1. Introduction

Metal–insulator–metal (MIM) capacitors are attractive passive elements for analog and mixed signal applications. Some sensitive circuits, such as A/D converter ICs, expect a very low variation of capacitance with voltage of less than 100 ppm/V². In this regard, high-k dielectric stack MIM capacitors have been proposed by many authors recently^[1–4]. It is observed that the voltage coefficient of capacitance (VCC) is higher at low frequencies than at high frequencies. This is due to migration and accumulation of charges at the interface of dielectrics. Such accumulated charges polarize for the applied field; such mechanism is referred as Maxwell–Wagner (MW) polarization^[5].

Maxwell–Wagner polarization is observed in many ferroelectric mixtures or heterostructures^[6–10]. The accumulation of charges at the interface of heterostructures for the applied field lead to enhancement of effective permittivity^[8]. A giant dielectric constant of ~1000 was achieved with ALD TiO₂/ Al₂O₃ nanolaminates by Wei Li et al.^[11]. Because of MW polarization, each TiO₂/Al₂O₃ interface accumulation yields a high dielectric constant of 750 times greater than that of Al₂O₃. Recently, MW effect was found in an MOS device by Jinesh et al.^[12]. It is worth noting that the MW charge accumulation is observed in forward bias only, i. e., the charge injection from high conductive to low conductive high-k material^[13]. The approach of imaginary permittivity ε" to ∞ in forward bias at low frequencies indicates the presence of MW polarization^[12]. However, models developed in these reports have ignored the dependence of applied potential across the dielectric stack.

Many reports are available on the modeling of dependence of capacitance with voltage for single-layer dielectric MIM capacitors^[13]. However, modeling of capacitance–voltage characteristics of bilayer or multilayer MIM capacitors are not reported yet as per our knowledge. In this paper, we have developed a model for voltage dependence of bilayer dielectrics MIM capacitor using MW polarization. This model accounts for the carrier tunneling probability of dielectric stack and MW relaxation time of accumulated interfacial charges. A good agreement was found between the model and the measured capacitance–voltage characteristics of TiO₂/Al₂O₃ MIM capacitors. It is observed that the MW polarization is significant in low frequencies of <10 kHz.

2. Experimental procedure

A 100 nm SiO₂ was grown by dry oxidation at a temperature of 1100 °C for about 30 min on Si substrate and thoroughly cleaned by deionized water. Over that, a bilayer of 15 nm Ti on 100 nm Al was deposited using an electron beam evaporator with tungsten filament at a pressure of 8 × 10⁻⁵ mBar. This Ti/Al film was anodized potentiostatically using non-aqueous solution of ammonium pentaborate dissolved in ethylene glycol (20 g/L) by the same size of platinum cathode. Oxidation was done for various anodization voltages of 25 V (Sample 1) and 30 V (Sample 2) till the anodization current density reduces to 1 μA/cm². Only three-quarters of the sample area were dipped in the electrolyte to avoid etching for the bottom electrode. This forms a barrier type anodic bilayer TiO₂/Al₂O₃ at lower and higher anodization voltages, respectively. After cleaning thoroughly by deionized water, a 50 nm thick Al top electrode was deposited on the samples using thermal evaporation with the shadow mask area of ~0.61 mm². Samples 1 and 2 are bilayer TiO₂/Al₂O₃ MIM capacitors. SEM cross-sections of the samples 1 and 2 are shown in Fig. 1. Due to the delamination of TiO₂ from bottom Al₂O₃ at higher anodization voltages, the anodization voltage was restricted to 30 V (> 30 V, not shown).

Fig. 2 shows the depth profile of the two samples using secondary ion mass spectrometry (SIMS) in positive mode with 1 kVCs. 30 kV gallium ion was used as the primary ion during the SIMS measurement. It shows ion distribution of Ti, Al, O, Si, Ti–O and Al–O. It is observed that the inward migration of oxygen ion increases, which forms a thin layer of Al₂O₃. It is also observed that outward migration of Al into TiO₂ is increased, which increases the thickness of AlTiO composite layer. Fig. 3 shows the X-ray diffraction patterns of the anodized samples. It is observed that the crystalline phases of TiO₂ anatase and rutile are present at all anodization voltages. Also, the crystalline Al₂O₃ (γ-Al₂O₃) emerges at 2θ = 65.5° for both the samples. The migration of oxygen and evolution of electrons into Al region increases and forms bilayer TiO₂/Al₂O₃ with a thin layer of crystalline Al₂O₃ near TiO₂/Al₂O₃ interface^[13]. The capacitance and leakage current density were measured using semiconductor parameter analyzer (HP4155C). The measured leakage current density as a function of applied voltage for all the samples is reported in our earlier work^[13].

