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Capacitance non-linearity study in Al2O3 MIM capacitors using an ionic polarization model
Microelectronic Engineering 83 (2006) 2422–2426 www.elsevier.com/locate/mee
Capacitance non-linearity study in Al2O3 MIM capacitors using an ionic polarization model S. Be´cu b
a,b,*
, S. Cre´mer a, J.L. Autran
b
a ST Microelectronics, 850 rue Jean Monnet, 38926 Crolles cedex, France L2MP – UMR CNRS 6137, Baˆtiment IRPHE, 49 rue Joliot-Curie, BP 146, 13384 Marseille cedex 13, France
Available online 20 October 2006
Abstract We have realized Al2O3 metal–insulator–metal (MIM) capacitors of three different thicknesses ranging from 7 to 20 nm, and performed capacitance versus voltage characterizations of the samples between 25 and 150 C. The MIM capacitors presented increasing capacitance densities with temperature and parabolic C–V curves with a positive curvature. These results are widely observed in MIM capacitors with high-j dielectrics such as HfO2 [1], Ta2O5 [2] or Y2O3 [3], but non-linearity origins are rarely reported in literature. In this article, an attempt has been made to estimate the non-linearity of the permittivity in Al2O3 dielectric, assuming that electronic polarization did not contribute to the non-linearity. 2006 Elsevier B.V. All rights reserved. Keywords: MIM capacitors; Non-linearity; Characterization; Polarization
1. Introduction Metal–Insulator–Metal (MIM) capacitors are widely used in analog and RF circuits to integrate decoupling or filtering functions, which require higher and higher capacitance density. At the same time, for more sensitive applications, such as accurate A/D converters, highly linear capacitance is required. There are two main options to fulfil the capacitance density increase: either decrease the dielectric thickness or introduce a dielectric with a high permittivity, but both of these options imply a voltage linearity degradation. For future generations, the trade-off between linearity and capacitance density should become more and more important for dielectric and MIM architecture choices. For a better engineering, it is of importance to estimate the non-linearity origins in order to optimize the dielectric deposition process. With this aim in view, in this article, we will give some experimental details before exposing the model *
Corresponding author. Address: ST Microelectronics, 850 rue Jean Monnet, 38926 Crolles cedex, France. Tel.: + 33 476 926 806; fax: +33 476 926 814. E-mail address: [email protected] (S. Be´cu). 0167-9317/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2006.10.049
we have developed to understand the physical origin of the capacitance variations with voltage and temperature. Finally, we will confront this model to experiment. 2. Materials and experiments 2.1. MIM stack integration We have realized 200 lm · 100 lm planar MIM capacitors with Al2O3 as dielectric integrated in the aluminum Back-End of 0.25 lm BiCMOS technology (see Fig. 1). Three different Al2O3 thicknesses have been deposited by Atomic Layer Deposition (ALD), between two TiN electrodes deposited by PVD. The targeted thicknesses were d = 7 nm, 13 nm and 20 nm, allowing to investigate the electric field effect in addition to the applied voltage effect on the MIM capacitance value. 2.2. Experimental details To study the MIM capacitance non-linearity, we have performed a precise C–V characterization using a HP
Capacitance variation (ppm)
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
2423
1500
1100 700
300 0 - 0 .8
-0.4
0.0
0 .8
0.4
Applied electric field (MV/cm) Fig. 3. Capacitance variations as a function of the applied electric field, measured at 25 C. The cross, square and circle symbols correspond, respectively, to the 7, 13 and 20 nm thick Al2O3. Fig. 1. Schematic of a sample containing the MIM capacitor integrated in the Al interconnects.
