De-embedding Theory

Chapter 3 Experimental Procedure of High-κ MIM Capacitor

3.3 De-embedding Theory

When circuits or devices work at high frequencies, many parasitic effects will happen. For example, a signal applied on one metal line, the potential of this metal line at any point is equal if the wavelength of signal is long enough, compared with metal line. But the potential of the metal line at any potential will be different when the wavelength of signal can compare with the metal line or shorter, i.e. high frequency signal. Hence, a metal line was regarded as a resistor at low frequency or resistor plus parasitic inductance and capacitance parameters at high frequency in an equivalent circuit model. In order to measure this MIM capacitance, we must layout additional probe pads and signal lines for measurement. However, these added potions will generate additional parasitic effects. So we must de-embed these parasitic

parameters to get the intrinsic high frequency capacitance. Fig. 3-12 shows the equivalent circuit of a RF MIM capacitor device at high frequency.

As devices were measured approach microwave frequency, we can not directly measure the lump circuit components, like RLC (resistance, inductance and capacitance), because of parasitic effects. Scattering-parameters (S-parameters) were obtained in general [3.9]. According to microwave theorem [3.9], we can transform the S-parameters into an equivalent circuit model to extract the component that we want.

Among added portions for measurement, the parasitic capacitance effects dominate in probe pads. We can transform both of the measured S-parameters of MIM capacitors and “OPEN” dummy device into admittance parameters (Y-parameters). Then we de-embed the parasitic capacitance effects form the MIM capacitor by YMIM-YOPEN. So we can get the de-embedded S-parameters from the transformation of de-embedded Y-parameters. We have de-embedded the parasitic shunt capacitance effects due to the probe pads of MIM capacitor. Then, we use an equivalent circuit method to simulate the de-embedded S-parameter shown in Fig. 3-9 to extract each components of the equivalent circuit model shown in Fig. 3-11 by Series Ⅳ. From this, we can obtain the RF capacitance of MIM capacitor.

Fig. 3-1. Typical layout suitable for coplanar probing with

ground-signal-ground (GSG) probe configuration

Fig. 3-2. Constant pitch, minimum pad size and minimum parallel-row

pad.

Fig. 3-3. Mask #1 (bottom electrode) of layout and process flow of the

RF MIM capacitor.

Fig. 3-4. Mask #2 (Active Region) of layout and process flow of the RF

MIM capacitor.

Fig. 3-5. Mask #3 (Via hole) of layout and process flow of the RF MIM

capacitor.

Fig. 3-6. Mask #4 (upper electrode) of layout and process flow of the RF

MIM capacitor.

Fig. 3-7. The RF MIM capacitor cross section view.

Fig. 3-8. The RF MIM capacitor layout view.

EOT = 2.0nm 2.4nm Measured

simulated

S11

S21 1.0

-2.0 -1.0

-0.5 0.5

1.0 2.0

Fig. 3-9. Measured and simulated scattering parameters of 2.0 and 2.4 nm

EOT Al doped TaO

MIM capacitors.

Fig. 3-10. The “OPEN” dummy structure for the device under test (DUT)

modeled in the shunt configuration.

R _P C

R _S L _S2 L _S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

R _P C

R _S L _S2 L _S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

Fig. 3-11. The equivalent circuit model and numerical values of elements

for capacitor simulation at RF regime.

Fig. 3-12. The equivalent circuit model of MIM capacitor at RF regime.

Chapter 4 The Characteristics of High-κ MIM Capacitor with Al doped TaO

Dielectrics

4.1 Capacitor Characteristics

For precision analog circuit applications, the MIM capacitors need to be useful over wide range of frequencies. To further study the RF frequency characteristics, we have measured the S-parameters up to 20 GHz. Fig. 4-1 shows the measured (de-embedded) and modeled S-parameters for Al doped TaOx MIM capacitors, where the modeled data is from the equivalent circuit model shown in Fig. 4-2. The RS, LS, RP, and C in the model are the parasitic series resistor, series inductor, parallel resistor, and capacitor, respectively. Good agreement between measured and modeled data is obtained that suggesting the good accuracy of physically based equivalent circuit model for both capacitors with different thickness, which can be used for capacitance extraction.

