# Extrinsic and Intrinsic Frequency Dispersion of High-k Materials in Capacitance-Voltage Measurements

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## Abstract

**:**

## 1. Introduction

_{2}as a gate dielectric [1,2,3]. However, according to the International Technology Roadmap for Semiconductors (ITRS), CMOS technology could be extended to 14 nm nodes by 2020 by adopting novel device structure and new materials. The physical gate length and printed gate length of the device can be scaled down to 6 nm and 9 nm, respectively [4]. The rapid shrinking of feature size of transistors has forced the gate channel length and gate dielectric thickness on an aggressive scale. As the thickness of SiO

_{2}gate dielectric thin films used in metal-oxide-semiconductor (MOS) devices was reduced towards about 1 nm, the gate leakage current level became unacceptable. Below the physical thickness of 1.5 nm, the gate leakage current exceeds the specifications. To overcome this leakage problem, high-k materials were introduced because they allow the physical thickness of the gate stack to be increased but keep the equivalent oxide thickness (EOT) unchanged. Hence, the gate leakage was found to be reduced by two to three orders of magnitude.

_{2}has frequently been observed in C-V measurements [10,11]. Several models and analytical formulae have been thoroughly investigated for correcting the data from measurement errors. Attention has been given to eliminate the effects of series resistance [12], oxide leakage, undesired thin lossy interfacial layer between oxide and semiconductor [13], surface roughness [14], polysilicon depletion [15,16,17] and quantum mechanical effect [18,19,20,21].

_{2}MOS devices [14] while the analysis of the La

_{x}Zr

_{1−x}O

_{2−δ}thin film and Ce

_{x}Zr

_{1−x}O

_{2−δ}thin film led to the conclusion that surface roughness was not responsible for the observed frequency dispersion for the thick high-k dielectric thin films. The polysilicon depletion effect and quantum confinement should be also considered. After taking into account all extrinsic causes of frequency dispersion mentioned above, the intrinsic effect (dielectric relaxation) of high-k dielectric thin films arose and several dielectric relaxation models were discussed. The dielectric relaxation results of Ce

_{x}Zr

_{1−x}O

_{2−δ}, LaAlO

_{3}, ZrO

_{2}and La

_{x}Zr

_{1−x}O

_{2−δ}thin films could be described by the Curie-von Schweidle (CS) law, the Kohlrausch-Williams-Watts (KWW) and the Havriliak-Negami (HN) relationship, respectively. The higher k-values were obtained from La

_{x}Zr

_{1-x}O

_{2-δ}and Ce

_{x}Zr

_{1-x}O

_{2-δ}thin films with the low lanthanide concentration levels (e.g., x ~ 0.1) where the more severe dielectric relaxation was observed. The causes of the dielectric relaxation were discussed in terms of this observation.

## 2. Experimental

**Figure 1.**Capacitance-voltage (C-V) measurement system of metal-oxide-semiconductor (MOS) devices. A MOS device was located on the manual probe station which was connected to the LCR meters (Agilent 4284A/4275A). The LCR meters were controlled by a desktop computer through a GPIB interface. The C-V measurement data extracted from the LCR meters were transferred back to the computer and saved to obtain the C-V curves automatically.

_{G}and V

_{G}are the charge area density and voltage on the metal electrodes, A is the metal electrode area, dV

_{ac}/dt is the AC voltage change, and i

_{ac}is the AC current. The capacitance of a MOS device was obtained by Agilent 4284A/4275A, which provided a small signal voltage variation rate (dV

_{ac}/dt) and measured the small signal current (i

_{ac}) flowing through the MOS device to calculate the differential capacitance of the MOS device according to Equation (1) [22,23]. For the Agilent 4284A/4275A precision LCR meters, there are two models used to calculate the device capacitance. One is the series model and the other is the parallel model, as shown in Figure 2. The parallel model was used in the following C-V and C-f measurements. In Figure 2, C

_{m}is the measured capacitance. R

_{m}and G

_{m}are the measured resistance and conductance respectively. C

_{D}is the depletion capacitance and Y

_{it}is the admittance due to interface states of the MOS device, respectively. C

_{ox}represents the actual frequency independent capacitance.

