Cromatograma analítico previo
3.6. Acoplamiento a detectores: espectrómetros de masas.
One of the major factors that degrade the dielectric properties of the thin films compared
to the bulk material is the interfacial capacitance or the dead layer. In this section, the interfacial
capacitance and the dead layer thickness of the doped and undoped BST films were determined
to understand the effect of the BMN dopant on the interface, if at all.
The interface capacitance is commonly estimated by measuring the capacitance of
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against the thickness to obtain a non-zero intercept [47, 178], interfacial capacitance. The active
and the “dead” portion of the BST film are commonly described by a series capacitor model as
shown in Figure 6.11. The regions I and III are the dead layers corresponding to the top and
bottom electrode interfaces, respectively, each with a width of Xd and constant permittivity of d.
The interior region II, of width of t-2Xd, t being the total thickness of the film, represents the
region of the BST film whose dielectric permittivity changes with electric field (b(E)), and
behaves similar to the bulk BST material. From this model, the total measured (𝐶𝑚𝑒𝑎𝑠)
capacitance of the film can be related to the capacitances from the bulk like region (II) and the
interfaces (I&III) as 𝐴 𝐶𝑚𝑒𝑎𝑠(𝐸)= 2𝑋𝑑 𝜀𝑑 + 𝑡−2𝑋𝑑 𝜀𝑏(𝐸), (6.6)
where, A is the area of the capacitors.
Figure 6.11. Schematic showing the two dead interfaces of the BST film of width Xd, and interior region, of width t-2Xd. The equivalent circuit is presented on the right
To use Eq. (6.6) in determining the interfacial capacitance density, one needs to deposit
multiple films with varying thickness and plot a graph of the term on the left hand side versus the
thickness of the film [47]. This approach is evidently expensive as it requires the deposition of
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be used in estimating the interfacial capacitance, dead layer thickness and permittivity, is the
phenomenological Landau-Ginzberg-Devonshire (LGD) theory [31, 179, 180].
As presented in section 2.2.3, in LGD theory, the Helmholtz free energy for the
ferroelectric materials is expressed in terms of polarization, P, (see Eq. (2.20)); from which the
equation of state(𝑖. 𝑒. ∂F 𝜕𝑃⁄ = 𝐸) leads to the relation between the polarization and electric field written as
E =𝑃 +𝑃3. (6.7)
From this equation the field dependent dielectric permittivity (b (E)) of the interior region (II)
can be defined as [31]
𝜀𝑏(𝐸) =𝜕𝑃𝜕𝐸 =𝛽+31𝑃2. (6.8)
Furthermore, using a simple hyperbolic identity
(sinh 𝜑)3+3
4sinh 𝜑 − 1
4sinh 3𝜑 = 0, (6.9)
Eq. (6.7) can be explicitly solved to express the polarization in terms of the electric field. To find
this, let’s define the polarization, P, with a sine hyperbolic function as
𝑃 = 𝐵 sinh 𝜑. (6.10)
Substituting Eq. (6.10) into Eq. (6.7) gives
(sinh 𝜑)3+
𝐵2sinh 𝜑 −
𝐸
𝐵3= 0. (6.11)
By comparing Eqs. (6.9) and (6.11) one can obtain
𝐵 = √4⁄ , 3 (6.12a)
95 φ =13sinh−1(√(27
43
⁄ ) 𝐸) , (6.12b)
which along with Eq. (6.10), results in an explicit expression for the polarization as a function of
electric field
𝑃(𝐸) = √(43) sinh (13sinh−1(√27
43 𝐸)). (6.13)
Finally, the expression for the measured capacitance density can be presented as
𝐴 𝐶𝑚𝑒𝑎𝑠(𝐸)= 2𝑋𝑑 𝜀𝑑 + 𝛽(𝑡 − 2𝑋𝑑) {1 + 4 (sinh ( 1 3sinh−1(√ 27 43 𝐸))) 2 }, (6.14)
by substituting Eq. (6.13) into (6.8), and the resulting expression into Eq. (6.6). This equation
can be fitted to the experimentally measured inverse capacitance density versus applied electric
field to extract the Xd, d, and parameters.
-600 -400 -200 0 200 400 600 60 80 100 120 140 160 180 200 Measured LDG theory 1/(C/A),m 2 /F Bias Field, KV/cm b a
Figure 6.12. Inverse capacitance density vs. electric field: a) pure and b) BMN doped BST film
Figure 6.12 shows the fitting of the measured data for the undoped and BMN doped BST
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extracted parameters are presented in Table 6.4. The coefficients, and , obtained from the fitting are comparable with values reported in the literature [179]. Wider dead layer thickness
accompanied by lower non-tunable permittivity for the BMN doped film leads to lower
capacitance density at the interface, which might be due the weakening of ferroelectricity as a
result of magnesium ions [68].
Table 6.4. Extracted fitting parameters for the doped and undoped BST films
Samples (m2/F) (m5/C2F) d Xd (nm) Interface capacitance (fF/um2)
Undoped BST 3.2x108 6.7X109 86.60 6.8 112.7
BMN doped BST 2.6x108 6.1X109 72.40 9.5 67.3
It is known that the presence of the dead layer at the interface reduces the overall
permittivity and tunability of the ferroelectric film [181]. In an ideal condition where there is no
dead layer (i.e. Xd=0) at the interfaces, meas = b, and the tunability can be written as 𝑛𝑖𝑑𝑒𝑎𝑙 = 𝜀𝑏(0)
𝜀𝑏(𝐸). (6.15a)
Here 𝑛𝑖𝑑𝑒𝑎𝑙 represents the tunability in the absence of the dead layer. In the presence of the dead layer (Xd
0), meas
b and rearranging Eq. (6.6), the measured permittivity can be written as𝜀𝑚𝑒𝑎𝑠(𝐸) =2𝑋 𝑡𝜀𝑑𝜀𝑏(𝐸)
𝑑𝜀𝑏(𝐸)+𝜀𝑑(𝑡−2𝑋𝑑). (6.15b)
The tunability of the non-ideal film can be expressed as
𝑛𝑚𝑒𝑎𝑠 = 𝜀𝑚𝑒𝑎𝑠(0)
𝜀𝑚𝑒𝑎𝑠(𝐸) = 𝜁(𝐸)𝑛𝑖𝑑𝑒𝑎𝑙, (6.15c)
where,
𝜁(𝐸) =2𝑋𝑑𝜀𝑏(𝐸)+𝜀𝑑(𝑡−2𝑋𝑑)
2𝑋𝑑𝜀𝑏(0)+𝜀𝑑(𝑡−2𝑋𝑑). (6.15d)
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For the particular devices used in this analysis, a tunability of 61 % for the undoped and
55 % for the BMN doped film was measured. Using the parameters in Table 6.4, the tunability
that would be obtained without dead layer is estimated to be 69 % for the undoped and 64 % for
the BMN doped films, showing an improvement by 11% and 16 %, respectively. Similarly, when
the dead layer is corrected, the dielectric constant at zero bias field for the undoped BST film
increases from 427 to 552 while that of the BMN doped film rises from 319 to 443—showing an
increase by 30 % and 40 %, respectively. The reduction in tunability and permittivity shows the
deteriorating effects of the dead layer on tunable devices. This influence is observed to be more
pronounced on the doped film which could be due to the presence of the Mg dopant.