The capacitance values were measured directly using the LCR meter from 10 KHz to 1 MHz and calculated from the measured (de-embedded) S-parameters up to 20 GHz using the equation [4.1]:

We can use the equation (1) to derive the capacitance densities at different frequencies. Z(C) in equation (2) is the total impedance in the equivalent circuit model of Fig. 4-2 and Z0 is the characteristic impedance of transmission line. The RF frequency ∆C/C in equation (1) is obtained by differentiating the measured S21 in equation (3), where shows the relation between the S21 and total impedance Z(C). Fig.

4-3 shows that the derived ∆C/C decreases rapidly with increasing frequency, which is advantageous for high frequency analog/RF circuits.

Figs. 4-4 and 4-5 show the C-V characteristics and ∆C/C for Al doped TaOx

MIM capacitors with physical thickness of 11.5 and 14.0 nm, respectively.

Capacitance densities at RF regime have been extracted in pervious work using equivalent modeling circuit fitting from measured S-parameters [4.2]-[ 4.4] and are also plotted in the figure. At 100 kHz frequency, high capacitor density of 17 and 15

fF/µm² are measured for 11.5 and 14 nm respectively, and a κ value of 22 is obtained

for Al doped TaOx dielectrics. The equivalent-oxide thickness (EOT, teq=κox*thigh-κ/κhigh-κ) values for these high-density capacitors are calculated to be 2.0 and 2.4 nm for 11.5 and 14.0 nm physical dielectric, respectively. Fig. 4-6 shows the J-V characteristics of Al doped TaOx MIM capacitors. The asymmetrical J-V and

breakdown voltages under positive and negative bias are due to the different work function of top Al and bottom Pt electrodes. The leakage current is increased by trading off the increasing capacitance density, and values of 4.5×10^-7 and 8.9×10^-7 A/cm² are measured at -2 V for respective EOT of 2.4 and 2.0 nm. The leakage current, 8.9×10^-7 A/cm², is low enough for circuit applications. This arises because the high capacitor density requires using only a small area. This is shown by the small leakage current of 5.2×10^-12 A of a large 10 pF capacitor, which is even smaller than the leakage current of a 0.13-µm MOSFET [4.5]. Note that ∆C/C in Fig. 4-3

decreases with increasing frequency. This is an advantage at the high operational frequencies of analog/RF circuits.

Fig. 4-7 shows the frequency dependence of both the capacitance and Q-factor for high-κ Al doped TaOx MIMcapacitors. Below 1 MHz, the Q-factor was derived from the measured loss tangent using 1/tanδ. At higher frequencies an equivalent

circuit model was used to determine the C value and the Q-factor from the measured S-parameters.

Only a small capacitance reduction (5%) occurs from 10 KHz to 10 GHz, which is attractive for RF applications. The Q-factor increase up to ~ 8 GHz is due to the resonance of the MIM capacitor with a parasitic inductor. The relatively low resonant frequency of ~8 GHz arises from the very large capacitance (42.5 pF) and the residual

inductance, even after de-embedding. A Q-factor of ~ 40 before resonance for the 17 fF/µm² capacitors is a desirable characteristic. The high capacitance density, low

leakage current, small frequency dependence and good Q-factor for these Al doped TaOx MIM capacitors are useful and important for analog and RF circuit applications.

4.2 The Normalized Capacitance Variation ∆C/C and Voltage Coefficient of Capacitance VCC

To further study the frequency dependence of ∆C/C and related quadratic VCC (α), we have plotted ∆C/C and α as a function of frequency in Fig. 4-8. The relation between α and ∆C/C is expressed in the following equation:

The β is the linear VCC, which is less important than α by using circuit cancellation method [4.6]. Again, the ∆C/C and α decrease monotonically with increasing frequency. Small ∆C/C ≤ 120 ppm and α ≤ 280 ppm/V² are obtained with increasing frequency into GHz regime indicating that the high-κ MIM capacitors can be used for

precision circuits with GHz operation frequency. Such high frequency is required since the operation speed of MOSFET continuously increases with device scaling down and the commercially available CPU circuit is already several GHz.

We have also plotted the ∆C/C and α as a function of 1/C (or equivalent to td since

1/C=td/ε0κ) in Fig. 4-9. Both ∆C/C and α decrease with increasing frequency similar to the trend shown in Fig. 4-8. In addition, the ∆C/C and α decrease with increasing

ln(1/C) or ln(td) regardless the using different high-κ dielectrics [4.7]. Although the detailed physics to explain such dependence is still under investigation, possible reason may be related to the leakage current through dielectric and/or carrier trapping or de-trapping inside the high-κ dielectric.

