Chapter 4 The Characteristics of High-κ MIM Capacitor
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
48
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
0
Fig. 4-1. Measured and simulated scattering parameters of 2.0 and 2.4 nm
EOT Al doped TaO
xMIM capacitors at RF regime.
50
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.
51
10
310
410
510
610
710
810
910
1010
1110
110
210
310
410
5∆C/C (ppm)
Frequency (Hz)
EOT 2.0nm EOT 2.4nm
Fig. 4-3. The ∆C/C of Al doped TaO
xMIM capacitors as a function of
frequency. It is notice that the ∆C/C decrease with increasing
frequency.
52
-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
xdielectrics at different frequencies from 10
KHz to 10 GHz. The voltage is applied to the bottom Pt
electrode. Measured area is 50mm × 50mm.
53
-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
xdielectrics at different frequencies from 10 KHz to 10 GHz. The voltage is applied to the bottom Pt electrode.
Measured area is 50mm × 50mm.
54
-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
xMIM
capacitors. The asymmetric J-V and breakdown voltages are due
to the different bottom Pt and top Al electrodes.
55
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
xMIM capacitors.
56
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.
57
∆C/C (ppm) Quadratic VCCα (ppm/V2 )
1/Capaictance Density (µm2/fF)
of 1/C. It is notice that the ∆C/C and αdecrease monotonically
with ln(1/C).
58
0 25 50 75 100 125 150
0 10000 20000 30000 40000 50000 60000 70000 80000
AlTaOx
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
xMIM
capacitors.
59
1k 10k 100k 1M 10M
101 102 103 104
AlTaOx 3.3 nm EOT AlTaOx 4.8 nm EOT
TCC (ppm/o C)
Frequency (Hz)
Fig. 4-11. The TCC as a function of frequency of Al doped TaO
xMIM
capacitors.
60
0.0 0.1 0.2 0.3 0.4 0.5
10 100 1000
AlTaOx 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
xMIM capacitors as a function of
1/C. TCC decreases monotonically with ln(1/C).
61
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
64
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
65
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
66
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:
)
67
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
68
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:
69
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.
70
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:
)
71
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 partiallyblocking, 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:
72
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.
73
Fig. 5-1. Switching of the macroscopic dipole by reversal of the applied
field.
74
Chapter 6
The Unified Understanding and Prediction of High-κ Al doped TaO
xMetal-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)
75
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
76
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.
77
= τ: relaxation time
µ: carrier mobility in insulator
n’: pre-factor of effective carrier concentration βs: 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.
78
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
xand (b) HfO
2[6.8] MIM capacitors. The
experimental data can be fitted by an emission current model.
79
Fig. 6-3. Measured and simulated normalized capacitance of high-κ (a) Al
doped TaO
xand (b) HfO
2[6.8] MIM capacitors as a function of
voltage. n
0and 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
xand (b) HfO
2[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
xand (b) HfO
2[6.8] MIM capacitors.
82
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
xand (b) HfO
2[6.8]
MIM capacitors as a function thickness.
83
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
xMIM capacitors as a
function thickness.
84
104 105 106
10-5 10-4
11.5 nm Al doped TaOx 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
xand (b) HfO
2[6.8] MIM capacitors as a function of
frequency.
85
-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
2MIM capacitors at for different frequencies of 10K,
100K, 500K, and 1MHz [6.8].
86
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
2MIM capacitor
[6.8].
87
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
2MIM capacitor
[6.8].
88
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
2MIM capacitor
[6.8].
89
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
2film MIM capacitor
[6.8].
90
Chapter 7
The Analysis of High-κ MIM Capacitor with Al doped TaO
xDielectrics
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
91
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
92
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
93
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.
94
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
xMIM
capacitors at different frequencies.
95
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
xMIM capacitors at different frequencies.
96
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
xMIM capacitors. The experimental data can be
fitted by an emission current model.
97
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 GHz
∆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.
98
10
410
510
610
710
810
910
1010
-910
-810
-710
-610
-510
-410
-3Relaxation time (sec)
Frequency (Hz)
Fig. 7-5. Modified carrier relaxation time as a function of frequency using
a [1+(f/f
0)2]-1/2 pre-factor.
99
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
xDielectrics
8.1 Conclusion
We have achieved record high 17 fF/µm2 capacitance 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/cm2 using high-κ Al doped TaOx MIM capacitors and processed at 400oC. 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
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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.
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Reference
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