4284 LCR meter. All the measurements have been done applying a 100 mV AC signal with a frequency of 100 kHz. We chose the amplitude for the voltage sweep in such a way that the amplitude of the applied electric field was less than 1 MV/cm. Thus, for the 7 nm-thick sample (respectively for the 13 and 20 nm thick-samples), we applied a sweep between 0.5 and 0.5 V (respectively between 1 and 1 V, and between 1.5 and 1.5 V). This low field amplitude allowed to study the intrinsic properties of the dielectric constant without being disturbed by leakage currents and charge trapping. The sufficiently large device area, 20,000 lm2, enabled to measure accurately the small capacitance variations as a function of the applied voltage without being limited by the tool resolution. 2.3. Capacitance measurements The voltage non-linearity is classically described by the polynomial law [4] CðV a Þ ¼ C 0 ð1 þ C 1 V a þ C 2 V 2a Þ
ð1Þ
15
10
5
0 0.04
DC 0 CðEa Þ þ CðEa Þ 2Cð0Þ ¼ C 2 d 2 E2a 0 ðE a Þ ¼ 2Cð0Þ C
0.08
0.12
ð2Þ
The applied electric field Ea corresponds to the ratio between the voltage drop across the dielectric Va and the dielectric thickness d. We have plotted the capacitance variation as a function of the applied electric field in Fig. 3. Whatever the thickness, the characteristics are merged. From Eq. (2), it proves that the product C2d2 is constant whatever the thickness. On the contrary, the C1/C0 distributions of Fig. 4, for the three Al2O3 thicknesses, are not merged at all, which means that theC1d value depends on the dielectric thickness. C1 is thus not an intrinsic material parameter but
Cumulative distribution (%)
Capacitance under 0 V (fF/µm2)
where Va is the applied voltage between the two electrodes. The three coefficients C0, C1 and C2 have been compared for all the samples.
First, Fig. 2 shows that the measured capacitance density at 0 V, C0, perfectly follows the classical e0e(0)/d law, where e0 is the vacuum permittivity and e(0) is the dielectric permittivity at 0 V. From Fig. 2, we extracted e(0) = 8.3 at 25 C. This is in perfect agreement with the alumina permittivity reported in [5]. Then as far as C2 is concerned, we defined the capacitance variation at a given applied electric field Ea by Eq. (2). It is defined in such a way that it does not depend on the C1 coefficient.
100 80 60 40 20 0 -0.1
-0.08
-0.06
-0.04
-0.02
C1/C0(in m2/(F.V) ) 0.16
d-1 (nm-1) Fig. 2. Capacitance at 0 V as a function of the inverse of the dielectric thickness, T = 25 C.
Fig. 4. C1/C0 cumulative distributions. The values have been measured at 25 C, on nine different capacitors for each thickness. The cross, square and circle symbols correspond, respectively, to the 7, 13 and 20 nm thick Al2O3.
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
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we can rather attribute the C1Va term, in Eq. (1), to the presence of charges in the dielectric. From Figs. 2–4, we can conclude that, only the constant and the quadratic terms of the C–Va polynomial law are intrinsic material parameters that define its dielectric permittivity. Thus, neglecting any first order term, the dielectric permittivity electric field dependence can be expressed by eðEa Þ ¼
aE2a
þ eð0Þ
ð3Þ
in which a is the quadratic coefficient of the dielectric constant. From experimental results of Fig. 3, a = 2.2 ± 0.2 · 103 · (MV/cm)2 at room temperature. The ±10% error has been estimated from the maximum difference between the three curves. 3. Dielectric constant model In pure Al2O3, for frequencies below the infrared domain, two polarization mechanisms coexist: the electronic and the ionic polarization. The first one is due to the polarization of the electronic cloud of all atoms which compose the material. This contribution is thus present in all materials, and gives rise to their refractive index RI. The second comes from the Al–O bonds distortion under the applied electric field. Assuming that neither electric field, nor temperature impacts the alumina refractive index compared to the ionic susceptibility vion, the relative dielectric constant takes the form v e ¼ ion þ RI 2 ð4Þ e0 In this section, we will describe an ionic susceptibility model based on a classical electrostatic approach to estimate both the capacitance at 0 V, C0, and the capacitance 0 variations, DC , due to the electric field. C0 3.1. Local environment and ionic susceptibility In amorphous alumina, most of the Al atoms are surrounded by oxygen atoms in a tetrahedral configuration so that the elementary unit of the system mainly consists
a
b
x Induced dipole
Fig. 6. Under a dc electric field the Al ion is displaced from its equilibrium position (a) which breaks the symmetry and induces a dipole (b).
