3.2 Electrical Permittivity

The electrical permittivity tensor $\tilde{\varepsilon}$ describes how the electric field ${\mathbf{{E}}}$ is related to the electric flux density ${\mathbf{{D}}}$ ,

$${\mathbf{{D}}} {=}\tilde{\varepsilon}\cdot{\mathbf{{E}}}.$$ (3.10)

For isotropic materials, the permittivity tensor reduces to a scalar-valued quantity and the direction of the electric flux density and the electric field is the same. Hence, for isotropic materials the electrical permittivity can be approximated

$$\varepsilon {=}{\varepsilon}_0 {{\varepsilon_{\mathrm{r}}}} \big(1 + \alpha_{\varepsilon} (T - T_0)\big),$$ (3.11)

where the temperature coefficient $\alpha_{\varepsilon}$ is determined by

$$\alpha_{\varepsilon} {=}{\frac{1}{{{\varepsilon_{\mathrm{r}}}}}} {\frac{\mathrm{d}{{{\varepsilon_{\mathrm{r}}}}}}{\mathrm{d}{T}}}$$ (3.12)

and can be derived from the CLAUSIUS^3.1-MOSOTTI^3.2equation [140]

$$\frac{N {\alpha^{\mathrm{pol}}}}{3 {{\varepsilon_0}}} {=}\frac{{{\varepsilon_{\mathrm{r}}}}- 1}{{{\varepsilon_{\mathrm{r}}}}+ 2},$$ (3.13)

where $N$ is the dipole density, ${\alpha^{\mathrm{pol}}}$ the polarizability, and ${\varepsilon_{\mathrm{r}}}$ the relative permittivity. This equation describes the relative permittivity as an implicit function of the polarizability and the dipole density. The latter is an inherent property of the material and does normally not change as long as the phase stage is not altered. Hence, the derivative with respect to the temperature $T$ is

$${\frac{\mathrm{d}{{{\varepsilon_{\mathrm{r}}}}}}{\mathrm{d}{T}}} ... ...hrm{pol}}}}}{\frac{\mathrm{d}{{\alpha^{\mathrm{pol}}}}}{\mathrm{d}{T}}}\right).$$ (3.14)

Due to mass conservation, the thermal volume expansion can be assumed to be equal to the negative temperature coefficient of the dipole density because the number of atom inside a atomic unit cell remains constant. The only assumption here is that the material persists in the phase stage. Hence, $\alpha_{\varepsilon}$ is

$$\alpha_{\varepsilon} {=}\frac{({{\varepsilon_{\mathrm{r}}}}-1)({{... ...d}{{\alpha^{\mathrm{pol}}}}}{\mathrm{d}{T}}} - {\alpha^{\mathrm{mech}}}\right).$$ (3.15)

According to the magnitude of the relative dielectric constant, materials can be divided into two groups: high-$\kappa$ materials, which have a larger ${\varepsilon_{\mathrm{r}}}$ than $\mathrm{SiO_2}$ , and low-$\kappa$ materials, which have a lower ${\varepsilon_{\mathrm{r}}}$ number. Materials with higher relative permittivities than $\mathrm{SiO_2}$ have a higher dielectric displacement field (electric flux density) as $\mathrm{SiO_2}$ if the same electric field is applied. Hence, the capacitive coupling between the two opposite sides of the materials is much tighter as with $\mathrm{SiO_2}$ . Therefore, such materials are used to increase the capacitance where the thickness of the materials can be reduced to obtain the same capacitance as with $\mathrm{SiO_2}$ .

However, capacitive coupling is often considered as a parasitic effect, for instance if an array of interconnect lines is considered, where each of them is used to transmit a different signal. Capacitive coupling means in this context that an electric signal in one of these lines influences an electric signal in the other lines. This is refered to as cross talking. If bus structures for data or addresses are considered too much cross talk might result in logic device failures. Hence, for these devices a reduction of the coupling is required. Materials with lower ${\varepsilon_{\mathrm{r}}}$ values ( low-$\kappa$ materials) offer an alternative to reduce the capacitive coupling.

