Thermo opto mechanical properties of AlN
IOP PUBLISHING
JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 41 (2008) 172001 (4pp)
doi:10.1088/0022-3727/41/17/172001
FAST TRACK COMMUNICATION
Thermo-opto-mechanical properties of
AlN nanostructures: a promising material
for NEMS applications
G Guisbiers and L Buchaillot
IEMN, UMR8520, Scientific City, Avenue Henri Poincaré, BP 60069, 59652 Villeneuve d’Ascq, France
E-mail: gregory.guisbiers@isen.iemn.univ-lille1.fr
Received 11 July 2008, in final form 14 July 2008
Published 8 August 2008
Online at stacks.iop.org/JPhysD/41/172001
Abstract
The properties of aluminium nitride (AlN) are investigated at the nanoscale for different shapes of
nanostructures. Spherical nanoparticles, cylindrical nanowires and nanofilms are the considered shapes.
The size and shape effects on the creep temperature, melting temperature, residual stress and energy
bandgap are discussed. The creep behaviour of AlN is analysed and compared with aluminium (Al) pure
metal. The higher creep resistance of AlN is demonstrated. The transition from inverse Hall–Petch to the
Hall–Petch relation is found to be around ∼15–18 nm in agreement with other authors. The energy
bandgap of AlN nanostructures is blue-shifted with the size reduction and the shape of the nanostructure,
according to the following relation: Eg (nanoparticle) > Eg (nanowire) > Eg (nanofilm).
(Some figures in this article are in colour only in the electronic version)
Table 1. Materials parameters for aluminium and aluminium
nitride [33].
1. Introduction
Aluminium nitride (AlN) is a III–V semiconductor material
and has many attractive properties [1] (table 1) such as high
melting temperature, high thermal stability, high thermal
conductivity, low thermal expansion coefficient and a wide
energy bandgap which make it a promising material for
nano/micro applications [2, 3] and bulk acoustic wave (BAW)
devices [4,5] as resonators and filters. A wide energy bandgap
as AlN is often desired for optical applications in deep ultraviolet (UV) spectra such as light emitting diodes (LEDs)
and laser diodes [6]. Indeed, to improve the resolution in
photolithography, sources with shorter emission wavelength
are required in nano/microelectronics. Moreover, creep is
an important issue for metallic parts in nano/micro-electromechanical systems (N/MEMS) [7]. Aluminium (Al) is
a material extensively used in N/MEMS, for example, as
bridge material in radio frequency MEMS (MEMS-RF).
Nevertheless, this material flows essentially by creep due to
its low melting point (a high melting point material has low
creep). Therefore, to replace Al and to stay compatible with
Al-etch processing which is in use and well characterized, AlN
0022-3727/08/172001+04$30.00
Material parameters
AlN
Al
Tm,∞ (K)
Hm,∞ (J m−3 )
γs (J m−2 )
γl (J m−2 )
E (GPa)
ν
α (K−1 )
η (Pa s)
Eg,∞ (eV)
3273
8.75 × 109
0.66
0.04
308
0.287
4.84 × 10−6
>1.73 × 10−3
6.2
933
1.08 × 109
1.15
0.86
70
0.340
23.03 × 10−6
1.73 × 10−3
0
seems to be an interesting alternative. Therefore, we propose in
this communication to investigate the thermo-opto-mechanical
properties of AlN at the nanoscale.
2. Melting temperature of AlN and
Al nanostructures
Using a thermodynamical model, fully described in [8–11],
we can calculate the size and shape effect on the melting
1
© 2008 IOP Publishing Ltd
Printed in the UK
J. Phys. D: Appl. Phys. 41 (2008) 172001
Fast Track Communication
Table 2. Shape parameters for aluminium and aluminium nitride.
Shape parameters
AlN
Al
αsphere (nm)
αcylinder (nm)a
αfilm (nm)
0.43
0.29
0.14
1.56
1.10
0.52
3. Higher creep resistance of AlN towards Al
Creep of materials is a time and temperature dependent
inelastic deformation process at constant stress [13–15]. The
evolution of creep with time can be separated into three
different stages: primary creep, secondary creep and ternary
creep. Each stage corresponds to a specific creep rate:
ε̇primary ∝ L−3 , ε̇secondary ∝ L0 , ε̇ternary ∝ L−2 , where L is
the dimension of the material. The primary, secondary and
ternary creep are, respectively, due to grain boundary diffusion
(Coble creep), dislocation movements (steady-state creep)
and bulk diffusion (Nabarro-Herring creep). At low sizes,
the creep is dominated by the diffusion process. The creep
temperature where creep becomes non-negligible is a function
of the melting temperature of the material as formulated here:
a
Calculated for a cylindrical nanowire with a length
equal to 100 nm.