of a slightly distorted (AlO4)5 tetrahedron [7]. Considering that dielectric properties are linked to the short-range environment of the material, we have modelled the whole system like it is represented in Fig. 5: the elementary tetrahedron can only take two positions with respect to the MIM electrodes. These two positions are referred by the numbers 1 and 2 in Fig. 5. For example, if we only consider the type 1 tetrahedrons, it is easy to describe the main polarization mechanism in alumina: it comes from the Al–O bonds distortion due to the voltage application between the metallic electrodes (see Fig. 6). This is a common phenomenon that exists in all non polar dielectrics: the electric dipole is induced by the electric field perturbation (see Fig. 6b). The full demonstration of the ionic susceptibility expression has been developed in [6] and it is given by Eqs. (5) and (6). vion ðEloc ; T Þ ¼
Nk B T o2 flnðJ 1 ðEloc ; T ÞÞ 2 oE2loc þ lnðJ 2 ðEloc ; T ÞÞg
J1 andJ2 are mathematical entities which express the fact that the Al ion displacement x can take all values allowed by the geometry. Z us ðxÞ 2eEloc J 1;2 ðEloc ; T Þ ¼ x dx ð6Þ exp exp kBT kBT x Eloc is the local field which is the superposition of the applied electric field Ea and the dipolar field due to surrounding molecules, T is the temperature, N is the Al cations
Aluminum
Self-energy us (eV)
2 Oxygen
4 0 -4 -8
Bottom electrode
ð5Þ
8
Top electrode
Al2O3 dielectric
Al3+
Eloc > 0
Eloc = 0
Top electrode
1
O2-
Bottom electrode
Fig. 5. The Al2O3 amorphous dielectric is modelled by tetrahedrons that can only take two positions with respect to the electrode direction.
-2
0
2
4
6
8
10
Bond distortion x (Å) Fig. 7. Self-energy as a function of the bond distortion x. T = 25 C, m = 9, and n = 1.
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
k¼
Eloc Ea
ð7Þ
According to Eqs. (5) and (6), we only need to express the self-energy to calculate the ionic susceptibility as a function of the electric field and the temperature. 3.2. Ionic bond With the quite simple elementary cell geometry of Fig. 6, and using a Mie–Gru¨neisen interatomic potential [9] given by Eq. (8), to model the Al–O bond interactions, the selfenergy us follows Eq. (9):
mn 1 r0 m 1 r0 n ; m>n mn m r n r 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 r 2 d OO A 0 x þ3 us ðxÞ uMG ðr0 þ xÞ þ 3 uMG @ 3 4 uMG ðrÞ ¼ u0
ð8Þ ð9Þ
In Eq. (8), u0 and r0 are, respectively, the Al–O bond energy and the Al–O bond length at equilibrium, and m and n are the repulsive and the attractive exponents of the potential, respectively. In Eq. (9), dOO is the distance between two oxygen neighbors, and xis the distortion of the bond between the central Al cation and the oxygen ion at the tetrahedron summit (see Fig. 6b). For geometrical reasons, considering an isolated elementary cell, the distortion amplitude can theoretically take all values between r0 and the infinity. In Fig. 7, we have plotted the self-energy as a function of x: the minimum potential is obviously reached for the equilibrium position x = 0; when the Al atom moves closer to the oxygen at the tetrahedron summit (x < 0) the self-energy rapidly increases, whereas it converges to 0 when x tends to infinity.
E loc= 0
9
6
E loc= 5 MV/cm 3 0 -1
- 0.5
0
0.5
m n r0 dOO RI at 546 nm N
9 1 1.9 4.1 1.77 3.2 · 1022 cm3
It is also interesting to represent the distortion amplitude distribution: the probability that a distortion x happens is given by Eq. (10). loc Þ exp W kðx;E T B ð10Þ P ðx; Eloc Þ ¼ R loc Þ exp W kðx;E x T B where W(x, Eloc) = us(x) ± 2eElocx is the total potential energy of a tetrahedron. In Fig. 8, one can notice that only very small bond distortions, of a tenth of angstrom order of magnitude, are likely to happen. If no perturbation is applied, the maximum probability is centred at equilibrium position x = 0, whereas when an electric field is applied the distribution changes and its maximum is shifted. Even under strong electric fields, the distribution changes are very small,
12
10
1
Bond distortion x (Å) Fig. 8. Distortion amplitude distributions: with no electric field (full line) and under a 5 MV/cm electric field (dots). T = 25 C, m = 9, and n = 1.
8
6 1
1.5
2
2.5
Al–O bond energy (eV) Fig. 9. Dielectric constant at 0 V and 25 C as a function of the Al–O bond energy.