4.3 The Temperature Coefficient of Capacitor TCC

In additional to the requirement of small ∆C/C and α dependence on voltage, the

small TCC is another important factor for precision analog circuit application. Figs.

4-10 and 4-11 show the respective ∆C/C versus temperature and TCC of the Al doped

TaOx capacitors at different frequencies with two different dielectric thicknesses of 3.3 and 4.8nm EOT. The temperature dependent ∆C/C decreases with increasing frequency that is consistent with the decreasing trend in Fig. 4-8. The ∆C/C in both high-κ capacitors increases with increasing temperature that shows the same trend

with other dielectric capacitors published in the literature [4.8].

We have further plotted the TCC as a function of 1/C (or equivalent to td) in Fig.

4-12 from the measured temperature dependence on ∆C/C. The TCC is also higher at

thinner thickness and decreases monotonically with increasing frequency from 10 kHz

to 1 MHz. It is noticed that the TCC in both high-κ dielectrics decreases rapidly with

increasing ln(1/C), and the extrapolated data is close to previous TCC data of HfO2

[4.7] at the same 1/C. Such exponential dependence of TCC on 1/C or td is similar to the dependence of ∆C/C and α on ln(1/C) in Fig. 4-9. Although the physical meanings

are currently under study, possible reason may still be related to the leakage current through dielectric and/or carrier trapping or de-trapping inside the high-κ dielectric

due to their temperature dependence in physics.

49 EOT = 2.0nm 2.4nm Measured

simulated

S11

S21 1.0

-2.0 -1.0

-0.5 0.5

1.0 2.0

Fig. 4-1. Measured and simulated scattering parameters of 2.0 and 2.4 nm

EOT Al doped TaO

MIM capacitors at RF regime.

R _P C

R _S L _S2 L _S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

R _P C

R _S L _S2 L _S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

Fig. 4-2. The equivalent circuit model and numerical values of elements

for capacitor simulation at RF regime.

10

⁴

10

⁵

10

⁶

10

⁷

10

⁸

10

⁹

10

¹⁰

10

¹¹

10

⁴

10

⁵

∆C/C (ppm)

Frequency (Hz)

EOT 2.0nm EOT 2.4nm

Fig. 4-3. The ∆C/C of Al doped TaO

MIM capacitors as a function of

frequency. It is notice that the ∆C/C decrease with increasing

frequency.

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 0

5 10 15 20

Voltage (V) Capacitance (fF/µm2 )

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

∆C/C (ppm)

10 KHz 100 KHz 1 MHz 1 GHz 10 GHz

Fig. 4-4 C-V and ∆C/C-V characteristics of MIM capacitors with 2.0

EOT Al doped TaO

dielectrics at different frequencies from 10

KHz to 10 GHz. The voltage is applied to the bottom Pt

electrode. Measured area is 50mm × 50mm.

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 0

4 8 12 16

Voltage (V) Capacitance (fF/µm2 )

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

10 KHz 100 KHz 1 MHz 1 GHz 10 GHz

∆C/C (ppm)

Fig. 4-5. C-V and ∆C/C-V characteristics of MIM capacitors 2.4 nm EOT Al doped TaO

dielectrics at different frequencies from 10 KHz to 10 GHz. The voltage is applied to the bottom Pt electrode.

Measured area is 50mm × 50mm.

-8 -6 -4 -2 0 2 4 6 8

10^-9 10^-8 10^-7 10^-6 10^-5 10^-4 10^-3

Voltage applied on bottom electrode (V) 2.0nm EOT

2.4nm EOT

area = 50µm x 50µm Leakage Current Density (A/cm2 )

Fig. 4-6. J-V characteristics of 2.0 and 2.4 nm EOT Al doped TaO

MIM

capacitors. The asymmetric J-V and breakdown voltages are due

to the different bottom Pt and top Al electrodes.

1k 10k 100k 1M 10M 100M 1G 10G 0

5 10 15 20

Quality Factor

2.0nm EOT 2.4nm EOT

Capacitance (fF/µm2 )

Frequency (Hz)

0 100 200 300 400 500 600 700

Fig. 4-7. The frequency dependent capacitance and Q-factor of 2.0 and

2.4 nm EOT Al doped TaO

MIM capacitors.

1k 10k 100k 1M 10M 100M 1G 10G 100G

10⁰

of frequency. It is notice that the ∆C/C and a decrease with

increasing frequency.

