Calculation of Polarization into Alternative Field

Chapter 5 The Theorem of the Variation of Dielectric Constant

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

(b)

Fig. 6-6. Quadratic VCC of high-κ (a) Al doped TaO

and (b) HfO

[6.8]

MIM capacitors as a function thickness.

10 20 30 40 50

0.0 0.5 1.0 1.5

∆C/C

Thickness (nm) at 5V bias

Fig. 6-7. Quadratic VCC of high-κ Al doped TaO

MIM capacitors as a

function thickness.

10⁴ 10⁵ 10⁶

10^-5 10^-4

11.5 nm Al doped TaO_x MIM Capacitor

Carrier mobility (cm Measured quadratic VCC at different frequency

Fitted carrier mobility based on the measured quadratic VCC

Frequency (Hz)

Carrier mobility (cm2 /vs)

(b)

Fig. 6-8. Quadratic VCC and fitted carrier mobility of high-κ (a) Al

doped TaO

and (b) HfO

[6.8] MIM capacitors as a function of

frequency.

-6 -4 -2 0 2 4 6

0 10000 20000 30000 40000

∆C/C (ppm)

Voltage (V)

Normallized capacitance decreases as the frequency increasing from 10 KHz, 100 KHz, 500 KHz, and 1MHz

Fig. 6-9. Simulated normalized capacitance as a function of voltage for

30 nm HfO

MIM capacitors at for different frequencies of 10K,

100K, 500K, and 1MHz [6.8].

0.0 0.5 1.0 1.5 2.0

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

10^-3 _Fresh

2000 s @1.5 V 4000 s @1.5 V 6000 s @1.5 V

Current density (A/cm2 )

Voltage (V)

Fig. 6-10. Stress induced leakage current of thick HfO

MIM capacitor

[6.8].

0 1000 2000 3000 4000 5000 6000 200

300 400 500 600 700 800

100 k 500 k 1 MHz

Quadratic VCC (ppm/V2 )

Stress time (s) at 1.5 V

Fig. 6-11. Stress induced quadratic VCC of thick HfO

MIM capacitor

[6.8].

0.0 0.5 1.0 1.5 2.0

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

Voltage (V)

Current density (A/cm2 ) ^Fresh 2000 s @1.5 V

4000 s @1.5 V 6000 s @1.5 V

Fig. 6-12. Stress induced leakage current of thin HfO

MIM capacitor

[6.8].

0 1000 2000 3000 4000 5000 6000 1500

1800 2100

2400 _{100 k}

500 k 1 MHz

Stress time (s) at 1.5 V

Quadratic VCC (ppm/V2 )

Fig. 6-13. Stress induced quadratic VCC of thin HfO

film MIM capacitor

[6.8].

Chapter 7 The Analysis of High-κ MIM Capacitor with Al doped TaO

Dielectrics

7.1 Reviewing the Work Before

The MIM capacitors were fabricated using high-κAl doped TaOx dielectrics described previously, where record high capacitance density of 17 fF/µm² was obtained. Fig. 7-1 shows the C-V characteristics as well as the ∆C/C values. These

data were obtained using an LCR meter from 10 KHz and 1 MHz, and calculated from measured S-parameters for the 1 to 10 GHz range, using

)

Fig. 7-2 shows that the ∆C/C decreases rapidly with increasing frequency, which is

advantageous for high frequency analog/RF circuits.

A free carrier injection model may be used to analyze the frequency dependence of ∆C/C [7.1]-[7.2]. The capacitor density is related to real part of the complex dielectric constant (ε^') and thickness (t)

C εt'

= (2)

The complex dielectric constant (ε^∗) is

where D is the diffusion coefficient.

The carriers in the MIM capacitor dielectric will follow an alternating signal

depending on τ , which is related to the carrier mobility (µ) and density (n) by

µqn

τ = ε (5)

with n derived from the leakage current and is a function of electric field (E)

2 )

7.2 Analysis of the Variation of Capacitance

To examine the field dependence of the carrier density, we first analyze the leakage current. Fig. 7-3 shows the measured and modeled leakage currents. The leakage current of ~9×10^-7 A/cm² at -2 V is low enough for circuit applications, since the capacitor area would be small and high density (17 fF/µm²). high. The J-V curve

can be fitted accurately using the emission current equation:

2 ) Thus the free carrier injection model should be validity in the following analysis.

In Fig. 7-4 we show the measured and calculated ∆C/C-V over the 10 KHz to 10

GHz frequency range. In the free carrier injection model, a nearly constant, weak

frequency dependent τ was used. This assumption is often used when analyzing

∆C/C-V data for MIM capacitors [7.1]-[7.2]. Although the agreement between measured and modeled ∆C/C-V at 10 KHz and 1 MHz is reasonable, the simulated curve deviates from the data at GHz frequencies and much over-estimates the ∆C/C-V

decrease.