Quadratic coefficient α (in units of 10-3× (MV/cm)-2)
Probability (%)
12
Table 1 Model parameters
Dielectric constant ε(0)
density, kB is the Boltzmann constant, and e is the elementary electric charge. In the latter equation, us is the self-energy of an elementary cell that is composed of one Al atom plus his first neighbors. The local field is generally a linear function of the external electric field [8]. In the following, we introduce a local field factor k, which is the ratio between the local and the applied electric field
2425
3.5 2.9 2.3 1.7 1.1 0.5
1
1.5
2
2.5
Local field factor Fig. 10. Quadratic coefficient a as a function of the local field factor at 25 C for u0 = 1.5 eV.
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
a
b
12.0
Capacitance variation (ppm)
Capacitance density at 0 V (fF/µm 2)
2426
10.5 9.0 7.5 6.0 4.5 3.0
0
25
50
75
100
125 150 175
2000 1600 1200
Increasing temperatures
800 400 0 -0.8
Temperature (°C)
-0.4
0.0
0.4
0.8
Applied electric field (MV/cm)
Fig. 11. Comparison between model (in full lines) and experiment (in symbols). (a) C0versus temperature for d = 7 nm in cross, 13 nm in squares and 20 nm in circles. (b) Capacitance variations versus applied electric field for d = 13 nm at three temperatures (T = 25 C in cross, 75 C in squares and 125 C in circles).
which lets augur that the ionic susceptibility field dependence is weak. 4. Results and discussion 4.1. Fit parameters choice at room temperature Among the physical parameters required to calculate the dielectric constant, both the Al-O bond energy u0 and the local field factor k are unknown for amorphous alumina. The other parameters have been tabulated in [6], and are reported in Table 1. In Fig. 9, the static dielectric constant e(0) at 25 C has then been plotted as a function of the bond energy. With u0 = 1.5 eV, the calculated static dielectric constant at 25 C is 8.3 like the experimental value. The local field factork has been chosen to best fit the quadratic coefficient a at 25 C. Taking into account the 10% error made on the experimental extraction of a (see Section 2.3), the local field factor has been found to be between 2 and 2.2. This local field factor value is between 17% and 25% higher than the value reported in [6], but it is still in between the Onsager (1.4) and Lorentz (3.3) model predictions [8] (see Fig. 10).
Based on results depicted in Fig. 3, this means that the model well predicts the capacitance variations as a function of the applied voltage whatever the thickness, from 25 to 125 C. 5. Conclusion We reported on the temperature and electric field effects on Al2O3 MIM capacitance for three dielectric thicknesses. Concerning the alumina dielectric permittivity, the thickness study has allowed to identify two intrinsic parameters: first the static term e(0) and secondly, the field quadratic coefficient a. The C–V characterizations showed the existence of a first order term that we have associated to extrinsic charges. Both intrinsic parameters have been successfully modelled using a classical electrostatic approach. The model fits very well the field and the temperature effects on alumina dielectric constant. As a consequence, our first assumption, which asserts that both the electric field and the temperature dependences of Al2O3 permittivity comes from the Al–O bond ionicity, is consistent with the experimental results.
4.2. Model versus experiment References To check the model validity, we have compared it to the experiment for different temperature conditions. From the previous considerations, we have fixed u0 = 1.5 eV, and k = 2. The results, in terms of capacitance, are depicted in Fig. 11. The temperature dependence of C0 is very well predicted between 25 and 150 C, whatever the dielectric thickness (Fig. 11a). The capacitance variations have only been compared to the model for the 13 nm-thick Al2O3 sample. The measured capacitance variations, as a function of the applied electric field, exactly follow the model predictions for the three calculated temperatures (Fig. 11b).
[1] H. Hu et al., IEEE Electron Dev. Lett. 23 (9) (2002) 514–516. [2] K.H. Allers, in: Proceedings of the 2003 BCTM , September 2003, pp. 35–38. [3] C. Durand et al., J. Vac. Sci. Technol. A 22 (3) (2004) 655–660. [4] M. Gros-Jean, in: Proceedings of 12th Workshop on Dielectrics in Microelectronics, 2002, pp. 73–76. [5] T.W. Hickmott, J. Appl. Phys. 93 (2003) 3461. [6] S. Be´cu et al., Appl. Phys. Lett. 88 (2006) 052904. [7] G. Gutie´rrez, Phys. Rev. B 65 (2002) 104202. [8] C.J.F. Bo¨ttcherTheory of Electric Polarization, Vol. 1, Elsevier, Amsterdam, 1973. [9] J.W. Mac Pherson, J. Appl. Phys. 95 (2004) 8101.