**Figure 2.**Conventional LCR meters typically measure the device capacitance based on (

**a**) Series capacitance model or (

**b**) Parallel capacitance model. C

_{m}is the measured capacitance. R

_{m}and G

_{m}are the measured resistance and conductance respectively. C

_{D}is the depletion capacitance and Y

_{it}is the admittance due to interface states of the MOS device, respectively.

_{ac}in the C-V and C-f measurements of MOS devices by the LCR meters should be taken into account. Especially crystalline thin ﬁlms exhibit signiﬁcantly higher leakage current than amorphous thin ﬁlms, which could be due to the leakage pathway introduced from the grain boundaries and the local defects [24,25]. An approximation for the percentage instrumental error was given by the formula $0.1\times \sqrt{1+{D}^{2}}$, where D is a dissipation factor. If the instrumentation error is less than 0.3%, the leakage current in the MOS device is negligible [13]. In the following C-V and C-f measurements, the leakage current in high-k thin films was so small that it was not a contributing factor to frequency dispersion [26].

_{3}, ZrO

_{2}, Ce

_{x}Hf

_{1−x}O

_{2−x}and La

_{x}Zr

_{1−x}O

_{2−δ}thin films, were deposited on n-type Si (100) substrates using liquid injection atomic layer deposition (ALD), carried out on an Aixtron AIX 200FE AVD reactor fitted with the “Trijet” ™ liquid injector system [27]. The doping level of Ce

_{x}Hf

_{1−x}O

_{2−x}thin film and La

_{x}Zr

_{1−x}O

_{2−δ}thin film was varied up to a concentration level of 63%, i.e., x = 0.63. The interfacial layer between the high-k thin film and silicon substrate was a ~1 nm native SiO

_{2}determined by cross-section transmission electron microscopy (XTEM). A thermal SiO

_{2}sample was grown using dry oxidation at 1100 °C to provide a comparison with the high-k stacks. MOS capacitors were fabricated by thermal evaporation of Au gates through a shadow mask with an effective area of 4.9 × 10

^{−4}cm

^{2}. The backside contact of selected Si wafers was cleaned with a buffer HF solution and subsequently a 200 nm thickness of Al film was deposited on it by thermal evaporation. Some selected samples of Ce

_{x}Hf

_{1−x}O

_{2−x}thin films and La

_{x}Zr

_{1−x}O

_{2−δ}thin films were annealed at 900 °C for 15 min in a N

_{2}ambient to crystallize the thin films before metallization. All the other samples were annealed in forming gas at 400 °C for 30 min. The C-V or C-f curves of Ce

_{x}Hf

_{1−x}O

_{2−δ}, La

_{x}Zr

_{1−x}O

_{2−δ}, ZrO

_{2}, LaAlO

_{3}and thermal SiO

_{2}thin films were measured to investigate their electrical properties. X-ray diffraction (XRD), XTEM and atomic force microscopy (AFM) of La

_{x}Zr

_{1−x}O

_{2−δ}thin films and Ce

_{x}Hf

_{1−x}O

_{2−δ}thin films were used to investigate their physical properties.

## 3. Results and Discussion

_{2}MOS capacitors (MOSC), the parasitic effect is introduced in Section 3.1.1. Dispersion could be avoided by depositing an Al thin film at the back of the silicon substrate. The correction models were able to minimize the dispersion as well. The existence of frequency dispersion in the LaAlO

_{3}sample is discussed in Section 3.1.2, which is mainly due to the effect of the lossy interfacial layer between the high-k thin film and silicon substrate on the MOSC. Relative thicker thickness of the high-k thin film than the interfacial layer significantly prevented frequency dispersion. Also, extracted C-V curves were reconstructed by mathematic correction models. Frequency dispersion from the effect of surface roughness was represented in an ultra-thin SiO

_{2}MOS device, which is discussed in Section 3.1.3. Furthermore, the surface property of the La

_{x}Zr

_{1−x}O

_{2−δ}thin films is studied. In Section 3.1.4 two further potential extrinsic causes: polysilicon depletion effect and quantum mechanical confinement, for frequency dispersion are considered. After careful considerations of extrinsic causes for frequency dispersion, intrinsic frequency dispersion is analyzed in Section 3.2. Section 3.2.1 describes the frequency dependence of k-value in La

_{x}Zr

_{1−x}O

_{2}/SiO

_{2}and Ce

_{x}Hf

_{1−x}O

_{2−δ}/SiO

_{2}stacks. In order to interpret intrinsic frequency dispersion, several dielectric relaxation models are introduced in Section 3.2.2 for high-k materials with specified fitting parameters. Last but not least, three possible causes of the dielectric relaxation for the La

_{x}Zr

_{1−x}O

_{2−δ}dielectrics are proposed in Section 3.2.3. The effects of the cation segregation caused by annealing and rapped electrons on the dielectric relaxation were negligible. However, a decrease in crystal grain size may be responsible for the increase in the dielectric relaxation.