It has been observed that most of the samples are showing a high degree of asymmetry at forward and reverse biases and leakage current of Sample 1 and Sample 2 drastically reduces. These are due to formation of AlTiO interfacial layer and TiO₂/Al₂O₃ stack. A detailed discussion of leakage characteristics and conduction mechanism of this bilayer MIM capacitors are reported in Ref. [13].

3. Results and discussion

Consider a bilayer MIM structure as shown in Fig. 4(a). The layers consist of two dielectric materials with distinct relative dielectric constant of ε_r1 and ε_r2 with thickness of d₁ and d₂ respectively. Each layer's conductivity and relaxation time are represented as σ_n and τ_n, respectively, for n = 1, 2. Fig. 4(b) shows the equivalent RC network of bilayer MIM structure. Here R1 & C₁ and R₂ & C₂ are individual resistance & capacitance of layer-1 and layer-2 respectively. C_MW is interfacial capacitance, also called “Maxwell–Wagner capacitance”, which is significant at low frequencies. To calculate the charge density of dielectric interface, Maxwell’s time varying accumulation process at pure insulator interface is considered. This charge density is used to calculate the interfacial capacitance, named C_MW. The values of R₁ and R₂ are ignored to calculate the total capacitance value due to their high values. The total capacitance can be expressed as,

${C_{\rm total}}\left( {{V_{\rm B}}} \right) = {\left( {\frac{1}{{{C_1}}} + \frac{1}{{{C_2}}}} \right)^{ - 1}} + {C_{\rm MW}}\left( {{V_{\rm B}}} \right),$

(1)

where MW capacitance can be calculated by C_MW = A q²N_MW where q is charge of an electron and A is top electrode area of capacitor^[14]. According to MW theory of double layer^{[14, 15]}, the accumulated charge density at the interface as a function of applied potential and time is expressed as^[14],

${N_{\rm MW}} = {\varepsilon _0} {\frac{{{\varepsilon _1}{\sigma _2} - {\varepsilon _2}{\sigma _1}}}{{{d_1}{\sigma _2} + {d_2}{\sigma _1}}}} {V_{\rm stack}}\left( {1 - {{\rm e}^{ { - t/{\tau _{\rm MW}}} }}} \right).$

(2)

Here V_stack is voltage across the bilayer dielectric stack for the applied bias voltage V_B; t is time of measurement and τ_MW is relaxation time of double layer. The accumulated interface charges and native traps build up potential, which opposes the applied field and reduces flow of charges, so the potential across the stack can be written as ${V_{\rm stack}} = \left| {{V_{\rm B}} - {V_{\rm bi}}} \right|$ . Here V_bi is built in potential at interface due to accumulated interface charges. This MW time constant (τ_MW) of double layer can be expressed as,

${\tau _{\rm MW}} = \frac{{{{\varepsilon _2}{d_1} + {\varepsilon _1}{d_2}}}}{{ {{d_1}{\sigma _2} + {d_2}{\sigma _1}} }}.$

(3)

If the measurement time t is greater than τ_MW, then the third term of Eq. (2) can be eliminated. Therefore, Eq. (2) can be rewritten as,

${N_{\rm MW}} = {\varepsilon _o}{\frac{{ {{\varepsilon _{r1}}{\sigma _2} + {\varepsilon _{r2}}{\sigma _1}}}}{{{d_2}{\sigma _1} + {d_1}{\sigma _2}}}} \left| {{V_{\rm B}} - {V_{\rm bi}}} \right|.$

(4)

We know ${\tau _1} = \frac{{{\varepsilon _1}}}{{{\sigma _1}}}$ , ${\tau _2} = \frac{{{\varepsilon _2}}}{{{\sigma _2}}}$ and ${G_1} = \frac{{{\sigma _1}}}{{{d_1}}}$ , ${G_2} = \frac{{{\sigma _2}}}{{{d_2}}}$ . By substituting these in Eq. (4),

${N_{\rm MW}} = \frac{{{G_1}{G_2}}}{{{G_1} + {G_2}}}\left[ {{\tau _1} - {\tau _2}} \right]\left| {{V_{\rm B}} - {V_{\rm bi}}} \right|.$

(5)

G₁, G₂, τ₁, τ₂ are defined as the conductance and relaxation time of layer 1 and layer 2. Here $\frac{{{G_1}{G_2}}}{{{G_1} + {G_2}}} = G$ , where G is the total conductance of bilayer. Therefore, Eq. (5) can be reduced to,

${N_{\rm MW}} = G\left[ {{\tau _1} - {\tau _2}} \right]\left| {{V_{\rm B}} - {V_{\rm bi}}} \right| .$

(6)

The total conductance G is a frequency sensitive term. It is expressed as ${{G = }}\,\, \omega {C_0}\varepsilon ''$ , where C₀ is the capacitance at zero bias, ε" is imaginary part of dielectric permittivity. ε" of bilayer dielectric stack can be expressed as^[5],