∆C/C (ppm) Quadratic VCCα (ppm/V2 )

1/Capaictance Density (µm²/fF)

of 1/C. It is notice that the ∆C/C and αdecrease monotonically

with ln(1/C).

0 25 50 75 100 125 150

0 10000 20000 30000 40000 50000 60000 70000 80000

AlTaO_x

3.3 nm 4.8 nm EOT 10 kHz 100 kHz 1 MHz

∆C/C(ppm)

Temperature (^oC)

Fig. 4-10. The ∆C/C as a function of temperature of Al doped TaO

MIM

capacitors.

1k 10k 100k 1M 10M

10¹ 10² 10³ 10⁴

AlTaO_x 3.3 nm EOT AlTaO_x 4.8 nm EOT

TCC (ppm/o C)

Frequency (Hz)

Fig. 4-11. The TCC as a function of frequency of Al doped TaO

MIM

capacitors.

0.0 0.1 0.2 0.3 0.4 0.5

10 100 1000

AlTaO_x HfO₂[4.7]

10 kHz 100 kHz 1 MHz

1/Capaictance Density (µm²/fF) TCC (ppm/o C)

Fig. 4-12. The TCC of Al doped TaO

MIM capacitors as a function of

1/C. TCC decreases monotonically with ln(1/C).

Chapter 5 The Theorem of the Variation of Dielectric Constant

5.1 Introduction of Free-carrier Relaxation

After recalling the background of Debye’s relation for dipole relaxation, we give a detail presentation for the derivation of the complex dielectric constant due to the contribution of space charges that made of free carriers able to move between blocking electrodes. Then we propose experimental tests of the model to discuss the formal analog between the relaxation of a collection of permanent dipoles and that of space charges.

5.2 Debye Formula

Dipole relaxation within the meaning of Debye is a purely viscous process without elastic forces of recovery. A typical equation for such a phenomenon is, for example, that which describes the speed ν of a particle of mass m after the application of a constant force F in a viscous medium. Exerting a friction force fν on the particle

to move:

) 1 ( fv

dt F mdv = −

If the particle, having charge Q, is in a field E, then the force F is qE. The relation

62 analogy, we can write for polarization by Por, orientation of whole dipoles in thermal balance:

here, τ indicates the dipolar relaxation time, Ps is the end value (static) of polarization

in a continuous-current field and Pi is the initial value which corresponds to quasi-instantaneous polarization due to the dependent electrons and ions (Por → Ps - Pi

in a constant field).

Let us imagine now that the particles (or, by analogy, dipoles) are in an alternate field of the form:

In alternative mode, all the variables, and in particular polarization by Por

orientation, oscillate sinusoidal with the pulsation ω, so that we can write:

)

We can directly obtain (if not rigorously) the formula of Debye while replacing in (3)

dPor/dt by his value iωPor drawn from (5):

63 We derive that:

)

so that the total polarization P(ω), sum of instantaneous polarization Pi and the

polarization of Por, is written:

)

By admitting that the local field of the dipoles is equal to the field applied. One has, by definition of the permittivities:

)

so that the relation (8) becomes:

)

the relation (12) is the famous formula of Debye.

5.3 Polarization by Space Charges

In the traditional treatments of the interfacial polarization of the Maxwell-Wagner type, one admits that the movement of the instantaneous charges in material does not affect the uniform distribution of the field. In fact, the changes

accumulate in the electrodes, in which they discharge more or less easily. It results from the gradients of concentration which tend to be opposed to accumulation. Briefly, the model treated initially by Mac Donald [5.1], who makes the following assumptions:

1. The material contains completely ionized fixed centers (for example, energy levels of impurities very close to the conduction band) and, mobility µ of free

electrons in equal concentration.

2. The electrodes are completely blocking so that the current of electrons at the balance of the electrodes (x =± d) are null.

In the absence of field applied, the free electrons, uniformly distributed in the sample, compensate for the load of the positive ionized centers everywhere, so that the sample is neutral. In the presence of a continuous-current field applied, the electrons are distributed in the sample under the combined action of the field and the diffusion which tends to be opposed to their accumulation to the electrodes.

If ρ(x) is the density of charge to balance with depth x in the sample index, average polarization is written:

)

If the direction of the field applied is then reversed, the electrons are distributed again

and the situation develops towards a new balance where ρ’(x) = - ρ(x). Therefore, the

new macroscopic dipole and polarization are opposed to the precedent.