#### 3.1. Extrinsic Causes of Frequency Dispersion During C-V Measurement

_{2}have been investigated, such as surface roughness [14], polysilicon depletion [15,16,17], quantum confinement (only for an ultra-thin oxide layer) [18,19,20,21], parasitic effect (including series resistance, back contact imperfection and cables connection) [28,29,30], oxide tunneling leakage current (direct tunneling current, F-N tunneling etc.) [31], unwanted interfacial lossy layer [13] and dielectric constant (k-value) dependence (dielectric relaxation) [26]. The extrinsic frequency dispersion is discussed firstly in Section 3.1. The extrinsic causes of frequency dispersion during C-V measurement in high-k thin film, which were investigated step by step before validating the effects of k-value dependence, were parasitic effect, surface roughness, and lossy interfacial layer. The other causes like tunneling leakage current and quantum confinement are negligible if the thickness of the high-k thin film is high enough. Polysilicon depletion effects were not considered due to the fact that metal gates were used here. The C-V results of high-k or SiO

_{2}based dielectrics are shown in Figure 3, Figure 4 and Figure 5, respectively. The parasitic effect (including back contact imperfection R

_{S}

^{’}, C

_{S}

^{’}, cables R

_{S}

^{”}, C

_{S}

^{”}and substrate resistance R

_{S}), the lossy interfacial layer effect C

_{i}, G

_{i}(between the high-k thin film and silicon substrate), polysilicon depletion effect and surface roughness on high-k thin films are summarized in detail in Figure 6.

**Figure 3.**Frequency dispersion in C-V measurements observed in the thermal oxide (SiO

_{2}) sample. In the absence of a substrate back Al contact, dispersion was evident in the sample with a small substrate area of 1cm

^{2}[32].

**Figure 4.**Presence of frequency dispersion in ZrO

_{2}samples at different frequencies (10kHz, 100kHz and 1MHz). The shadowed boxes indicate the presence of metal Al contact at the back of silicon substrates with an effective area of 6 cm

^{2}and the capacitance equivalent thickness (CET) is 2.7 nm. C

_{acc}is the capacitance in the accumulation range [32].

**Figure 5.**C-V curves from a Ce

_{x}Hf

_{1−x}O

_{2−δ}thin film at different frequencies (from 100 Hz to 200 kHz). Frequency dispersion could still be observed regardless of the interfacial layer effect of MOS structures and parasitic effects (caused by substrate resistance, back contact imperfection and cables). This kind of dispersion was caused by the frequency dependence of the k-value (dielectric relaxation) [33].

**Figure 6.**Causes of frequency dispersion during C-V measurement in the high-k thin film were the parasitic effect (including back contact imperfection resistance R

_{S}

^{’}and capacitance C

_{S}

^{’}, cables resistance R

_{S}

^{”}and capacitance C

_{S}

^{”}, substrate series resistance R

_{S}and depletion layer capacitance of silicon C

_{D}) and the lossy interfacial layer effect (interfacial layer capacitance C

_{i}and conductance G

_{i}). The dashed box includes surface roughness effect, polysilicon depletion effect, high-k capacitance C

_{h}, high-k conductance G

_{h}, the lossy interfacial layer capacitance C

_{i}and conductance G

_{i}. The oxide capacitance C

_{ox}consists of the high-k capacitance C

_{h}and the lossy interfacial layer capacitance C

_{i}.

#### 3.1.1. Parasitic Effect

_{S}of the quasi-neutral silicon bulk between the back contact and the depletion layer edge at the silicon surface underneath the gate; and (2) the imperfect contact of the back of the silicon wafer. Frequency dispersion caused by the parasitic effect is shown in Figure 3.