Capacitance non-linearity study in Al2O3 MIM capacitors using an ionic polarization model S. Be´cu b
a,b,*
, S. Cre´mer a, J.L. Autran
b
a ST Microelectronics, 850 rue Jean Monnet, 38926 Crolles cedex, France L2MP – UMR CNRS 6137, Baˆtiment IRPHE, 49 rue Joliot-Curie, BP 146, 13384 Marseille cedex 13, France
Available online 20 October 2006
Abstract We have realized Al2O3 metal–insulator–metal (MIM) capacitors of three different thicknesses ranging from 7 to 20 nm, and performed capacitance versus voltage characterizations of the samples between 25 and 150 C. The MIM capacitors presented increasing capacitance densities with temperature and parabolic C–V curves with a positive curvature. These results are widely observed in MIM capacitors with high-j dielectrics such as HfO2 [1], Ta2O5 [2] or Y2O3 [3], but non-linearity origins are rarely reported in literature. In this article, an attempt has been made to estimate the non-linearity of the permittivity in Al2O3 dielectric, assuming that electronic polarization did not contribute to the non-linearity. 2006 Elsevier B.V. All rights reserved. Keywords: MIM capacitors; Non-linearity; Characterization; Polarization
1. Introduction Metal–Insulator–Metal (MIM) capacitors are widely used in analog and RF circuits to integrate decoupling or filtering functions, which require higher and higher capacitance density. At the same time, for more sensitive applications, such as accurate A/D converters, highly linear capacitance is required. There are two main options to fulfil the capacitance density increase: either decrease the dielectric thickness or introduce a dielectric with a high permittivity, but both of these options imply a voltage linearity degradation. For future generations, the trade-off between linearity and capacitance density should become more and more important for dielectric and MIM architecture choices. For a better engineering, it is of importance to estimate the non-linearity origins in order to optimize the dielectric deposition process. With this aim in view, in this article, we will give some experimental details before exposing the model *
Corresponding author. Address: ST Microelectronics, 850 rue Jean Monnet, 38926 Crolles cedex, France. Tel.: + 33 476 926 806; fax: +33 476 926 814. E-mail address: [email protected] (S. Be´cu). 0167-9317/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2006.10.049
we have developed to understand the physical origin of the capacitance variations with voltage and temperature. Finally, we will confront this model to experiment. 2. Materials and experiments 2.1. MIM stack integration We have realized 200 lm · 100 lm planar MIM capacitors with Al2O3 as dielectric integrated in the aluminum Back-End of 0.25 lm BiCMOS technology (see Fig. 1). Three different Al2O3 thicknesses have been deposited by Atomic Layer Deposition (ALD), between two TiN electrodes deposited by PVD. The targeted thicknesses were d = 7 nm, 13 nm and 20 nm, allowing to investigate the electric field effect in addition to the applied voltage effect on the MIM capacitance value. 2.2. Experimental details To study the MIM capacitance non-linearity, we have performed a precise C–V characterization using a HP
Capacitance variation (ppm)
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
2423
1500
1100 700
300 0 - 0 .8
-0.4
0.0
0 .8
0.4
Applied electric field (MV/cm) Fig. 3. Capacitance variations as a function of the applied electric field, measured at 25 C. The cross, square and circle symbols correspond, respectively, to the 7, 13 and 20 nm thick Al2O3. Fig. 1. Schematic of a sample containing the MIM capacitor integrated in the Al interconnects.
