MODELING OF CV CHARACTERISTICS OF HIGH-k MIM CAPACITORS

Transcription

1 Chapter 6 MODELING OF CV CHARACTERISTICS OF HIGH-k MIM CAPACITORS 6.1 Introduction and Motivation Capacitance-Voltage characteristics and voltage linearity of MIM capacitor are important performance parameters for design of AMS ICs. Achieving low voltage coefficient of capacitance (VCC) is still a challenge to meet ITRS recommendations. However, the origin of nonlinear behavior of capacitance with voltage is not clear. Some authors attempted to model the voltage nonlinearity using orientation polarization [Phung et al., 2011], electrode polarization [Gonon and Vallae, 2007], electrostriction [Wenger et al., 2008] and ionic polarization [Be cu et al., 2006]. However, most of them are either complex or biased to particular materials. For instance, Phung et al demostrated the modeling of negative VCC for SiO 2 MIM capacitors using orientation polarization of polar dielectrics [Phung et al., 2011]. This model can not be used for nonpolar dielectrics, such as Al 2 O 3 and Ta 2 O 5. Also the dipole moment of O-Si-O bond was calculated using complex techniques. Kim et al reprted that the VCC is a linear function of temperature [Kim et al., 2004]. However, the models mentioned above did not considered the effect of temperature. 82

2 In this chapter, a generalized model of voltage nonlinearity for MIM capacitors is presented using microscopic and macroscopic ionic polarizations. The model was verified with fabricated MIM capacitors with low and high dielectric constant materials such as Al 2 O 3 and TiO 2, respectively, at various temperatures. The model maps the dependable elements of voltage linearity, such as dielectric thickness and dielectric constant, to meet the recommendations of International Technology Roadmap for Semiconductor (ITRS) [ITRS, 2011]. Bilayer and multilayer dielectric MIM capacitors are proposed to solve the issues in single layer cases, such as high VCC and high leakage current density. However, no report was found on modeling of CV characteristics of such capacitors as per our knowledge. In our observations in leakage characteristics of bilayer MIM capacitors, the interfacial traps and migration of charges were predicted. Therefore, the modeling of CV characteristics of bilayer dielectric stack MIM capacitors should account space charge accumulation or interfacial polarization. This phenomena is also referred as Maxwell-Wagner polarization. In this chapter, a CV model for bilayer MIM capacitors is proposed which shows a good fit with experimental results of anodic TiO 2 /Al 2 O 3 MIM capacitors. 6.2 Modeling of C-V characteristics of single layer MIM capacitor Dielectric materials are largely influenced by their polarization properties with applied electric field. In MIM capacitors, the formation of capacitance is due to various polarization mechanisms, namely electronic polarization (P e ), ionic polarization (P i ), orientation polarization (P o ) and space charge polarization (P sc ). The total polarization of dielectric layer can be expressed as P = P e + P i + P o + P sc. Among these, ionic and electronic polarization are almost independent of applied field. Electronic polarization is due to deformation of electrons in molecules which lead to dipole formation. For the applied field, the induced shift of cations (metal) of molecule with respect to neighbor atoms (oxide) of 83

3 Figure 6.1: Dielectric response and polarization mechanisms with frequency dielectric layer causes induced dipole moment. The polarization of these induced dipoles are called ionic polarization. It occurs for ionically bonded materials (Al 2 O 3, HfO 2 and Ta 2 O 3 ), sometimes called as non-polar dielectrics. Dipolar/orientation polarization is due to polarization of inherent/permanent polar molecules with applied dielectric field. For the applied AC field, the accumulation of charges/carriers at the insulating boundaries (interface and interface traps) is called as space charge polarization or interfacial polarization. Electrode polarization is highly sensitive to frequency because of its time dependency. Fig shows the occurrence of various polarization mechanism in the frequency scale In this model, the susceptibility for ionic and electric polarizations are considered as field independent. The orientation of induced dipoles for the applied field is considered as macroscopic case where the field is uniform through out the dielectric layer. However, according to Lorentz approach, the field can t be uniform within the dipole molecule or a cluster of dipole molecules. For this case, the internal field or local field inside dipole is considered which orients the electronics charges of dipole molecules. This electronic orientation of charges are treated as microscopic case which is known as Clausius Mossotti 84