_{2}MOS capacitors, since in this case the effect of the lossy interfacial layer between the bulk dielectric and silicon substrate can be neglected. The thickness of thermal SiO

_{2}was thick enough to allow the tunneling leakage current to be neglected. [36,37]. Frequency dispersion in the SiO

_{2}capacitor was only observed in samples with small substrate effective areas as depicted in Figure 7a (closed symbols extracted from Figure 3). In addition, the measured results were also no longer reproducible for small samples in the absence of Al back contacts, as shown in Figure 7b (the closed symbols). It therefore impacted the measurement reliability.

**Figure 7.**Frequency dispersion in C-V measurements observed in thermal oxide (SiO

_{2}) samples. (

**a**) In the absence of substrate back Al contact, dispersion was evident only in the sample with a smaller substrate area (denoted by s1); (

**b**) The reproducibility of the tested devices in both the presence and absence of back metal contact. Both of the sample sets were measured three times within 24 hours. Closed symbols (e.g., ▲) signified the C-V results from the sample without back Al contact (indicated by a blank square), while the opened symbols (e.g., ○) showed the C-V results from the other sample with back Al contact (indicated by a shadow square) [32].

_{2}, one must take into account the parasitic components that may arise due to the silicon series resistance and the imperfection of the back contact. A correction may then be applied for the measured C-V curves in order to obtain their true values. Figure 8a shows an equivalent circuit of an actual case in comparison with the measurement mode, where C

_{ox}represents the actual frequency independent capacitance across the SiO

_{2}gate dielectric, R

_{S}includes both the bulk resistance in the silicon substrate and contributions from various contact resistances and cable resistances. The presence of the back contact capacitance and contributions from cable capacitance were also modeled by a capacitance. C

_{S}, C

_{C}, G

_{C}, C

_{m}, G

_{m}refer to corrected (without the effect of the parasitic components R

_{S}and C

_{S}) measured capacitance and conductance, respectively. Following Kwa [13], the corrected capacitance C

_{C}was given by [32]:

_{ma}and G

_{ma}are the capacitance and conductance measured in strong accumulation. The measured capacitance can be recovered, independently of the measured frequencies, by applying the correction according to the model as depicted in Figure 8b. Alternatively, the parasitic effects can simply be minimized by depositing an Al thin film at the back of the silicon substrate (open symbols in Figure 7b and solid line in Figure 8b). In summary, it has been demonstrated that once the parasitic components are taken into account, it is possible to determine the true capacitance values free from errors. Therefore, the measurement system reliability can be maintained.

**Figure 8.**Effects of series resistance and back contact imperfection. (

**a**) Equivalent circuit model, taking into account the presence of parasitic components from series resistance, cables and back contact imperfection (with the addition of the C

_{S}and R

_{S}). C

_{D}is the depletion capacitance of silicon and Y

_{it}is the admittance due to the interface states between SiO

_{2}and silicon substrate, respectively. C

_{ox}is the oxide capacitance; (

**b**) Extracted C

_{C}-Vg curves based on measured data C

_{m}and G

_{m}using Equation (2). Dispersions disappear after considering C

_{S}and R

_{S}or depositing back Al contact (solid line). The blank square shows the tested device without back Al contact on silicon substrate. The effective substrate area is 1cm

^{2}[32].

#### 3.1.2. Lossy Interfacial Layer Effect

_{h}<< C

_{i}) and the effect of C

_{i}on C

_{m}was eliminated. Furthermore the effect of the lossy interfacial layer conductance G

_{i}on frequency dispersion can be suppressed by replacing the native SiO

_{2}by a denser SiO

_{2}thin film. In Figure 4, the frequency dispersion effect was significant even with the Al back contact and the bigger substrate area. In this case, C

_{h}(CET = 2.7 nm) was comparable with C

_{i}(~1 nm native SiO

_{2}) and the frequency dispersion effect was attributed to losses in the interfacial layer capacitance, caused by interfacial dislocation and intrinsic differences in bonding coordination across the chemically abrupt ZrO

_{2}/SiO

_{2}interface.

**Figure 9.**High frequency C-V results of LaAlO

_{3}thin film. The absence of frequency dispersion in the LaAlO

_{3}sample is observed with an effective area of 6 cm

^{2}with back Al contact. C

_{acc}is the capacitance in the accumulation range [32].