3300 AlN
3200
3100
Tm (K)
3000
2900
1000
Tc = kTm ,
(2)
Al
where k depends on the nature of the material, metal
or semiconductor (k ∼ 0.3 for metals and ∼0.4 for
semiconductors) [16]. At the macroscale, creep can be
neglected when the following condition is satisfied, T <
kTm,∞ . At the nanoscale, this condition becomes T < kTm
[17]. Here we have Tm (AlN) > Tm (Al), which means that the
creep temperature of AlN is higher than the corresponding one
of Al for any considered shape of nanostructures. Therefore,
AlN is more creep resistant than Al whatever the size and shape
of the material.
900
800
Bulk
Nanofilm
Nanocylinder
Nanosphere
700
600
500
10
100
L (nm)
Figure 1. Melting temperature versus the dimension of the
nanostructure for aluminium (Al) and aluminium nitride (AlN)
materials. The thick solid lines represent the bulk behaviour of these
materials. The thin solid, dashed and dotted lines represent the
nanofilms, cylindrical nanowires and spherical nanoparticles
respectively.
4. Residual stress and hardness of AlN and Al
nanostructures
Size is known to have a significant effect on the mechanical
behaviour of materials, in particular, on the residual stress,
yield stress and hardness [14,18,19]. Residual stresses appear
during the fabrication of nanostructures even without applying
any external stress [2, 19–26]. The yield stress corresponds
to the stress where the material quits the elastic deformation
regime. And the yield stress is approximately one-third of the
Vickers hardness [27].
Residual stresses have two origins: intrinsic and thermal.
Intrinsic stress, also called growth stress, depends on the
deposition process. Yacaman [28] showed that the intrinsic
stress increases with the size until a given size where a stress
release mechanism should dominate. Thermal stress appears
during the cooling phase of the fabrication process due to
the difference between the thermal expansion coefficients of
the deposited material and the substrate. Considering freestanding nanostructures (no substrate), we have only to take
into account the intrinsic stress origin. For metals deposited
by the evaporation process, we can calculate the intrinsic stress
with the following equation [19]:
temperature of free-standing materials for sizes higher than
L ∼ 4 nm to keep the thermodynamics arguments valid. The
melting temperature of a nanostructure can be calculated by
Tm /Tm,∞ = 1 + [(γl − γs )/Hm,∞ ][A/V ], where Tm,∞ is
the bulk melting temperature, γl and γs are, respectively, the
surface energy in the liquid and solid phases, Hm,∞ is the
bulk melting enthalpy and A/V is the area over volume ratio.
To quantify the size effect on the melting temperature, for a
specific shape, with only one parameter, a shape parameter
called αshape is defined as αshape = [(γs −γl )/Hm,∞ ](AL/V ).
Therefore, the melting temperature of a nanostructure can be
rewritten as
Tm /Tm,∞ = 1 − αshape /L,
(1)
where L is the diameter or thickness of the nanostructure.
For free-standing nanostructures, the ratios αfilm /αsphere and
αcylinder /αsphere are, respectively, equal to 1/3 and 2/3 (table 2)
as already discussed by Wautelet [12]. In figure 1, the
nanoscale melting temperature of Al and AlN is plotted versus
the size. As already noted for transition metals, the size effect
is lower on high melting point materials compared with low
melting point materials [11]. For any shape considered here,
the nanoscale melting temperature of AlN is always higher
than the corresponding one of Al.
σ = (E/1 − ν)α(Tm − Troom )e−Eh/ηkTm ,
(3)
where h and k have the usual meaning. Troom is the room
temperature considered to be around ∼300 K. E, ν, α and η
are, respectively, Young’s modulus, the Poisson ratio, the
2
J. Phys. D: Appl. Phys. 41 (2008) 172001
Fast Track Communication
thermal expansion coefficient and the dynamic viscosity of
the material.