We thus attend a phenomenon similar to a dipolar relieving of Debye type, occurring in a sample that becomes from initially homogeneous (statistically) to heterogeneous under the action of the field. This is represented on figure 5-1.

5.4 Calculation of Polarization into Alternative Field

Following, we will study the dynamic balance of the system when the sample is subjected to an alternate field of pulsation ω and rather low amplitude so that the

system remains linear, and consequently that the variables (time and space) are separable. In fact, it is more about a limit on the tension applied (at ordinary temperature).

We concern for the uniform concentration n0 and their mobility µ of free electrons (and centers) in the absence of alternating field. Under the terms of the assumptions, the total number of electrons per unit area of the sample, thickness 2d, is equal to 2dn0 in the presence of the alternate field.

That is to say:

) 15

t (

i a

a A e

E = ^ω

the alternate field applied, whose amplitude Aa assumes lower than KT/2. The

concentration of charge at x-coordinate x is not very different from n0, so that the

difference n - n0 oscillates with the pulsation ω and we can write:

)

In the same way, the potential V(x,t) and the field E(x,t) take the respective forms:

Of course, the factors ν(x), ϕ(x) and A(x) are complex quantities. The complex permittivity ε*(ω) of the sample is by definition:

)

where polarization P, defined by (14), results from the excess of density ν(x), given by

(16).

Calculation of ν(x) is the current of particles and the number of charges crossing

to x-coordinate x, by unit time, unit area of the x-plane. It is the sum of the drift current nµE and the diffusion current -- D∇n:

)

While introducing the values of n and V into (21), given from (16) and (17) respectively, we can obtain:

)

Since ν is small compared with n0, the third term of the member of right-hand

side of (22) is negligible compared with the first two terms. In addition, the Poisson's equation is written here:

)

and, while combining (23) with the simplified equation (22), we can obtain:

)

written in the form:

)

By using the variable complexes reduced:

L x

but the conservation of the instantaneous charges implies:

∫

−^d =

and the constant A0 results from the boundary condition imposed by the potential

applied:

∫

−^d =

dA(x) dx 2Aad which is written:

)

was left that (28) takes the form:

)

is null, since the electrodes are blocking:

Taking (27) and (30) into account, (31) is written:

)

By remembering that n ~ n0 and using the written accounts above between L, D, µ and n, (32) takes the form:

and like LZ – L/Z = iωτL/Z, the relation above gives ν1, and consequently:

)

Consequently, the polarization given by (14) becomes:

)

After integration of the numerator:

)

While returning to the definition of P(ω) (19), we have finally obtained:

)

The relation (37) was extended by Meaudre and Mesnard [5.2]-[5.3] if we take account of the existence of traps in material. Y must be modified.

5.5 Distribution of the Field

It results from the relations (31) and (33) that:

)

The module of A(X) is various with δ and ωτ. We observe that for a given value of δ, the field is more distorted (especially close to the electrodes) when ωτ is small.

The following relations give ϕ(x), A(x) and ν(x):

)

It results from the preceding formulas that ϕ(x) and ν(x) are dependent using the

relation:

At low frequency (ωτ << 1), the member of right-hand side of (40) is very small,

so that:

With high frequency (ωτ >> 1), it becomes comparable with xAa. As the field

inside the sample is distorted a little at high frequency, we can pose:

)

The relations (41) and (41’) show that ϕ is proportional to ν whatever the

frequency except in the vicinity of the electrodes.

The electrodes are partially blocking

. -- If the electrodes are partially

blocking, one can admit that the current with x = d (and, by symmetry, with x = -d) is proportional to the local density of load at the interface:

)

where γ is a coefficient without dimension and characterizing the ability of the electrode blocking [5.4] (γ = 0 for a blocking electrode). Then the equation (31)

becomes:

While using (27) and (30), and replacing ν1 by ν’in these relations, (43)

becomes:

Having ν(x), we can obtain P(ω) like the work previously we do and find ε* : )

Therefore, we have obtained the useful equations (37) and (46) to analyze the frequency-dependent permittivity ε* in the next chapter.

Fig. 5-1. Switching of the macroscopic dipole by reversal of the applied

field.