4284 LCR meter. All the measurements have been done applying a 100 mV AC signal with a frequency of 100 kHz. We chose the amplitude for the voltage sweep in such a way that the amplitude of the applied electric field was less than 1 MV/cm. Thus, for the 7 nm-thick sample (respectively for the 13 and 20 nm thick-samples), we applied a sweep between 0.5 and 0.5 V (respectively between 1 and 1 V, and between 1.5 and 1.5 V). This low field amplitude allowed to study the intrinsic properties of the dielectric constant without being disturbed by leakage currents and charge trapping. The sufficiently large device area, 20,000 lm2, enabled to measure accurately the small capacitance variations as a function of the applied voltage without being limited by the tool resolution. 2.3. Capacitance measurements The voltage non-linearity is classically described by the polynomial law [4] CðV a Þ ¼ C 0 ð1 þ C 1 V a þ C 2 V 2a Þ
ð1Þ
15
10
5
0 0.04
DC 0 CðEa Þ þ CðEa Þ 2Cð0Þ ¼ C 2 d 2 E2a 0 ðE a Þ ¼ 2Cð0Þ C
0.08
0.12
ð2Þ
The applied electric field Ea corresponds to the ratio between the voltage drop across the dielectric Va and the dielectric thickness d. We have plotted the capacitance variation as a function of the applied electric field in Fig. 3. Whatever the thickness, the characteristics are merged. From Eq. (2), it proves that the product C2d2 is constant whatever the thickness. On the contrary, the C1/C0 distributions of Fig. 4, for the three Al2O3 thicknesses, are not merged at all, which means that theC1d value depends on the dielectric thickness. C1 is thus not an intrinsic material parameter but
Cumulative distribution (%)
Capacitance under 0 V (fF/µm2)
where Va is the applied voltage between the two electrodes. The three coefficients C0, C1 and C2 have been compared for all the samples.
First, Fig. 2 shows that the measured capacitance density at 0 V, C0, perfectly follows the classical e0e(0)/d law, where e0 is the vacuum permittivity and e(0) is the dielectric permittivity at 0 V. From Fig. 2, we extracted e(0) = 8.3 at 25 C. This is in perfect agreement with the alumina permittivity reported in [5]. Then as far as C2 is concerned, we defined the capacitance variation at a given applied electric field Ea by Eq. (2). It is defined in such a way that it does not depend on the C1 coefficient.
100 80 60 40 20 0 -0.1
-0.08
-0.06
-0.04
-0.02
C1/C0(in m2/(F.V) ) 0.16
d-1 (nm-1) Fig. 2. Capacitance at 0 V as a function of the inverse of the dielectric thickness, T = 25 C.
Fig. 4. C1/C0 cumulative distributions. The values have been measured at 25 C, on nine different capacitors for each thickness. The cross, square and circle symbols correspond, respectively, to the 7, 13 and 20 nm thick Al2O3.
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
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we can rather attribute the C1Va term, in Eq. (1), to the presence of charges in the dielectric. From Figs. 2–4, we can conclude that, only the constant and the quadratic terms of the C–Va polynomial law are intrinsic material parameters that define its dielectric permittivity. Thus, neglecting any first order term, the dielectric permittivity electric field dependence can be expressed by eðEa Þ ¼
aE2a
þ eð0Þ
ð3Þ
in which a is the quadratic coefficient of the dielectric constant. From experimental results of Fig. 3, a = 2.2 ± 0.2 · 103 · (MV/cm)2 at room temperature. The ±10% error has been estimated from the maximum difference between the three curves. 3. Dielectric constant model In pure Al2O3, for frequencies below the infrared domain, two polarization mechanisms coexist: the electronic and the ionic polarization. The first one is due to the polarization of the electronic cloud of all atoms which compose the material. This contribution is thus present in all materials, and gives rise to their refractive index RI. The second comes from the Al–O bonds distortion under the applied electric field. Assuming that neither electric field, nor temperature impacts the alumina refractive index compared to the ionic susceptibility vion, the relative dielectric constant takes the form v e ¼ ion þ RI 2 ð4Þ e0 In this section, we will describe an ionic susceptibility model based on a classical electrostatic approach to estimate both the capacitance at 0 V, C0, and the capacitance 0 variations, DC , due to the electric field. C0 3.1. Local environment and ionic susceptibility In amorphous alumina, most of the Al atoms are surrounded by oxygen atoms in a tetrahedral configuration so that the elementary unit of the system mainly consists
a
b
x Induced dipole
Fig. 6. Under a dc electric field the Al ion is displaced from its equilibrium position (a) which breaks the symmetry and induces a dipole (b).