For ceramics, as the relaxation process is different from
metals, we cannot use equation (4) to calculate the intrinsic
stress due to the exponential factor which corresponds to a
creep-yielding relaxation process. Indeed ceramics do not have
a yielding part in the stress–strain curve and they relax directly
by cracks [27]. Therefore, we propose equation (4) to calculate
the intrinsic stress in ceramics before cracking:
σ = (E/1 − ν)α(Tm − Troom ).
(4)
To be in the condition to use equation (4), we consider
AlN nanostructures built from evaporated Al material under
nitrogen atmosphere.
For a spherical nanoparticle with a size equal to 10 nm,
we get an internal stress around ∼100 MPa for Al and ∼6 GPa
for AlN. In a previous paper [11], we have indicated, for sizes
below ∼15 nm, the link between the yield stress (hardness)
of material and the intrinsic residual stress. As also noted
by Meyers [14], the stress first increases before decreasing as
the grain size increases. The grain size corresponding to the
maximum stress value depends on the strain rate.
Furthermore, Lucas et al [29] have studied the nitrogen
diffusion effect on the hardness of aluminium coating. Their
results are interesting; they noted an increase in hardness
with nitrogen incorporation into aluminium coating. Nitrogen
forms with aluminium precipitates of aluminium nitride into
the coating. These AlN precipitates act like a barrier to
dislocation motions: cutting or bowing mechanisms. In the
aluminium coating, the stress required to bend dislocations
around an AlN precipitate of diameter 2 nm, called the bowing
stress, is around 7.8 GPa [29]. This value can be compared
with the internal stress value (∼6 GPa) which is quite similar.
The transition between the two dislocations motions occurs
when the shear stress equals the bowing stress. This transition
occurs at L ≈ 18 nm [29]. The maximal strength is obtained
when the shear stress is equal to the bowing stress. This
transition can explain the inverse Hall–Petch behaviour where
dislocation motion seems to play a very important role.
Below ∼15–18 nm, there is no relaxation process, and the
stress/hardness increases with the size. Above ∼15–18 nm,
the dislocations decrease the hardness of the material. For
Zhao et al [30], the transition between the inverse Hall–Petch
relation and the Hall–Petch relation occurs between 15 and
30 nm, which is quite the same as ours.
As a harder material is more creep resistant, this
calculation confirms the superior creep resistance behaviour
of AlN towards Al. Furthermore, a harder material exhibits
a higher fatigue endurance limit [31]; therefore, AlN is
particularly suitable as bridge material in MEMS-RF.
Figure 2. Energy bandgap versus the diameter of the aluminium
nitride (AlN) nanoparticle, nanocylinder (h = 100 nm) and versus
the thickness of the nanofilm. The bulk energy bandgap is indicated
at 6.2 eV. This figure highlights the blue shift with size reduction.
where Eg,∞ is the energy bandgap of the bulk semiconductor.
It can be easily understood by writing the energy bandgap as a
function of the enthalpy and entropy between the conduction
and valence electronic bands: Eg = Hcv − T Scv .
In figure 2, the energy bandgap of AlN has been plotted
versus the size of free-standing nanostructures. A blue shift
appears with size reduction for all considered nanostructures.
For a given size, the blue shift magnitude varies with the
shape of the nanostructure, according to the following relation:
Eg (nanoparticle) > Eg (nanowire) > Eg (nanofilm).
6. Conclusions
The following properties of aluminium nitride: melting
temperature, creep temperature, intrinsic residual stress and
energy bandgap, have been investigated at the nanoscale. Not
only the size but also the shape of the nanostructure has an
influence on the material properties. From a thermodynamical
analysis, the higher creep resistance of AlN towards Al has
been demonstrated for all sizes and shapes of nanostructures.
The free-standing AlN nanostructures, with an energy bandgap
of >6.2 eV, are particularly well adapted for deep UV
nano/micro-applications. With the same shape parameter,
αshape , it allows one to calculate the melting temperature,
creep temperature, intrinsic residual stress and energy bandgap
for a given material at the nanoscale. Nevertheless, some
other effects such as segregation and composition can slightly
modify the surface properties such as surface tension and then
the related nano-properties of AlN. These effects will be the
subject another work.