$\varepsilon '' = \frac{{1 + {\omega ^2}\left[ {{{\tau _{\rm MW}}{\tau _1}} + {{\tau _{\rm MW}}{\tau _2}} - {{\tau _1}{\tau _2}}} \right]}}{{\omega {C_0} ({{R_1} + {R_2}}) \left[ {1 + {\omega ^2}\tau _{\rm MW}^2} \right]}},$

(7)

thus Eq. (5) becomes,

${N_{\rm MW}} = \frac{{1 + {\omega ^2}\left[ { {{\tau _{MW}}{\tau _1}} + {{\tau _{\rm MW}}{\tau _2}} - {{\tau _1}{\tau _2}}} \right]}}{{{{R_1} + {R_2}}\left[ {1 + {\omega ^2}\tau _{\rm MW}^2} \right]}}\left[ {{\tau _1} - {\tau _2}} \right]\left| {{V_{\rm B}} - {V_{\rm bi}}} \right|.$

(8)

The compatibility of proposed model with measured capacitance for the applied voltages at various frequencies is shown in Figs. 5(a) and 5(b) for samples 1 and 2 respectively.

Table 1 shows the parameters adopted for fitting this model with measured capacitance. Since the Eq. (8) has linear relationship of applied voltage with accumulated charge density, it has not got a good fit with measured C–V characteristics. This is due to the ignorance of charge migration between dielectric materials. According to Maxwell and Wagner, the dielectric layers are thick and pure insulators that do not conduct. But the fabricated bilayer nanostructured thin films have sufficiently large conductivity. The migration of charges for the applied potential can be incorporated into Eq. (8). To incorporate this realistic situation, tunneling probability has been added using trap-assisted tunneling model^[17]. Therefore Eq. (8) can be rewritten as,

$\begin{array}{*{20}{l}}{{N_{{\rm{MW}}}}\left( {{V_{\rm{b}}}} \right) = {q^2}\displaystyle \frac{{ {1 + {\omega ^2}{\tau _{{\rm{MW}}}}{\tau _1} + {\tau _{{\rm{MW}}}}{\tau _2} - ({\tau _1} - {\tau _2})} }}{{\left( {{R_1} + {R_2}} \right)\left( {1 + {\omega ^2}\tau _{MW}^2} \right)}} \left( {{\tau _1} - {\tau _2}} \right)}\\[18pt]\text{×}\,\, { \left| {\left( {{V_{\rm{b}}} - {V_{{\rm{bi}}}}} \right)\left\{ {1 - \exp \left[ {\left( {q{V_{{\rm{stack}}}} - {\phi _1} + {\phi _2} + {\phi _t} - {\phi _{ii}}} \right)/{k_{\rm{B}}}T} \right]} \right\}} \right|},\end{array}$

(9)

where ϕ₁, ϕ₂, ϕ_ii, ϕ_t, k_B, and T are the barrier height at Al/TiO₂ interface, barrier height at Al₂O₃/TiO₂ interface, trap barrier height of insulator/insulator interface traps, barrier height of traps, Boltzmann constant and temperature respectively. Compatibility of the model is shown in Figs. 6(a) and 6(b) along with model with tunneling probability. The proposed model with tunneling probability is showing a better fit with measured data. MW effect of permittivity enhancement in bilayer MIM capacitors leads to increase in VCC at low frequencies. It is observed that the ratio of permittivity of both dielectric materials determines the MW capacitance. For instance, TiO₂/Al₂O₃ shows a capacitance enhancement of twice compared to series capacitance of bilayer in our experiments. At the same time, sandwich of TiO₂ and Al₂O₃ multilayer stack shows a giant dielectric constant of > 500 times of single layer MIM structure ^[11]. Therefore, the dependence of capacitance with voltage can be reduced by choosing the materials with less R_di, such as ZrO₂/HfO₂ (29/25 = 1.6) and HfO₂/Al₂O₃ (29/9 = 3.22).

4. Conclusion

Capacitance–voltage characteristics of bilayer MIM capacitors are deduced from Maxwell approach on accumulation of charges at dielectric interface. With Wagner equation on space charge polarization, the voltage dependence of dielectric enhancement is derived. The model shows good agreement with experiment. It was observed that the Maxwell– Wagner polarization occurs at low frequencies and largely depends on field direction. The charge built-up due to tunneling and accumulation has added more accuracy compared to the ideal case. Physics and modeling of capacitance–voltage characteristics of MIM capacitors are useful to analyze the origin of nonlinearities, formation of capacitance, and frequency dependence, and dielectric relaxation in MIM capacitors.

Acknowledgment

The authors would like to acknowledge the Science and Engineering Research Board, India for financial support.