4 Figure 6.2: (a) Al 2 O 3 Molecule at equilibrium (b) Induced dipole due to bond distortion for the applied field model Macroscopic model In dielectric materials, the voltage dependency of dielectric constant is based on the ionic polarization of induced dipoles and orientation polarization of permanent dipole. The polar dielectric materials have permanent dipoles which offer the polarization due to field. The paraelectric materials, such as Al 2 O 3, TiO 2 and Ta 2 O 3, are called non-polar oxides since they do not have permanent dipoles. The metal and oxygen atoms of paraelectric materials are coupled by ionic bond. This metal-oxygen bond is distorted due to applied field which changes the inter-atomic distance between metal and oxygen atoms. This distortion creates induced dipole in such dielectrics [Talebian and Talebian, 2012]. Fig 6. 1 shows the distortion of metal ions (cations) in Al 2 O 3 which results induced dipole for the applied field. In macroscopic model, the polarization of nonpolar dielectrics is modeled using orientation of induced dipole. If angle between the induced dipole µ = α ie E and the applied electric field is θ, then the average dipole moment can be expressed as M = N di α ie cos 2 θe loc [Neumann, 1986]. Here α ie is the internal electronic polarizability which is a microscopic quantity. E loc is the local electric field within the dipole sphere which changes with respect to the position in dielectric layer from electrode. This is expressed as E loc = λe, for the applied external field E and field correction factor λ. In 85

5 Onsager Model, the neighboring molecules are also affecting the polarization and the field correction factor was derived as λ = 3ε r /[2ε r + 1] [Neumann, 1986]. Using Boltzmann statistics, the mean value of cos 2 θ can be given as, cos 2 θ = ( ) 0 exp αie cos 2 θe loc k B T cos 2 θsinθdθ ( ) π 0 exp αie cos 2 θe loc k B T sinθ dθ π (6.2.1) which can be reduced to 2nd order Langevin function with axial symmetry [Gubin, 2009], cos 2 θ = L 2 (β) = 1 2 L(β) (6.2.2) β where, L(β) = eβ + e β e β e β 1 β = cothβ 1 β (6.2.3) L(β) is referred as Langevin function [Gubin, 2009]. Here β = µe loc k B T. Thus, the average dipole moment can be expressed as, M = N di α e L 2 (β)e loc = N di α e [ 1 2 β L(β) ]E loc (6.2.4) The total orientation polarizability of induced dipoles is α oi = N di M E. Therefore, α o = N di α e L 2 (β) = λn di α e [ 1 2 β L(β) ] (6.2.5) where N di = n di Ad is the number of induced dipoles in dielectric layer with uniform applied electric field E, n di is the density of induced dipoles per cubic volume, A and d are area and thickness of dielectric layer, respectively Microscopic model The molecular electronic orientation is influenced by associated internal field within dipole and the external field. In Lorentz approach, the dipoles are modeled as sphere with few molecule/atoms inside where the total internal electric field for isotropic materials is 86

6 expressed as E int = E ext + E pol [Talebian and Talebian, 2012]. Here, E pol is the field due to polarized charge distribution, expressed as E pol = P 3ε 0 with polarization P. The total polarization per dipole is expressed as, p total = ε 0 αe total [Talebian and Talebian, 2012], therefore, Also, ( P ie = ε 0 N di α ie E + P ) ie 3ε 0 P ie = ε 0N di α ie E 1 N di α ie /3 (6.2.6) (6.2.7) where α ie is internal/induced electronic polarizability of charges within the dipole sphere. Since P = ε 0 (ε r 1)E = ε 0 χ ie E, this equation can be written as, α ie = 3ε 0 N di ( ) εr 1 ε r + 2 (6.2.8) This is called Clausius Mossotti equation. This equation relates the macroscopic element ε r with microscopic element α ie. Using this microscopic and macroscopic models of ionic polarization, the total polarization can be expressed as, P = ( ε 0 χ e + N di [ 3ε0 N di ( )] ) εr 1 L 2 (β) E (6.2.9) ε r + 2 The over all permittivity of dielectric layer of thickness d can be expressed in term of applied bias V (= Ed). The second order Langevin function can be approximated to L 2 (β) β 2 15 for β << 1 or λ µe << k BT [Gubin, 2009], ε (V ) = ε 0 + ε 0 χ e + N di [ 3λε0 N di ( )] εr 1 3 E 2 ε r (k B T ) 2 (6.2.10) This equation can be compared with empirical relation, C (V ) = C 0 ( αv 2 + βv + 1 ), where C 0 is the capacitance at zero bias. Therefore, the quadratic (α) coefficient of 87