of a slightly distorted (AlO4)5 tetrahedron [7]. Considering that dielectric properties are linked to the short-range environment of the material, we have modelled the whole system like it is represented in Fig. 5: the elementary tetrahedron can only take two positions with respect to the MIM electrodes. These two positions are referred by the numbers 1 and 2 in Fig. 5. For example, if we only consider the type 1 tetrahedrons, it is easy to describe the main polarization mechanism in alumina: it comes from the Al–O bonds distortion due to the voltage application between the metallic electrodes (see Fig. 6). This is a common phenomenon that exists in all non polar dielectrics: the electric dipole is induced by the electric field perturbation (see Fig. 6b). The full demonstration of the ionic susceptibility expression has been developed in [6] and it is given by Eqs. (5) and (6). vion ðEloc ; T Þ ¼
Nk B T o2 flnðJ 1 ðEloc ; T ÞÞ 2 oE2loc þ lnðJ 2 ðEloc ; T ÞÞg
J1 andJ2 are mathematical entities which express the fact that the Al ion displacement x can take all values allowed by the geometry. Z us ðxÞ 2eEloc J 1;2 ðEloc ; T Þ ¼ x dx ð6Þ exp exp kBT kBT x Eloc is the local field which is the superposition of the applied electric field Ea and the dipolar field due to surrounding molecules, T is the temperature, N is the Al cations
Aluminum
Self-energy us (eV)
2 Oxygen
4 0 -4 -8
Bottom electrode
ð5Þ
8
Top electrode
Al2O3 dielectric
Al3+
Eloc > 0
Eloc = 0
Top electrode
1
O2-
Bottom electrode
Fig. 5. The Al2O3 amorphous dielectric is modelled by tetrahedrons that can only take two positions with respect to the electrode direction.
-2
0
2
4
6
8
10
Bond distortion x (Å) Fig. 7. Self-energy as a function of the bond distortion x. T = 25 C, m = 9, and n = 1.
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
k¼
Eloc Ea
ð7Þ
According to Eqs. (5) and (6), we only need to express the self-energy to calculate the ionic susceptibility as a function of the electric field and the temperature. 3.2. Ionic bond With the quite simple elementary cell geometry of Fig. 6, and using a Mie–Gru¨neisen interatomic potential [9] given by Eq. (8), to model the Al–O bond interactions, the selfenergy us follows Eq. (9):
mn 1 r0 m 1 r0 n ; m>n mn m r n r 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 r 2 d OO A 0 x þ3 us ðxÞ uMG ðr0 þ xÞ þ 3 uMG @ 3 4 uMG ðrÞ ¼ u0
ð8Þ ð9Þ
In Eq. (8), u0 and r0 are, respectively, the Al–O bond energy and the Al–O bond length at equilibrium, and m and n are the repulsive and the attractive exponents of the potential, respectively. In Eq. (9), dOO is the distance between two oxygen neighbors, and xis the distortion of the bond between the central Al cation and the oxygen ion at the tetrahedron summit (see Fig. 6b). For geometrical reasons, considering an isolated elementary cell, the distortion amplitude can theoretically take all values between r0 and the infinity. In Fig. 7, we have plotted the self-energy as a function of x: the minimum potential is obviously reached for the equilibrium position x = 0; when the Al atom moves closer to the oxygen at the tetrahedron summit (x < 0) the self-energy rapidly increases, whereas it converges to 0 when x tends to infinity.
E loc= 0
9
6
E loc= 5 MV/cm 3 0 -1
- 0.5
0
0.5
m n r0 dOO RI at 546 nm N
9 1 1.9 4.1 1.77 3.2 · 1022 cm3
It is also interesting to represent the distortion amplitude distribution: the probability that a distortion x happens is given by Eq. (10). loc Þ exp W kðx;E T B ð10Þ P ðx; Eloc Þ ¼ R loc Þ exp W kðx;E x T B where W(x, Eloc) = us(x) ± 2eElocx is the total potential energy of a tetrahedron. In Fig. 8, one can notice that only very small bond distortions, of a tenth of angstrom order of magnitude, are likely to happen. If no perturbation is applied, the maximum probability is centred at equilibrium position x = 0, whereas when an electric field is applied the distribution changes and its maximum is shifted. Even under strong electric fields, the distribution changes are very small,
12
10
1
Bond distortion x (Å) Fig. 8. Distortion amplitude distributions: with no electric field (full line) and under a 5 MV/cm electric field (dots). T = 25 C, m = 9, and n = 1.
8
6 1
1.5
2
2.5
Al–O bond energy (eV) Fig. 9. Dielectric constant at 0 V and 25 C as a function of the Al–O bond energy.
Quadratic coefficient α (in units of 10-3× (MV/cm)-2)
Probability (%)
12
Table 1 Model parameters
Dielectric constant ε(0)
density, kB is the Boltzmann constant, and e is the elementary electric charge. In the latter equation, us is the self-energy of an elementary cell that is composed of one Al atom plus his first neighbors. The local field is generally a linear function of the external electric field [8]. In the following, we introduce a local field factor k, which is the ratio between the local and the applied electric field
2425
3.5 2.9 2.3 1.7 1.1 0.5
1
1.5
2
2.5
Local field factor Fig. 10. Quadratic coefficient a as a function of the local field factor at 25 C for u0 = 1.5 eV.
