Dielectric Polarization and Relaxation of High-k MIM Capacitors

MIM C

2.2-1 Dielectric Polarization

In an MIM capacitor system composed of two parallel metallic plates with an insulator thickness d (in meter, m), the capacitance CMIM (in farad, F) is calculated by using

MIM

Q S

C V d

ε×

= = (2-18)

, where V (in volt) is the applied voltage of the capacitor, Q (coulomb, C) is the amount of storage charges of the capacitor, ε (in F/m) is the permittivity of the insulator, and S (in m²) is the area of the plate. The capacitance is proportional to the permittivity which signifies the charge storage capability of the capacitor, and the large permittivity of the high-k dielectric contributes from the lattice structure and the polarizability under the external bias. The lattice structure of the dielectric determined by the crystal phase has been formed as the dielectric film deposited or grown during the fabrication process. Therefore, the changes in permittivity of high-k MIM capacitor during measurement could be contributed to the changes in polarization behaviors.

In general, the total polarizability of a dielectric could be divided into four kinds of possible compositions, as sketched in Fig. 2-10 and denoted below [96].

(1) The electronic polarizability, pe, is caused by the slight displacement of the negatively charged electron cloud in an atom relative to the positively charged nucleus. Electronic polarizability occurs in all solids and in some, such as diamond, and it is the only contributor to the dielectric constant since ionic, dipolar, and space charge polarizabilities are absent.

(2) The ionic polarizability, pi, arises from a slight relative displacement or a separation of anions and cautions in a solid.

(3) The dipolar polarizability, pd, arises in materials, such as HCl and H2O, which contain permanent electric dipoles or are induced by the defects and polar molecules in the dielectric. These dipoles may change their orientation and they tend to align themselves with an applied electric field. The effect of dipolar polarizability has large temperature dependence since the dipoles may be “frozen” at low temperature.

(4) The space charge polarizability, ps, occurs in unperfected dielectric with a long range charge migration.

Consequently, the total polarization (p) of dielectric is the sum of all contributions and

expresses as follows:

e i d s

p= p +p +p + p (2-19)

2.2-2 Dielectric Relaxation

As described above, the total polarizability in dielectric composed of electronic polarizability, ionic polarizability, dipolar polarizability, and space charge polarizability can contribute to the permittivity. However, as the polarity of external force changes, all the polarization species, like electrons, ions, dipoles, and space charges, need a period of time to displace from the original site or rotate from the earlier orientation to the new equilibrium state, resulting in the other polarization status. This lag response process of the dielectric is the dielectric relaxation, and the period of time when the polarization specie needs to follow the alternating force is the relaxation time. Moreover, four kinds of polarization species have various response times, and Fig. 2-11 reveals the relation between the polarization mechanism and the alternating frequency of external force that the polarization can follow [97]. We can recognize from this diagram that the space charges have the longest relaxation time than any other polarization species, and their polarized phenomenon could be neglected as the alternating frequency of external force is higher than 1 MHz. But, the electrons can rapidly response the changes of external force to result in the shortest relaxation time, and hence such rapidly response of electrons can contribute to the polarization even at very high frequency. In a trap-rich dielectric, such as many high-k materials, the dipolar relaxation and the space charge relaxation are emphasized at our testing frequencies in the range from 10 kHz to 500 kHz. Thus, we give a brief description about these two types relaxation behaviors as following.

Space charges in dielectric of the high-k MIM capacitor are made of free carriers that inject from the biased electrode, and then they become the excess mobile charges moving in the dielectric film [98]. The excess mobile charge in the insulator is expected to follow the

alternating signal with a relaxation time τs depending on the mobility μ, the mobile charge concentration n(E), and the permittivity ε. According to Coelho [99], the complex permittivity ε^* is given by

, where ε’ is the real part of the complex permittivity and ε’’ is the imaginary part of the complex permittivity. And, A is given by

1

, where d is the thickness of the dielectric, ω is angular frequency, and DE is the diffusion coefficient according to Einstein’s law (DE=μkT/e). The relaxation time (τs) is given by

s ( ) en E τ ε

= μ . (2-22)

Here, n(E) is mobile charge concentration dependant on the electric field E. A higher relaxation time means the mobile charges are more difficult to follow the alternating signals.

The frequencies that space charge can follow are low, we would observe the tail of the space charge relaxation at the test frequencies (10 kHz to 500 kHz). If the conduction of the MIM capacitor is given by Schottky effect, the n(E) could be expressed as the following formula

12

, where n0 is uniform concentration of mobile charges without any applied voltage.

Accordingly, the amount of excess mobile charges in the insulator could be affected by the electric field, the temperature, and the barrier height at the metal/dielectric interface.

The dipole relaxation effect can be observed even at infrared frequency while the free carrier effects become negligible. The dipolar relaxation time (τd) shows a wide distribution range of 10^-10 sec to 10³ sec, which results in a broad frequency range where the dipole can contribute to the dielectric relaxation. It has been reported that the defect dipole relaxation

dominates the lower frequency part or the longer relaxation time region of the total dipolar relaxation behaviors [100]. Besides, as the temperature increases, the τd decreases to manifest a faster response for alternating signals.