Chapter 6 The Unified Understanding and Prediction of High-κ Al doped TaO

Metal-Insulator-Metal Capacitors

6.1 Introduction

In MIM capacitors, one of the great challenges is to achieve small voltage coefficient of capacitance (VCC). Though experimental results of VCC variation such as thickness effects have been reported [6.1]-[6.6], the mechanism of VCC dependence remains unclear. Here we present a unified understanding of voltage based on the free carrier injection model. And the model can also be used to understand other high-κ materials. So we will use this model to characterize the high-κ Al doped TaOx and compare with the most popular high-κ HfO2.

6.2 The Free Carrier Injection Model

Fig. 6-1 shows the free carrier injection model, which attributes capacitance variation to injected carriers [6.7]. The excess charges in the insulator layer will

follow the alternating signal with a relaxation time (τ) that depends on the mobility of carrier (µ), excess carrier density (n), and dielectric constant (ε). A higher relaxation time means the carriers are more difficult to follow the alternating signal. Fig. 6-2 (a)

and (b) show the leakage current of 11.5 nm Al doped TaOx and 30 nm HfO2 MIM capacitors [6.3], respectively. Fig. 6-3 (a) and (b) show that the model fits very well with measured C-V curve of Al doped TaOx and HfO2 MIM capacitors, respectively.

6.3 Thickness Dependence

Fig. 6-4 (a) and (b) show the linear dependence of carrier concentration pre-factor (n0) of Al doped TaOx and HfO2 MIM capacitors on thickness, respectively.

Simulated normalized capacitances (∆C/C) as a function of voltage with different

thickness are shown in Fig. 6-5 (a) and (b). Fig. 6-6 (a) and (b) suggest thickness dependence of quadratic VCC of Al doped TaOx and HfO2 MIM capacitors have a

relation of α∝t⁻ⁿ shown in Fig. 6-7, which is consistent with the report in [6.6]. This thickness effect is due to E-field reduction with increased thickness. It is also found from Fig. 6-6 (b) that, after accounting the thickness dependence of n0, the decreasing of quadratic VCC with thickness is slower, which implies the limitation of high-κ

materials for analog applications where small VCC is needed.

6.4 Frequency Dependence

From the model itself, there is no frequency dependence of VCC. To fit the frequency dependence of VCC, the change of mobility at different frequencies need to

be considered. Fig. 6-8 (a) and (b) show the measured quadratic VCC and fitted carrier mobility at different frequencies. It is found that both the quadratic VCC and the fitted carrier mobility decrease with the frequency. Simulated normalized capacitances as a function of voltage at different frequencies are shown in Fig. 6-9. It is believed that the carrier mobility becomes smaller with increasing frequency, which leads to a higher relaxation time and a smaller capacitance variation.

6.5 Stress Induced Voltage Coefficient of Capacitance

Fig. 6-10 and Fig. 6-11 show that, for thick HfO2 MIM capacitors, both stress induced leakage current and quadratic VCC changes slightly with stress time. In comparison, Fig. 6-12 and Fig. 6-13 show that, for thin HfO2 MIM capacitors, both the leakage current and quadratic VCC changes quite significantly with stress time.

The results imply that both stress induced leakage current (SILC) and the variation of quadratic VCC are correlated to each other. With the increase of stress time, carrier

mobility (µ) could be modulated to be smaller by the stress generated traps, and further leads to a higher relaxation time and a smaller capacitance variation.

= τ: relaxation time

µ: carrier mobility in insulator

n’: pre-factor of effective carrier concentration β_s: β factor in current emission

T: temperature

t: thickness of insulator ω: frequency (=2πf) J: leakage current density

Fig. 6-1. Free carrier injection model to analyze the frequency-dependent

∆C/C.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 10^-8

10^-7 10^-6 10^-5 10^-4

Leakage Current Density (A/cm2 )

Voltage (V) Measurements

Schottky emission model

(a)

0 1 2 3 4 5 6

10^-9 10^-8 10^-7 10^-6 10^-5 10^-4

Current density (A/cm

2 )

Voltage (V)

(b)

Fig. 6-2. The measured and simulated J-V characteristic of high-κ (a) Al

doped TaO

and (b) HfO

[6.8] MIM capacitors. The

experimental data can be fitted by an emission current model.

Fig. 6-3. Measured and simulated normalized capacitance of high-κ (a) Al

doped TaO

and (b) HfO

[6.8] MIM capacitors as a function of

voltage. n

and µ are extracted by fitting the measured data.