_{C}, was [32]:

_{m}and G

_{m}are the measured capacitance and conductance and ω is the measurement angular frequency. At an angular frequency ω

_{j}(j = 1 or 2), the measured capacitance and conductance are C

_{mj}and G

_{mj}respectively. Since the expression of C

_{C}with respect to ω

_{j}, C

_{mj}and G

_{mj}is complicated, three abstract parameters, Δ, I

_{mj}, and R

_{mj}have been introduced to reduce the expression of C

_{C}. Figure 10b shows the corrected C-V curves from Figure 4, extracted using Equation (5). All of the extracted C-V curves closely align with one another over the three different frequency pairs to reconstruct the true capacitance values. This indicates that the presence of a lossy interfacial layer is also responsible for the effect of frequency dispersion in high-k stacks.

**Figure 10.**Effect of the lossy interfacial layer on high-k stacks. (

**a**) Four-element equivalent circuit model for high-k stacks, taking into account the presence of the interfacial layer with the additional capacitance, C

_{i}, and conductance, G

_{i}, parallel circuit components. C

_{h}and G

_{h}represent the actual capacitance and conductance across the high-k dielectric. C

_{D}is the depletion capacitance and Y

_{it}is the admittance due to interface states, respectively; (

**b**) Extracted C

_{C}-Vg curves based on dual-frequency data from Figure 3 and the equivalent circuit model from Figure 10a [32].

#### 3.1.3. Surface Roughness Effect

_{2}MOS device [14]. In the following discussion, the effects of direct tunneling, series resistance and surface roughness on the capacitance were taken into account without considering quantum confinement and the polysilicon depletion effect. From Figure 11, the measured capacitance C

_{m}is given by [38]:

_{S}is the series resistance. From Equation (6), the real capacitance taking into account the surface roughness, C

_{ideal}, can be calculated and it is free of frequency [38]. It was found that the surface roughness affects frequency dispersion when the thickness of ultra-thin oxides is ~1.3nm. To investigate whether the unwanted frequency dispersion of the high-k materials in Figure 5 is caused by the surface roughness or not, the surface properties of the La

_{x}Zr

_{1−x}O

_{2−δ}thin films was studied using AFM. The typical AFM micrographs of the La

_{x}Zr

_{1−x}O

_{2−δ}annealed thin films (x = 0.35 and x = 0.09) are shown in Figure 12.

**Figure 11.**Equivalent circuit of the parallel mode of the measurement system. G is the conductance due to pure tunneling effect. g is the conductance due to the surface roughness effect. R

_{S}is the series resistance. Figure is taken from Reference 36.

**Figure 12.**AFM micrographs of the surface of La

_{x}Zr

_{1−x}O

_{2}annealed thin films. (

**a**) x = 0.35; (

**b**) x = 0.09 [33].

**Figure 13.**C-V results at different frequencies from the annealed La

_{x}Zr

_{1−x}O

_{2−δ}samples after back Al contact deposition and the effective substrate area was 6 cm

^{2}: (

**a**) x = 0.35; and (

**b**) x = 0.09. Signiﬁcant frequency dispersion was observed for the x = 0.09 annealed sample, but not for the x = 0.35 annealed sample [26].

#### 3.1.4. Other Effects

#### 3.2. Intrinsic Causes of Frequency Dispersion During C-V Measurements

#### 3.2.1. Frequency Dependence of k-Value

**Figure 14.**(

**a**) Frequency dispersion in C-V measurements observed from La

_{x}Zr

_{1−x}O

_{2}samples after back Al contact deposition and the effective substrate area was 6 cm

^{2}. Therefore, all the extrinsic causes of frequency dispersion were excluded; (

**b**) A summary of frequency dependence of k-value extracted from Figure 14a, Figure 7 (SiO

_{2}), Figure 9 (LaAlO

_{3}), and Figure 10 (ZrO

_{2}). No frequency dependence of k-value was observed for the LaAlO

_{3}/SiO

_{2}and ZrO

_{2}/SiO

_{2}stacks. The frequency dependence of the k-value was observed for the La

_{x}Zr

_{1−x}O

_{2}/SiO

_{2}stacks [32].

**Figure 15.**Frequency dependence of the k-value was extracted from C-f measurements of La

_{0.35}Zr

_{0.65}O

_{2−δ}and La

_{0.09}Zr

_{0.91}O

_{2−δ}thin films annealed at 900 °C, or extracted from Figure 13 (a,b). Frequency dependence of the Ce

_{x}Hf

_{1−x}O

_{2−δ}thin film was extracted from Figure 5 [33].