7 capacitance is, ( ) 2 [ ε0 α 1.8 λ N di k B T ( )] εr ε r + 2 d 2 (6.2.11) Here, N di is the number of induced dipoles and ε r is the static dielectric constant of material. The model shows that α is inversely proportional to square of the thickness which shows a good agreement with the model proposed by Wenger et al and Phung et al [Phung et al., 2011, Wenger et al., 2008]. It also shows a large dependence with ε r, which indicates that higher dielectric constant materials shall lead to large α. Eq. 6 shows that the α is inversely proportional to temperature, but it is observed that the α shows a linear relation with temperature T in many recent works [Wenger et al., 2008, Kim et al., 2004]. This is due to increase in induced dipole moment with temperature which can be introduced as M (T ) = MηT. Here η is the increment factor of linear relation for the temperature T. This yields, ( ) 2 [ ε0 α 1.8 λη N di k B ( )] εr 1 3 T ε r + 2 d 2 (6.2.12) The polarization due to permanent dipoles of paraelectric materials can be modeled in accordance with [Phung et al., 2011]. Therefore, we can introduce the orientation polarization P o = N pd µ d L(β) in Eq as described in [Phung et al., 2011], where N pd is the number of permanent dipoles and µ pd is the dipole moment. It is observed that L(β) converges to unity much faster than L 2 (β) [Gubin, 2009] which indicates that the polar dielectric material show a high α than that of paraelectrics Model verification Quadratic coefficient of capacitance α is extracted from measured C-V characteristics using ( empirical relation C (V ) = C 0 αv 2 + βv + 1 ). For both TiO 2 and Al 2 O 3 MIM capacitors, the modeled and extracted α are plotted as a function of dielectric thickness in Fig Measured values of α of HfO 2 and Y 2 O 3 MIM capacitors for various thicknesses are available in [Wenger et al., 2008], which is also included in Fig to validate the model. Fitting parameters of the model for all materials are presented in Table Al 2 O 3 shows 88

8 α-vcc (ppm/v 2 ) Model(Alumina) Measured (Alumina) Model(Titania) Measured (Titania) Model(Y2O3) Measured (Y2O3) Model(Hafinia) Measured (Hafinia) Thickness (nm) Figure 6.3: Measured and modeled quadratic coefficient of capacitance α of various MIM capacitors [Kannadassan et al., 2012a,b, Wenger et al., 2008]. Material Al 2 O 3 TiO 2 HfO 2 Y 2 O 3 Dielectric constant (ε r ) Induced dipole density (N di ) (10 22 /cm 3 ) Table 6.1: Material parameters for modeling low α and good agreement with measured data. The crystalline state and strong ionic bond of anodic Al 2 O 3 also support in reduction of α. It is observed that for the anodic TiO 2 capacitor, with same thickness of 15nm, the α decreases for higher anodization voltages due to transformation of amorphous to crystalline state which reduces the dependency of field. HfO 2 and Y 2 O 3 MIM capacitors show a good fit with our model. Fig shows the fitting compatibility of model with measured quadratic coefficient of capacitance at various temperatures. It is observed that titania MIM capacitor shows a strong dependence with temperature. This is due to week ionic bond which lead to large distortion of metal atoms. MIM capacitors with alumina shows a low dependence with temperature as thickness increases. It is due to decrease in field as thickness increases which intern reduces the local field. Also the ionic bond is strong and less sensitive to temperature. The model can be used to explore the limitation of thickness and dielectric constant to meet the ITRS requirement. Fig shows the required physical thickness of dielectric layer to meet α = 100ppm/V 2, assuming the average density of dipoles N di = per cm 3. It is observed that required thickness d increases with dielectric constant ε r. However, 89

