculate the average values
only. The initial distribution of polar regions determines the initial conditions of the system (1)–(6), which
do not affect the equilibrium hysteresis loop shape [9].
Surely, the domain formation can be caused by many other factors besides the considered charged
defects, e.g. by local inhomogeneous stresses and elastic defects. In particular, the presence of elastic defects
or other pinning centers undoubtedly causes additional domain splitting, domain walls movement and
pinning. Allowing for piezoelectric effect, the displacement fluctuations caused by random elastic defects
could be included in the system of coupled equations. Thus, one could assume that their contribution leads
to additional smearing of hysteresis loop, changes the coercive field and saturation law at high external
fields [1].
Thus, the modelling based on the coupled equations (1)–(6) gives realistic coercive field values and a typ-
ical pyroelectric hysteresis loop shape. Taking into account that the inhomogeneous reversal of spontaneous
88
Pyroelectric response of inhomogeneous ferroelectric-semiconductor films
polarization and pyroelectric response occurs in the doped or inhomogeneous ferroelectrics-semiconductors,
the proposed coupled equations could be more relevant in the phenomenological description of their pyro-
electric properties than the models based on Landau-Khalatnikov phenomenology.
References
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versity Press, Oxford, 1978.
2. Lang S.B. Physics Today, 2005, 58(8), 31–36.
3. Cross L.E. Ferroelectric Ceramics: Tailoring Properties for Specific Application. Birkhauser Verlag,
Basel, 1993.
4. Scott J.F. Ferroelectric Memories. Springer, Berlin and Heidelberg, 2000.
5. Fridkin V.M. Ferroelectrics Semiconductors. Consultant Bureau, New-York and London, 1980.
6. Sandomirskii V.B., Khalilov Sh.S., Chenskii E.V. Physics and technics of semiconductors, 1982, 16,
440.
7. Baudry L., Tournier J. J. Appl. Phys., 2005, 97, 024104–11.
8. Bratkovsky A.M., Levanyuk A.P. Phys. Rev. Lett., 2005, 94, 107601–4.
9. Morozovska A.N., Eliseev E.A. J. Phys.: Condens. Matter., 2004, 16, 8937–8956; Preprint cond–
mat/0408647.
10. Morozovska A.N., Eliseev E.A. Physica B, 2005, 355, 236–243.
11. Morozovska A.N., Eliseev E.A. Phys. Stat. Sol. (b), 2005, 242, 947–961.
12. Morozovska A.N. Ferroelectrics, 2005, 317, 37–42.
13. Morozovska A.N., Eliseev E.A., Remiens D., Soyer C. J. Appl. Phys., 2006, 100, 014109–1–12.
14. Bravina S.L., Cattan E., Morozovsky N.V., Remiens D. Semiconductor Physics, Quantum Electronics
& Optoelectronics, 2004, 7, 263–271.
15. Kostsov E.G. Ferroelectrics, 2005, 314, 169–187.
16. Haccart T., Cattan E., Remiens D. Semiconductor Physics, Quantum Electronics & Optoelectronics,
2002, 5 (1), 78–88.
Пiроелектричний вiдгук неоднорiдних
сегнетоелектрично-напiвпровiдникових плiвок
Г.М.Морозовська
Iнститут фiзики напiвпровiдникiв iм. В. Лашкарьова, НАН України, пр. Науки 41, 03028 Київ, Україна
Отримано 4 вересня 2006 р., а остаточному виглядi – 17 листопада 2006 р.
Модифiковано пiдхiд Ландау-Халатнiкова та показано, що пiроелектричний вiдгук неоднорiдної
сегнетоелектрично-напiвпровiдникової плiвки з зарядженими дефектами може бути описаний за
допомогою шести зв’язаних рiвнянь для шести параметрiв порядку: середня електрична iндукцiя,
її середньоквадратичне вiдхилення, корелятор флуктуацiй iндукцiї та густини заряду дефектiв, пi-
роелектричний коефiцiєнт, його середньоквадратичне вiдхилення та корелятор з густиною заряду
дефектiв.
Зв’язанi рiвняння описують неоднорiдне переключення пiроелектричного вiдгуку на вiдмiну вiд рiв-
нянь типу Ландау-Халатнiкова, якi вiдповiдають випадку однорiдного переключення з рiзким ма-
ксимумом пiроелектричного вiдгуку поблизу коерцитивного поля. Запропонована модель пояснює
типовi петлi пiроелектричного гiстерезису у Pb(Zr,Ti)O3 плiвках.
Ключовi слова: плiвка сегнетоелектрика-напiвпровiдника, пiроелектричний вiдгук, зарядженi
дефекти
PACS: 77.80.-e, 77.84.Dy, 68.03.Cd, 68.35.Gy
89
90
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