**Figure 16.**Frequency dependence of the k-value was extracted from C-f measurements observed in four La

_{x}Zr

_{1−x}O

_{2−δ}thin films. The square-symbols were measured from the La

_{0.09}Zr

_{0.91}O

_{2−δ}sample. The diamond-symbols were measured from the La

_{0.35}Zr

_{0.65}O

_{2−δ}sample. The triangle-symbols were measured from the La

_{0.22}Zr

_{0.78}O

_{2−δ}sample. The circle-symbols were measured from the La

_{0.63}Zr

_{0.27}O

_{2−δ}sample. Solid lines are from fitting results from the Cole-Davidson equation, while the dashed line is from the HN equation. The parameters α, β and τ are parameters from the Cole-Davidson or HN equation [32,33].

_{0.22}Zr

_{0.78}O

_{2}and La

_{0.63}Zr

_{0.37}O

_{2}) are given in Figure 14a. Figure 14b showed no frequency dependence of the k-value in LaAlO

_{3}/SiO

_{2}and ZrO

_{2}/SiO

_{2}stacks. However, the frequency dependence of the k-value was observed in La

_{x}Zr

_{1}

_{–x}O

_{2}/SiO

_{2}stacks. The k-values of La

_{0.22}Zr

_{0.78}O

_{2}and La

_{0.63}Zr

_{0.37}O

_{2}were observed and separately decreased from 13.5, 10.5 to 12 and 9.5 as the frequency increased from 1 kHz to 1 MHz. A constant frequency response was observed in thermal SiO

_{2}, as shown in Figure 14b.

_{x}Hf

_{1−x}O

_{2−δ}, La

_{0.35}Zr

_{0.65}O

_{2−δ}and La

_{0.09}Zr

_{0.91}O

_{2−δ}thin films are given in Figure 15. The zirconia thin film with a lanthanum (La) concentration of x = 0.35 showed that a k-value slowly decreased from 18 to 15 as the frequency increased from 100 Hz to 1 MHz. In contrast the lightly doped 9% sample had a sharp decreased k-value and suffered from a severe dielectric relaxation. A k-value of 39 was obtained at 100Hz, but this value was reduced to a k-value of 19 at 1 MHz. The 10% Ce doped hafnium thin film also had a k-value change from 33 at 100 Hz to 21 at 1 MHz. Figure 16 summarizes the frequency dependence of k-value of four La

_{x}Zr

_{1–x}O

_{2}thin films from Figure 14 and Figure 15.

**Table 1.**The fitted parameters of the dielectric relaxation models for Figure 16.

Models | Cole-Cole | Cole-Davidson | Havriliak-Negami | ||||
---|---|---|---|---|---|---|---|

Parameters | α | τ (s) | β | τ (s) | α | β | τ (s) |

La_{x}Zr_{1–x}O_{2–δ} | 0.75 | 3.9 × 10^{−7} | 0.0721 | 0.0028 | 0.6535 | 0.3458 | 7.3 × 10^{−5} |

x=0.09 | |||||||

La_{x}Zr_{1–x}O_{2–δ} | 0.866 | 4.6 × 10^{−11} | 0.028 | 0.006 | 0 | 0.028 | 0.006 |

x=0.35 | |||||||

La_{x}Zr_{1–x}O_{2–δ} | 0.815 | 3.8 × 10^{−11} | 0.0186 | 0.0013 | 0 | 0.0186 | 0.0013 |

x=0.22 | |||||||

La_{x}Zr_{1–x}O_{2–δ} | 0.82 | 5.2 × 10^{−12} | 0.0143 | 0.0012 | 0 | 0.0143 | 0.0012 |

x=0.63 |

#### 3.2.2. Dielectric Relaxation Models and Data Fitting

_{s}, which is usually denoted as “static dielectric constant”. ε

_{s}is also deﬁned as the zero-frequency limit of the real part, ε', of the complex permittivity. ε

_{∞}is the dielectric constant at ultra-high frequency. ε' is the k-value.

_{x}Zr

_{1–x}O

_{2}and Ce

_{x}Hf

_{1–x}O

_{2–δ}) occurred over a wide frequency range. The data was unable to be fitted with the Debye equation because the high-k materials have more than one relaxation time.