9 Capacitance density (ff/μm 2 ) Thickness d (nm) α-vcc (ppm/v 2 ) Model(14nm-Alumina) Measured (14nm-Alumina) Model(22nm-Alumina) Measured (22-Alumina) Model(15nm-Titania) Measured (15nm-Titania) η=2.7 η=3.9 η= Temperature ( o C) Figure 6.4: Measured and modeled quadratic coefficient of capacitance α of various Temperature α = 100ppm /V 2 α = 200ppm /V ITRS α = 100ppm α = 200ppm ε r Dielectric Constant ε r Figure 6.5: Required thickness of dielectric to meet the ITRS recommendations for various dielectric constants. it saturates to 100nm after ε r 30. This indicates that the high dielectric constant MIM capacitors require > 100nm thickness of dielectric layer to meet the ITRS recommendation. Fig also shows the required thickness if α = 200ppm/V 2 is acceptable, which shows 20% reduction in thickness. Inset of Fig shows the maximum achievable capacitance density for the extracted thickness to meet α = 100ppm/V 2 and α = 200ppm/V 2. Fig and its inset are highly useful to select the material thickness and electrode area as per the IC design requirement. For vertical or thickness miniaturization, one should go for low dielectric material with higher electrode area to achieve high capacitance. For horizontal or area reduction of ICs, the thicker and higher dielectric constant layer is preferable. However, the technology limitations such as oxidation rate, deposition rate, defect density and ionic bonding of material play a major role in the performance of MIM capacitors. 90

10 6.3 Modeling of C-V characteristics of bilayer MIM capacitors High-k multilayer dielectric stack MIM capacitors were proposed by many authors in recent years [Kaynak et al., 2011, Kim et al., 2004, Wu et al., 2012, Tsui and Cheng, 2010]. It is observed that the voltage coefficient of capacitance (VCC) is higher at low frequencies than high frequencies. This is due to migration and accumulation of charges at the interface of dielectrics. Such accumulated charges are polarized due to the applied field and the mechanism is called as Maxwell-Wagener polarization [Sillars, 1937]. Maxwell Wagner (MW) polarization was observed in many ferroelectric heterostructures [Erbil et al., 1996, Qu et al., 1998, O Neill et al., 2000, Catalan et al., 2000, Shen et al., 2001]. The grains of two different ferroelectric materials form heterostructures which lead to accumulation of charges for applied field. This accumulation of charges enhances the dielectric constant of the entire material stack. Giant dielectric constant ( 1000) from TiO 2 /Al 2 O 3 multilayer was achieved by Wei Li et al [Li et al., 2010]. This multilayer laminate structure was fabricated with ALD deposition of 1A o Al 2 O 3 and 0.3A o TiO 2. The interface accumulation yields a high dielectric constant because of MW polarization. Recently, MW effect has been found in MOS device by Jinesh et al [Jinesh et al., 2009]. It can be noted that the MW charge accumulation is observed in forward bias only, i. e., the charge injection from high conductive semiconductor to low conductive high-k material. The approach of imaginary permittivity (ε ) to in forward bias at low frequencies indicates the presence of MW polarization [Jinesh et al., 2009]. However, these reports were ignored the dependence of applied potential across the dielectric stack. The measured capacitance voltage characteristics of samples AT3 and AT4 at low frequencies ( 10kHz) are shown in Fig The characteristics are asymmetric and frequency dependent. This is due to the accumulation of charges for reverse bias which enhances the capacitance. This indicates the MW polarization of accumulated charges is voltage dependent. It is also observed that the maximum capacitance at reverse bias 91

11 Capacitance density (ff/μm 2 ) Capacitance density (ff/μm 2 ) Measured(1KHz) Measured(1KHz) Measured(5KHz) Measured(5KHz) Measured(10KHz) Measured(10KHz) Applied Voltage (V) Applied Voltage (V) Figure 6.6: Measured Capacitance Voltage at low frequencies for two bilayer samples at room temperature decreases with increment in frequency. This is due to large relaxation time of accumulated interfacial charges. In this chapter, the interfacial charge density and capacitance due to applied AC field are studied in detail. A model for capacitace-voltage characteristics is proposed using MW polarization model of interfacial charges and compared with measured results at various frequencies C-V modeling with Maxwell-Wagner polarization James Clerk Maxwell was the first man who coined the physics of conduction through heterogeneous media [Maxwell, 1873]. He considered a dielectric material consists of sphere of another dielectric material which forms heterogeneous or discontinuity. He mathematically proved that the fields on both the spheres are not equal. He also claimed that when a set of sheets of alternative dielectric materials kept transverse to field, charge accumulation at the interfaces will take place. This theory was extended by Wagner and Sillars later with proper experimental setup [Sillars, 1937]. It was observed that the effective dielectric constant of dielectric stacks was enhanced by accumulation of charges at interfaces of materials which is named as Maxwell-Wagner polarization. In this part of the chapter, we assumed a simple dielectric stack of two materials, TiO 2 and Al 2 O 3. A CV model is proposed using MW polarization theory to examine the interface of anodic bilayer system at low frequencies. 92