Table: Typical relative dielectric constants for various high-$\kappa$ materials.

Material	${{\varepsilon_{\mathrm{r}}}}$	${{\mathcal{E}_{\mathrm{G}}}}$	References
	[1]	eV
$\mathrm{SiO_2}$	$3.9$	$9.0$	[182,183,184]
	$3.9$	$8.0 - 9.0$	[185,186,25]
	$3.9 - 4.6\:$(nitridation)	$-$	[127,187,188]
$\mathrm{SiO_2}$ (TEOS)	$4.1$	$-$	[21]
${\mathrm{Si_3N_4}}$	$7.5$	$5.0$	[184,189]
	$7.0 - 7.9$	$5.0 - 5.3$	[79,185,182,183]
	$4.0\:$(Si-rich)$- 8.0\:$(N-rich)$$	$-$	[25,79]
SiCN	$5.0$	$-$	[20]
${\mathrm{TiO_2}}$	$40.0$	$3.5$	[189,184]
	$39.0 - 170.0$	$3.0 - 3.5$	[190,182,186]
${\mathrm{ZrO_2}}$	$23.0$	$5.8$	[183]
	$12.0 - 25.0$	$5.0 - 7.8$	[182,184,190,185]
${\mathrm{HfO_2}}$	$25.0$	$5.7$	[184,185]
	$16.0 - 40.0$	$4.5 - 6.0$	[182,183,190]
${\mathrm{Al_2O_3}}$	$9.0$	$8.7$	[185]
	$5.0 - 12.0$	$8.7 - 9.0$	[190,182,25]
${\mathrm{Y_2O_3}}$	$15.0$	$5.6$	[185,183]
	$4.4 - 18.0$	$5.5 - 6.0$	[186,190]
${\mathrm{Ta_2O_5}}$	$25.0$	$4.4$	[189,184]
	$23.0 - 26.0$	$4.4 - 4.5$	[190,185,182,183]
${\mathrm{ZrSiO_4}}$	$12.6$	$6.0$	[190]
	$3.8 - 12.6$	$4.5 - 6.0$	[186,183,182]

A major drawback of all high-$\kappa$ and low-$\kappa$ materials is their mechanical weakness and the complicated and expensive fabrication. The high ${\varepsilon_{\mathrm{r}}}$ value of high-$\kappa$ materials vanishes at a certain temperature, where a phase change takes place, reducing ${\varepsilon_{\mathrm{r}}}$ to 1. This temperature is called CURIE temperature in analogy to thermo-magnetic effects. Hence, the thermal budget for these materials is very limited. The low-$\kappa$ materials are either porous and very hard or soft, where both properties are extrema for the fabrication. For instance, if a soft interlayer dielectric material is processed by a chemical mechanical polishing (CMP) process step in Cu technology, the abrasion of the new materials is considerable higher than that of Cu and $\mathrm{SiO_2}$ . The mechanical weakness of this soft material requires additional protective layers for a normalized CMP procedure. Because many of the low-$\kappa$ materials are compounds of severals material, they also limit the thermal budget for the fabrication processes. Otherwise the amorphous and porous materials re-anneal and the advantageous properties vanish.

Table: Typical relative dielectric constants for various low-$\kappa$ materials.

Material	${\varepsilon_{\mathrm{r}}}$	References
	[1]
$\mathrm{SiO_2}$	$3.9$	[182,183,184,185,186,25]
${\mathrm{SiOC}}$	$2.4 - 3.4$	[150,148,20,149]
porous ${\mathrm{SiOC}}$	$2.5$	[150]
SiOCH	$2.7$	[152]
organo-silcate glasses (OSGs)	$2.3 - 3.1$	[21]
aromatic polymers	$2.7$	[21]
with air gaps	$1.6 - 2.5$	[154,155,156]