To fit the measured ∆C/C-V over the whole frequency range requires that

decreases with increasing frequency. In Fig. 7-5 we show the relaxation time needed to explain the data and the curve

where f0 is a constant frequency. Using this variation of τ the simulated curves in Fig.

7-6 agree well with the ∆C/C-V data. This is due to the fact that in the free carrier

injection model, we neglected the contribution of the dipole relaxation to the capacitance. The dipole effects are known to be particular important in ferroelectrics crystals and contribute strongly to the nonlinear dielectric response [7.2] and thus to

the ∆C/C variations. The contribution of dipole effects can be observed even at infrared frequency, while the free carrier effects become negligible at frequencies

higher than about 1 MHz. The frequency dependence of τ represented by [1+(f/f0)²]^-1/2 has the same form as that for a FET’s voltage gain [7.3]-[7.4]. This

suggests that the reduction of ∆C/C with increasing frequency can be understood

simply by similar physics - the injected carriers into the MIM capacitor cannot follow the increasing frequency and cause ∆C/C or the transistor’s gain to decrease.

0.000 0.25 0.50 0.75 1.00

5 10 15 20

Voltage (V) Capacitance (fF/µm2 )

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

∆C/C (ppm)

10 KHz 1 MHz 1 GHz 10 GHz

Fig. 7-1. C-V and ∆C/C-V characteristics of high-k Al doped TaO

MIM

capacitors at different frequencies.

10³ 10⁴ 10⁵ 10⁶ 10⁷ 10⁸ 10⁹ 10¹⁰ 10¹¹ 10²

10³ 10⁴ 10⁵

∆C/C (ppm)

Frequency (Hz)

Fig. 7-2. ∆C/C of Al doped TaO

MIM capacitors at different frequencies.

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

Fig. 7-3. The measured and simulated J-V characteristic of high-k Al

doped TaO

MIM capacitors. The experimental data can be

fitted by an emission current model.

0.0 0.2 0.4 0.6 0.8 1.0

10^-4 10^-2 10⁰ 10² 10⁴ 10⁶

10⁸ simulation measurement 10 KHz 1 MHz 1 GHz 10 GH^z

∆C/C (ppm)

Voltage (V)

Fig. 7-4. Measured and simulated ∆C/C-V data assuming a nearly

constant t. Although good agreement is obtained at 10 KHz

and 1 MHz, the assumption fails to account for the 1 GHz and

10 GHz data.

10

⁴

10

⁵

10

⁶

10

⁷

10

⁸

10

⁹

10

¹⁰

10

^-9

10

^-8

10

^-7

10

^-6

10

^-5

10

^-4

10

^-3

Relaxation time (sec)

Frequency (Hz)

Fig. 7-5. Modified carrier relaxation time as a function of frequency using

a [1+(f/f

)2]-1/2 pre-factor.

0.0 0.2 0.4 0.6 0.8 1.0

10¹ 10² 10³ 10⁴ 10⁵

simulation measurement 10 KHz 1 MHz 1 GHz 10 GHz

∆C/C (ppm)

Voltage (V)

Fig. 7-6. Measured and simulated ∆C/C using the frequency dependent

relaxation time shown in Fig. 7-5.

100

Chapter 8 The Conclusion of High-κ MIM Capacitor with Al doped TaO

Dielectrics

8.1 Conclusion

We have achieved record high 17 fF/µm²capacitance density, small 5%

capacitance reduction to RF frequency range, small ∆C/C ≤ 120 ppm at 1 GHz, and

low leakage current of 8.9×10^-7 A/cm²using high-κ Al doped TaOx MIM capacitors and processed at 400^oC. This high capacitance density with good device integrity can greatly reduce the chip size of RF circuits and be useful for precision circuit application at high frequencies.

We also show the space charge relaxation in the dielectrics and use it to derive the free carrier injection model to describe the frequency-dependence and voltage-dependence of capacitance (∆C/C) for high-κ MIM capacitors.

Although the free carrier injection model can fit well the voltage-dependence of capacitance (∆C/C) for high-κ MIM capacitors, the nearly constant relaxation time (τ) assumption can not match the measured fast reduction of ∆C/C as frequency increases

into GHz regime. We using a modified free carrier injection model with an additional pre-factor to account for the frequency dependence of the relaxation time, good

101

agreement between the frequency dependence of the measured and modeled voltage dependence of the capacitance, ∆C/C-V, has been obtained for high-κ Al doped TaOx

MIM capacitors. This simple model should be helpful in simulations of circuits that include MIM high-κ dielectric capacitors.

102

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Chapter 3

[3.1] S. B. Chen, J. H. Chou, A. Chin, J. C. Hsieh, and J. Liu, “High density MIM

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 81-0)