S. Be´cu et al. / Microelectronic Engineering 83 (2006) 2422–2426
a
b
12.0
Capacitance variation (ppm)
Capacitance density at 0 V (fF/µm 2)
2426
10.5 9.0 7.5 6.0 4.5 3.0
0
25
50
75
100
125 150 175
2000 1600 1200
Increasing temperatures
800 400 0 -0.8
Temperature (°C)
-0.4
0.0
0.4
0.8
Applied electric field (MV/cm)
Fig. 11. Comparison between model (in full lines) and experiment (in symbols). (a) C0versus temperature for d = 7 nm in cross, 13 nm in squares and 20 nm in circles. (b) Capacitance variations versus applied electric field for d = 13 nm at three temperatures (T = 25 C in cross, 75 C in squares and 125 C in circles).
which lets augur that the ionic susceptibility field dependence is weak. 4. Results and discussion 4.1. Fit parameters choice at room temperature Among the physical parameters required to calculate the dielectric constant, both the Al-O bond energy u0 and the local field factor k are unknown for amorphous alumina. The other parameters have been tabulated in [6], and are reported in Table 1. In Fig. 9, the static dielectric constant e(0) at 25 C has then been plotted as a function of the bond energy. With u0 = 1.5 eV, the calculated static dielectric constant at 25 C is 8.3 like the experimental value. The local field factork has been chosen to best fit the quadratic coefficient a at 25 C. Taking into account the 10% error made on the experimental extraction of a (see Section 2.3), the local field factor has been found to be between 2 and 2.2. This local field factor value is between 17% and 25% higher than the value reported in [6], but it is still in between the Onsager (1.4) and Lorentz (3.3) model predictions [8] (see Fig. 10).
Based on results depicted in Fig. 3, this means that the model well predicts the capacitance variations as a function of the applied voltage whatever the thickness, from 25 to 125 C. 5. Conclusion We reported on the temperature and electric field effects on Al2O3 MIM capacitance for three dielectric thicknesses. Concerning the alumina dielectric permittivity, the thickness study has allowed to identify two intrinsic parameters: first the static term e(0) and secondly, the field quadratic coefficient a. The C–V characterizations showed the existence of a first order term that we have associated to extrinsic charges. Both intrinsic parameters have been successfully modelled using a classical electrostatic approach. The model fits very well the field and the temperature effects on alumina dielectric constant. As a consequence, our first assumption, which asserts that both the electric field and the temperature dependences of Al2O3 permittivity comes from the Al–O bond ionicity, is consistent with the experimental results.
4.2. Model versus experiment References To check the model validity, we have compared it to the experiment for different temperature conditions. From the previous considerations, we have fixed u0 = 1.5 eV, and k = 2. The results, in terms of capacitance, are depicted in Fig. 11. The temperature dependence of C0 is very well predicted between 25 and 150 C, whatever the dielectric thickness (Fig. 11a). The capacitance variations have only been compared to the model for the 13 nm-thick Al2O3 sample. The measured capacitance variations, as a function of the applied electric field, exactly follow the model predictions for the three calculated temperatures (Fig. 11b).
[1] H. Hu et al., IEEE Electron Dev. Lett. 23 (9) (2002) 514–516. [2] K.H. Allers, in: Proceedings of the 2003 BCTM , September 2003, pp. 35–38. [3] C. Durand et al., J. Vac. Sci. Technol. A 22 (3) (2004) 655–660. [4] M. Gros-Jean, in: Proceedings of 12th Workshop on Dielectrics in Microelectronics, 2002, pp. 73–76. [5] T.W. Hickmott, J. Appl. Phys. 93 (2003) 3461. [6] S. Be´cu et al., Appl. Phys. Lett. 88 (2006) 052904. [7] G. Gutie´rrez, Phys. Rev. B 65 (2002) 104202. [8] C.J.F. Bo¨ttcherTheory of Electric Polarization, Vol. 1, Elsevier, Amsterdam, 1973. [9] J.W. Mac Pherson, J. Appl. Phys. 95 (2004) 8101.