12 ε 1, σ 1 ε 2, σ 2 E R 1 C 1 C MW R 2 C 2 (a) (b) Figure 6.7: Schematic of bilayer configuration (a) Layer specification, (b) Equivalent circuit at low frequencies. A bilayer dielectric MIM structure is shown in Fig (a). This two dielectric materials have permittivities of ε r1 and ε r2 with thickness of d 1 and d 2 respectively. Conductivity and relaxation time of these layers are represented as σ n and τ n, respectively. Fig (b) shows the equivalent RC network of bilayer MIM structure. C MW is interfacial capacitance due to accumulation of charges, also called Maxwell-Wagner capacitance. This capacitance is significant at low frequencies. The total capacitance can be expressed as, ( 1 C tot (V B ) = + 1 ) 1 +C MW (V B ) (6.3.1) C 1 C 2 where C MW = A.q 2.N MW [Sze and Ng, 2006] where q is charge of an electron and A is top electrode area of capacitor. According to MW theory of double layer [Maxwell, 1873, Morshuis et al., 2007], the accumulated charge density at the interface as a function of applied potential and time is expressed as, [ ] [ ( (εr1 σ 2 ε r2 σ 1 ) N MW = ε 0.V stack. 1 e σ 1 d 2 + σ 2 d 1 t τ MW )], (6.3.2) where V stack = V B V bi is voltage across the bilayer dielectric stack for the applied bias voltage V B over a time period of t. V bi is built in potential at interface due to accumulated interface charges and native interface traps. τ MW relaxation time of double layer which can be expressed as 93

13 τ MW = (d 1ε 2 + d 2 ε 1 ) (d 1 σ 2 + d 2 σ 1 ) (6.3.3) τ MW is in the order of micro-seconds. If the time t is very greater than τ MW, then the third term vanishes. This yields This equation can be rearranged as, [ ] (εr1 σ 2 ε r2 σ 1 ) N MW = ε 0. V B V bi, (6.3.4) σ 1 d 2 + σ 2 d 1 N MW = ( ) σ 1.σ ε1 2 σ 1 ε 2 σ 2 d 1.d 2 ( σ1 d 1 + σ 2 d 2 ) V B V bi. (6.3.5) We know, τ 1 = ε 1 σ 1 & τ 2 = ε 2 σ 2 and G 1 = σ 1 d 1 & G 2 = σ 2 d 2. By replacing these, ( ) G1 G 2 N MW =.(τ 1 τ 2 ). V B V bi (6.3.6) G 1 + G 2 here G 1G 2 G 1 +G 2 = G, where G is total conductance of bilayer. Therefore, Eq can be reduced to, N MW = G.(τ 1 τ 2 ). V B V bi (6.3.7) The total conductance G is a frequency sensitive term. It is expressed as G = ωc 0 ε, where ε is imaginary part of dielectric permittivity. ε of bilayer dielectric stack was derived by Wagner [O Neill et al., 2000], ε = 1 + ω2 ((τ MW.τ 1 ) + (τ MW.τ 2 ) (τ 1.τ 2 )) ωc 0 (R 1 + R 2 ) ( 1 + ω 2 τmw 2 ) (6.3.8) and, thus Eq. ( ) can be rewritten as N WM = 1 + ω2 ((τ MW.τ 1 ) + (τ MW.τ 2 ) (τ 1.τ 2 )) (R 1 + R 2 ) ( 1 + ω 2 τmw 2 ).(τ 1 τ 2 ). V B V bi (6.3.9) 94