5. Energy bandgap of AlN nanostructures
It has been detailed in [32] that with the same αshape parameter
(defined in section 1), we can describe the size effect on the
energy bandgap of semiconductors, Eg , with the following
equation:
Eg /Eg,∞ = 1 + αshape /L,
(5)
Acknowledgment
G Guisbiers thanks the ANR PNANO M&NEMS project for
financial support.
3
J. Phys. D: Appl. Phys. 41 (2008) 172001
Fast Track Communication
[18] Arzt E 1998 Acta Mater. 46 5611
[19] Guisbiers G, Van Overschelde O and Wautelet M 2007 Acta
Mater. 55 3541
[20] Ohring M 2002 Materials Science of Thin Films Deposition
and Structure (New York: Academic)
[21] Thornton J A and Hoffman D W 1989 Thin Solid Films 171 5
[22] Thompson C V and Carel R 1996 J. Mech. Phys. Solids
44 657
[23] Nix W D and Clemens B M 1999 J. Mater. Res. 14 3467
[24] Koch R 1994 J. Phys.: Condens. Matter 6 9519
[25] Doerner M F and Nix W D 1988 Crc Crit. Rev. Solid State
Mater. Sci. 14 225
[26] Freund L B and Chason E 2001 J. Appl. Phys. 9 4866
[27] Gere J M 2002 Mechanics of Materials (Cheltenham, UK:
Nelson Thornes)
[28] Yacaman M J, Ascencio J A, Liu H B and Gardea-Torresdey J
2001 J. Vac. Sci. Technol. B— Microelectron. Nanometer
Struct.—Process. Meas. Phenom. 19 1091
[29] Lucas S, Chevallier J and Chechenin N G 1994 Surf. Coat.
Technol. 66 334
[30] Zhao M, Li J C and Jiang Q 2003 J. Alloys Compounds
361 160
[31] Wei G, Bhushan B and Jacobs SJ 2004 J. Vac. Sci. Technol. A
22 1397
[32] Guisbiers G, Van Overschelde O and Wautelet M 2008 Appl.
Phys. Lett. 92 103121
[33] Martienssen W and Warlimont H 2005 Springer Handbook of
Condensed Matter and Materials Data (Berlin: Springer)
References
[1] Vurgaftman I, Meyer J R and Ram-Mohan L R 2001 J. Appl.
Phys. 89 5815
[2] Madou M J 2002 Fundamentals of Microfabrication: the
Science of Miniaturization (Boca Raton, FL: CRC Press)
[3] Rao C N R, Deepak F L, Gundiah G and Govindaraj A 2003
Prog. Solid State Chem. 31 5
[4] Loebl H P, Klee M, Metzmacher C, Brand W, Milsom R and
Lok P 2003 Mater. Chem. Phys. 79 143
[5] Löbl H P, Klee M, Milsom R, Dekker R, Metzmacher C,
Brand W and Lok P 2001 J. Eur. Ceram. Soc. 21 2633
[6] Taniyasu Y, Kasu M and Makimoto T 2006 Nature 441 325
[7] Spengen W M V 2003 Microelectron. Reliab. 43 1049
[8] Guisbiers G and Wautelet M 2006 Nanotechnology 17 2008
[9] Abudukelimu G, Guisbiers G and Wautelet M 2006 J. Mater.
Res. 21 2829
[10] Guisbiers G, Abudukelimu G, Clement F and Wautelet M
2007 J. Comput. Theor. Nanosci. 4 309
[11] Guisbiers G, Kazan M, Van Overschelde O, Wautelet M and
Pereira S 2008 J. Phys. Chem. C 112 4097
[12] Wautelet M 1998 Phys. Lett. A 246 341
[13] Tjong S C and Chen H 2004 Mater. Sci. Eng. R 45 1
[14] Meyers M A, Mishra A and Benson D J 2006 Prog. Mater. Sci.