80 Carrier conc. pre-factor (x1015 cm-3 )

qµn0(V/t)=A*T2exp(-qφb/kT) so, n0/t ~ constant

(a)

Carrier conc. pre-factor (x1014 cm-3 )

Thickness (nm) qµn₀(V/t)=A*T²exp(-qφb/kT)

so, n₀/t ~ constant

(b)

Fig. 6-4. Dependence of carrier concentration pre-factor of high-κ (a) Al

doped TaO

and (b) HfO

[6.8] MIM capacitors on thickness.

81 as thickness increased from 10 to 50 nm by step of 10nm

normallized capacitance decreases as thickness increased from 20 to 60 nm by step of 10 nm

(b)

Fig. 6-5. Simulated normalized capacitance as a function of voltage for

different thickness of 20, 30, 40, 50, and 60 nm of high-κ (a) Al

doped TaO

and (b) HfO

[6.8] MIM capacitors.

10 20 30 40 50

10^-3 10^-2 10^-1 10⁰

Quadratic VCC (1/V2 )

Thickness (nm)

(a)

20 30 40 50 60 70

10¹ 10² 10³ 10⁴

Quadratic VCC (ppm/V2 )

Thickness (nm)

Without accounting thickness depedence of carrier concentration pre-factor

Accounting thickness depedence of carrier concentration pre-factor

在文檔中高介電常數射頻金屬-絕緣層-金屬電容及其電容值變動之研究 (頁 45-0)

Chapter 3 Experimental Procedure of High-κ MIM Capacitor

3.3 De-embedding Theory

Fig. 3-1. Typical layout suitable for coplanar probing with

ground-signal-ground (GSG) probe configuration

Fig. 3-2. Constant pitch, minimum pad size and minimum parallel-row

pad.

Fig. 3-3. Mask #1 (bottom electrode) of layout and process flow of the

RF MIM capacitor.

Fig. 3-4. Mask #2 (Active Region) of layout and process flow of the RF

MIM capacitor.

Fig. 3-5. Mask #3 (Via hole) of layout and process flow of the RF MIM

capacitor.

Fig. 3-6. Mask #4 (upper electrode) of layout and process flow of the RF

MIM capacitor.

Fig. 3-7. The RF MIM capacitor cross section view.

Fig. 3-8. The RF MIM capacitor layout view.

Fig. 3-9. Measured and simulated scattering parameters of 2.0 and 2.4 nm

EOT Al doped TaO

MIM capacitors.

Fig. 3-10. The “OPEN” dummy structure for the device under test (DUT)

modeled in the shunt configuration.

R P C

R S L S2 L S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

R P C

R S L S2 L S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

Fig. 3-11. The equivalent circuit model and numerical values of elements

for capacitor simulation at RF regime.

Fig. 3-12. The equivalent circuit model of MIM capacitor at RF regime.

Chapter 4

The Characteristics of High-κ MIM Capacitor with Al doped TaO

Dielectrics

4.1 Capacitor Characteristics

4.2 The Normalized Capacitance Variation ∆C/C and Voltage Coefficient of Capacitance VCC

4.3 The Temperature Coefficient of Capacitor TCC

Fig. 4-1. Measured and simulated scattering parameters of 2.0 and 2.4 nm

EOT Al doped TaO

MIM capacitors at RF regime.

R P C

R S L S2 L S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

R P C

R S L S2 L S1

= 0.01nH = 0.01nH

= 41.0/36.5pF

= 10Ω = 1.5/2.0kΩ

Fig. 4-2. The equivalent circuit model and numerical values of elements

for capacitor simulation at RF regime.

10

10

10

10

10

10

10

10

10

10

10

10

10

10

Fig. 4-3. The ∆C/C of Al doped TaO

MIM capacitors as a function of

frequency. It is notice that the ∆C/C decrease with increasing

frequency.

Fig. 4-4 C-V and ∆C/C-V characteristics of MIM capacitors with 2.0

EOT Al doped TaO

dielectrics at different frequencies from 10

KHz to 10 GHz. The voltage is applied to the bottom Pt

electrode. Measured area is 50mm × 50mm.

Fig. 4-5. C-V and ∆C/C-V characteristics of MIM capacitors 2.4 nm EOT Al doped TaO

R _P C

R _S L _S2 L _S1

R _P C

R _S L _S2 L _S1

R _P C

R _S L _S2 L _S1

R _P C

R _S L _S2 L _S1