_{0.91}Zr

_{0.09}O

_{2}, La

_{0.22}Zr

_{0.78}O

_{2}, La

_{0.35}Zr

_{0.65}O

_{2}and La

_{0.63}Zr

_{0.37}O

_{2}thin films and the fitted parameters are shown in Table 1. All of the data perfectly fitted, but the relaxation time was too small (e.g., 10

^{−11}s), as shown in Table 1.

_{0.91}Zr

_{0.09}O

_{2}thin films was not acceptable.

_{0.91}Zr

_{0.09}O

_{2}, La

_{0.22}Zr

_{0.78}O

_{2}, La

_{0.35}Zr

_{0.65}O

_{2}and La

_{0.63}Zr

_{0.37}O

_{2}thin films more accurately than the Cole-Cole and Cole-Davidson equations which have only one distribution parameter. The fitting curves are shown in Figure 16. The fitting parameters of the La

_{x}Zr

_{1–x}O

_{2}(x = 0.09, 0.22, 0.35 and 0.63) dielectrics are provided in Table 1.

^{−5}s.

_{K}is the characteristic relaxation time, β

_{K}is a stretching parameter, whose magnitude could vary from 0 to 1. For β

_{K}= 1 the Debye process is obtained. In order to analyze the KWW law in the frequency domain, a Fourier transform is needed. The KWW function in the frequency domain is [63]:

_{K}[70]. However, a possible relationship between α, β and β

_{K}was hinted at by the results in Reference [71,72,73], where the following analytical relations could be derived:

_{k}is

_{K}is the distribution parameter of the KWW equation. For the shape parameters, there is a direct transformation from the HN parameters into the KWW parameter. It is well known that a Fourier transform is needed to analyze the KWW law in the frequency domain. However, there is no analytic expression for the Fourier transform of the KWW function in the frequency domain. Any Fourier transform of the KWW function in the frequency domain can be approximated by a HN function which has a more complex relaxation form, but not vice versa [74].

^{-n}behavior, 0 ≤ n ≤ 1) [75,76]. After a Fourier transform, the complex susceptibility CS relation is:

_{∞}is the high frequency limit of the permittivity, χ

_{CS}= [ε

_{CS}× (ω) − ε

_{∞}]/(ε

_{s}− ε

_{∞}) is the dielectric susceptibility related to the CS law [53]. The value of the exponent (n) indicates the degree of dielectric relaxation [63,77]. The values obtained for the exponent n, showed that a weak dependence of the permittivity on frequency was observed [78]. A n−1 value of zero would indicate that the dielectric permittivity is frequency independent (no dielectric relaxation) [79].

_{∞}is the high frequency limit permittivity, ε

_{s}is the permittivity of free space, σ is the dc conductivity, χ

_{KWW}= [ε

_{KWW}*(ω) − ε

_{∞}]/(ε

_{s}− ε

_{∞}) is the dielectric susceptibility related to the KWW law [53].

_{x}Zr

_{1−x}O

_{2}(x = 0.22, 0.35 and 0.63) dielectrics clearly show a power-law dependence on frequency known as the CS law, $k\propto {f}^{n-1}$, (0 ≤ n ≤ 1) [33,34]. For La

_{x}Zr

_{1−x}O

_{2−δ}thin films with x = 0.63, x = 0.35 and x = 0.22 La content, the dielectric relaxation response could be ﬁtted by the pure CS law, the n values were 0.981, 0.98 and 0.985 when the composition of La, x, were 0.22, 0.35 and 0.63, respectively.

_{k}and n were 0.53 and 0.91, respectively, and the relaxation time τ

_{K}was 3 × 10

^{−3}s, as shown in Figure 17. From Figure 17, the exponent value n decreased within creasing k-values.

_{k}= 0.53 and n = 0.91) could be substituted by the HN function where α and β were 0.6535 and 0.3458, respectively, in Figure 16 because both the HN function and the combined CS+KWW relationship both have two distribution parameters.

#### 3.2.3. Dielectric Relaxation Mechanisms

_{0.09}Zr

_{0.91}O

_{2−δ}thin film and Ce

_{0.1}Hf

_{0.9}O

_{2−δ}thin film, respectively, as shown in Figure 15; (2) the tetragonal or cubic phase was formed, but was accompanied by significant dielectric relaxation. For high levels of doping (such as 35%), (1) no significant enhancement of the k-value was achieved; (2) the tetragonal or cubic phase was formed and no significant dielectric relaxation was observed, as shown in Figure 16 [33].