51 427
[15] Bull S J 1998 Surf. Coat. Technol. 107 101
[16] Driver J H 2004 Scr. Mater. 51 819
[17] Guisbiers G and Buchaillot L 2008 Nanotechnology submitted
4
JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 41 (2008) 172001 (4pp)
doi:10.1088/0022-3727/41/17/172001
FAST TRACK COMMUNICATION
Thermo-opto-mechanical properties of
AlN nanostructures: a promising material
for NEMS applications
G Guisbiers and L Buchaillot
IEMN, UMR8520, Scientific City, Avenue Henri Poincaré, BP 60069, 59652 Villeneuve d’Ascq, France
E-mail: gregory.guisbiers@isen.iemn.univ-lille1.fr
Received 11 July 2008, in final form 14 July 2008
Published 8 August 2008
Online at stacks.iop.org/JPhysD/41/172001
Abstract
The properties of aluminium nitride (AlN) are investigated at the nanoscale for different shapes of
nanostructures. Spherical nanoparticles, cylindrical nanowires and nanofilms are the considered shapes.
The size and shape effects on the creep temperature, melting temperature, residual stress and energy
bandgap are discussed. The creep behaviour of AlN is analysed and compared with aluminium (Al) pure
metal. The higher creep resistance of AlN is demonstrated. The transition from inverse Hall–Petch to the
Hall–Petch relation is found to be around ∼15–18 nm in agreement with other authors. The energy
bandgap of AlN nanostructures is blue-shifted with the size reduction and the shape of the nanostructure,
according to the following relation: Eg (nanoparticle) > Eg (nanowire) > Eg (nanofilm).
(Some figures in this article are in colour only in the electronic version)
Table 1. Materials parameters for aluminium and aluminium
nitride [33].
1. Introduction
Aluminium nitride (AlN) is a III–V semiconductor material
and has many attractive properties [1] (table 1) such as high
melting temperature, high thermal stability, high thermal
conductivity, low thermal expansion coefficient and a wide
energy bandgap which make it a promising material for
nano/micro applications [2, 3] and bulk acoustic wave (BAW)
devices [4,5] as resonators and filters. A wide energy bandgap
as AlN is often desired for optical applications in deep ultraviolet (UV) spectra such as light emitting diodes (LEDs)
and laser diodes [6]. Indeed, to improve the resolution in
photolithography, sources with shorter emission wavelength
are required in nano/microelectronics. Moreover, creep is
an important issue for metallic parts in nano/micro-electromechanical systems (N/MEMS) [7]. Aluminium (Al) is
a material extensively used in N/MEMS, for example, as
bridge material in radio frequency MEMS (MEMS-RF).
Nevertheless, this material flows essentially by creep due to
its low melting point (a high melting point material has low
creep). Therefore, to replace Al and to stay compatible with
Al-etch processing which is in use and well characterized, AlN
0022-3727/08/172001+04$30.00
Material parameters
AlN
Al
Tm,∞ (K)
Hm,∞ (J m−3 )
γs (J m−2 )
γl (J m−2 )
E (GPa)
ν
α (K−1 )
η (Pa s)
Eg,∞ (eV)
3273
8.75 × 109
0.66
0.04
308
0.287
4.84 × 10−6
>1.73 × 10−3
6.2
933
1.08 × 109
1.15
0.86
70
0.340
23.03 × 10−6
1.73 × 10−3
0
seems to be an interesting alternative. Therefore, we propose in
this communication to investigate the thermo-opto-mechanical
properties of AlN at the nanoscale.
2. Melting temperature of AlN and
Al nanostructures
Using a thermodynamical model, fully described in [8–11],
we can calculate the size and shape effect on the melting
1
© 2008 IOP Publishing Ltd
Printed in the UK
J. Phys. D: Appl. Phys. 41 (2008) 172001
Fast Track Communication
Table 2. Shape parameters for aluminium and aluminium nitride.
Shape parameters
AlN
Al
αsphere (nm)
αcylinder (nm)a
αfilm (nm)
0.43
0.29
0.14
1.56
1.10
0.52
3. Higher creep resistance of AlN towards Al
Creep of materials is a time and temperature dependent
inelastic deformation process at constant stress [13–15]. The
evolution of creep with time can be separated into three
different stages: primary creep, secondary creep and ternary
creep. Each stage corresponds to a specific creep rate:
ε̇primary ∝ L−3 , ε̇secondary ∝ L0 , ε̇ternary ∝ L−2 , where L is
the dimension of the material. The primary, secondary and
ternary creep are, respectively, due to grain boundary diffusion
(Coble creep), dislocation movements (steady-state creep)
and bulk diffusion (Nabarro-Herring creep). At low sizes,
the creep is dominated by the diffusion process. The creep
temperature where creep becomes non-negligible is a function
of the melting temperature of the material as formulated here:
a
Calculated for a cylindrical nanowire with a length
equal to 100 nm.