14 Capacitance density (ff/μm 2 ) Capacitance density (ff/μm2) Model(1KHz) Measured(1KHz) Model(5KHz) Measured(5KHz) Model(10KHz) Measured(10KHz) Model(1KHz) Measured(1KHz) Model(5KHz) Measured(5KHz) Model(10KHz) Measured(10KHz) Applied Voltage (V) Applied Voltage (V) Figure 6.8: MW Capacitance Voltage Model (without tunneling probability) and Measured C-V fitting compatibility for samples (a) AT-3, (b) AT-4 Sample σ d 1 (nm) d 2 (nm) ε r1 ε 1 σ 2 φ 1 (ev ) φ 2 (ev ) V bi (V ) name r2 (ps/cm) (ps/cm) AT AT Table 6.2: Fitting parameters of Maxwell-Wagener capacitance model The compatibility of the proposed model with measured capacitance for the applied voltages at various frequencies is shown in Fig (a) and 6. 8 (b) for AT-3 and AT-4 respectively. Table 6. 1 shows the parameters adopted for fitting this model with measured capacitance. The model is not compatible with measured CV characteristics. This is due to linear relation of applied voltage with accumulated charge density. According to Maxwell and Wagner, the dielectric layers are thick and pure insulators which do not conduct. But the fabricated bilayer has nanostructured thin films which has sufficiently large conductivity. One can observed this from Fig This migration of charges for the applied potential can be incorporated to Eq. ( ). Tunneling probability of charges across the bilayer can be introduced using trap assisted tunneling model [Houssa et al., 2000]. Therefore, Eq. ( ) can be rewritten as, N MW (V B ) = q ω2 ((τ MW.τ 1 ) + (τ MW.τ 2 ) (τ 1.τ 2 )) (R 1 + R 2 ) ( 1 + ω 2 τmw 2 ).(τ 1 τ 2 ). (V B V bi ){1 exp[(qv stack φ 1 + φ 2 )/k B T ]} (6.3.10) 95

15 Capacitance density (ff/μm 2 ) Capacitance density (ff/μm 2 ) Model(1KHz) Model(1KHz) Measured(1KHz) Model(5KHz) Measured(1KHz) Model(5KHz) Measured(5KHz) Model(10KHz) Measured(10KHz) Measured(5KHz) Model(10KHz) Measured(10KHz) Applied Voltage (V) Applied Voltage (V) Figure 6.9: MW Capacitance Voltage Model (with tunneling probability) and Measured C-V fitting compatibility for samples (a) AT-3, (b) AT-4 Fig (a) and 6. 9 (b) show the fitting compatibility of model with tunneling probability which shows a better fit with measured data. The capacitance enhancement is not observed in forward bias because the accumulation of charges is not happened. This indicate that the MW polarization occurs only when the field travels from high conductive material to low. Similar effect is observed in MOS devices [Li et al., 2010]. This is due to both the permittivity and bandgap of dielectric materials. 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 material (R di = ε r1 /ε r2 ) determines the MW capacitance. For instance, TiO 2 /Al 2 O 3 has R di = 90/9 = 10 which shows a capacitance enhancement of twice compared to series capacitance of bilayer in our experiments. At the same time, sandwich of TiO 2 and Al 2 O 3 multilayer stack shows a giant dielectric constant of > 500 times of single layer MIM strucutre [Li et al., 2010]. Therefore, the dependence of capacitance with voltage can be reduced by choosing the materials with less R di, such as ZrO 2 /HfO 2 (29/25=1.16) and HfO 2 /Al 2 O 3 (29/9=3.22). 6.4 Summary In this chapter, the modeling of capacitance-voltage characteristics and it s nonlinearity for single and multilayer MIM capacitors are presented in detail. The ionic polarization and 96

16 orientation polarization are statistically incorporated to formulate the dielectric constant of single layer dielectrics. The quadratic coefficient of capacitance, α, is modeled with microscopic and macroscopic ionic polarization of induced dipole. It is observed that the distortion of ionic bond in dielectric molecule for the applied field leads to such dependence of capacitance with applied voltage. This indicates the strong ionic bond between metal-oxygen which should be enhanced during fabrication process with some additional techniques, such as annealing or re-anodization. It predicts that the coefficient α is inversely proportional to thickness of dielectric layer and directly proportional to dielectric constant. These observations have good agreement with experimental data of anodic Al 2 O 3 and TiO 2 MIM capacitors. The model also helps to find the required thickness of high-k dielectric materials in MIM capacitors to meet the ITRS recommendations. C-V characteristics of bilayer MIM capacitors are derived from Maxwell approach on accumulation of charges at dielectric interface. The voltage dependence of permittivity enhancement with Wagner equation on space charge polarization is derived. The model shows good agreement with experimental results. It is observed that the Maxwell-Wagener polarization occurs at low frequencies and largely depends on field direction. The charge built-up due to tunneling and accumulation have added more accuracy compared to the ideal case. 97