**Figure 18.**XRD data from La

_{x}Zr

_{1−x}O

_{2−δ}thin films (bottom) and Ce

_{x}Hf

_{1−x}O

_{2−δ}thin films (top), as-deposited and following annealing at 900 °C. As-deposited thin films were amorphous. In the annealed thin films, diffraction peaks were from the tetragonal or cubic phase. Data from the monoclinic phase was labeled, m [33].

_{x}Zr

_{1−x}O

_{2−δ}dielectric are possible: (1) ion movement of unbounded La

^{+}or Zr

^{+}ions in the metal-oxide lattice resulting in dielectric relaxation [83]; (2) the combination of unbound metal ions with electron traps, generating dipole moments and inducing dielectric relaxation [84]; (3) a decrease in crystal grain size, causing an increase in the dielectric relaxation due to increased stresses [85,86]. It has been shown that the effect of the cation segregation caused by annealing and rapped electrons on the dielectric relaxation were negligible [26]. However, it has been reported that a decrease in crystal grain size can cause an increase in the dielectric relaxation in ferroelectric relaxor ceramics and this relaxation effect has been attributed to higher stresses in the smaller grains [85,86]. After annealing, the doping level affected the phase of the thin film crystallization and the size of the crystal grains formed that cause the dielectric relaxation. For a La concentration of x = 0.35 dielectric thin films with the 900 °C N

_{2}-annealed containing ~15 nm crystals did not suffer from severe dielectric relaxation and a similar effect appeared to occur with the 900 °C air annealed, producing ~4 nm diameter equiaxed nanocrystallites within the thin film, which suffered from severe dielectric relaxation [87]. So, the cause of the dielectric relaxation is believed to be related to the size of the crystal grains formed during annealing and doping affects the size of the crystal grains formed.

## 4. Conclusions

_{0.09}Zr

_{0.91}O

_{2−δ}thin film and Ce

_{0.1}Hf

_{0.9}O

_{2−δ}thin film, respectively, at 100 Hz; while no significant enhancement of the k-value was achieved with high levels of doping (such as 35%).

_{x}Zr

_{1−x}O

_{2−δ}, LaAlO

_{3}, ZrO

_{2}and La

_{x}Zr

_{1−x}O

_{2−δ}thin films may be described by either the combined CS+KWW laws or the HN relationship. The fitting results of the HN equation showed that the asymmetry of the dielectric loss peak β increases with decreasing concentration levels of La x. For a severe dielectric relaxation (for example, the significant decrease of the k-value with increasing frequency for the La

_{0.09}Zr

_{0.91}O

_{2−δ}thin film), the width change of the loss peak α played an important role during data fitting. For the La

_{0.09}Zr

_{0.91}O

_{2−δ}thin film, it was found that the combined CS+KWW relaxation process (β

_{k}= 0.53 and n = 0.91) can be substituted by the HN function where distribution parameters α and β were 0.6535 and 0.3458, respectively because both the HN function and the combined CS+KWW relationship had two distribution parameters.

## Acknowledgement

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Tao, J.; Zhao, C.Z.; Zhao, C.; Taechakumput, P.; Werner, M.; Taylor, S.; Chalker, P.R.
Extrinsic and Intrinsic Frequency Dispersion of High-*k* Materials in Capacitance-Voltage Measurements. *Materials* **2012**, *5*, 1005-1032.
https://doi.org/10.3390/ma5061005

**AMA Style**

Tao J, Zhao CZ, Zhao C, Taechakumput P, Werner M, Taylor S, Chalker PR.
Extrinsic and Intrinsic Frequency Dispersion of High-*k* Materials in Capacitance-Voltage Measurements. *Materials*. 2012; 5(6):1005-1032.
https://doi.org/10.3390/ma5061005

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Tao, J., C. Z. Zhao, C. Zhao, P. Taechakumput, M. Werner, S. Taylor, and P. R. Chalker.
2012. "Extrinsic and Intrinsic Frequency Dispersion of High-*k* Materials in Capacitance-Voltage Measurements" *Materials* 5, no. 6: 1005-1032.
https://doi.org/10.3390/ma5061005