3300 AlN
3200
3100
Tm (K)
3000
2900
1000
Tc = kTm ,
(2)
Al
where k depends on the nature of the material, metal
or semiconductor (k ∼ 0.3 for metals and ∼0.4 for
semiconductors) [16]. At the macroscale, creep can be
neglected when the following condition is satisfied, T <
kTm,∞ . At the nanoscale, this condition becomes T < kTm
[17]. Here we have Tm (AlN) > Tm (Al), which means that the
creep temperature of AlN is higher than the corresponding one
of Al for any considered shape of nanostructures. Therefore,
AlN is more creep resistant than Al whatever the size and shape
of the material.
900
800
Bulk
Nanofilm
Nanocylinder
Nanosphere
700
600
500
10
100
L (nm)
Figure 1. Melting temperature versus the dimension of the
nanostructure for aluminium (Al) and aluminium nitride (AlN)
materials. The thick solid lines represent the bulk behaviour of these
materials. The thin solid, dashed and dotted lines represent the
nanofilms, cylindrical nanowires and spherical nanoparticles
respectively.
4. Residual stress and hardness of AlN and Al
nanostructures
Size is known to have a significant effect on the mechanical
behaviour of materials, in particular, on the residual stress,
yield stress and hardness [14,18,19]. Residual stresses appear
during the fabrication of nanostructures even without applying
any external stress [2, 19–26]. The yield stress corresponds
to the stress where the material quits the elastic deformation
regime. And the yield stress is approximately one-third of the
Vickers hardness [27].
Residual stresses have two origins: intrinsic and thermal.
Intrinsic stress, also called growth stress, depends on the
deposition process. Yacaman [28] showed that the intrinsic
stress increases with the size until a given size where a stress
release mechanism should dominate. Thermal stress appears
during the cooling phase of the fabrication process due to
the difference between the thermal expansion coefficients of
the deposited material and the substrate. Considering freestanding nanostructures (no substrate), we have only to take
into account the intrinsic stress origin. For metals deposited
by the evaporation process, we can calculate the intrinsic stress
with the following equation [19]:
temperature of free-standing materials for sizes higher than
L ∼ 4 nm to keep the thermodynamics arguments valid. The
melting temperature of a nanostructure can be calculated by
Tm /Tm,∞ = 1 + [(γl − γs )/Hm,∞ ][A/V ], where Tm,∞ is
the bulk melting temperature, γl and γs are, respectively, the
surface energy in the liquid and solid phases, Hm,∞ is the
bulk melting enthalpy and A/V is the area over volume ratio.
To quantify the size effect on the melting temperature, for a
specific shape, with only one parameter, a shape parameter
called αshape is defined as αshape = [(γs −γl )/Hm,∞ ](AL/V ).
Therefore, the melting temperature of a nanostructure can be
rewritten as
Tm /Tm,∞ = 1 − αshape /L,
(1)
where L is the diameter or thickness of the nanostructure.
For free-standing nanostructures, the ratios αfilm /αsphere and
αcylinder /αsphere are, respectively, equal to 1/3 and 2/3 (table 2)
as already discussed by Wautelet [12]. In figure 1, the
nanoscale melting temperature of Al and AlN is plotted versus
the size. As already noted for transition metals, the size effect
is lower on high melting point materials compared with low
melting point materials [11]. For any shape considered here,
the nanoscale melting temperature of AlN is always higher
than the corresponding one of Al.
σ = (E/1 − ν)α(Tm − Troom )e−Eh/ηkTm ,
(3)
where h and k have the usual meaning. Troom is the room
temperature considered to be around ∼300 K. E, ν, α and η
are, respectively, Young’s modulus, the Poisson ratio, the
2
J. Phys. D: Appl. Phys. 41 (2008) 172001
Fast Track Communication
thermal expansion coefficient and the dynamic viscosity of
the material.
For ceramics, as the relaxation process is different from
metals, we cannot use equation (4) to calculate the intrinsic
stress due to the exponential factor which corresponds to a
creep-yielding relaxation process. Indeed ceramics do not have
a yielding part in the stress–strain curve and they relax directly
by cracks [27]. Therefore, we propose equation (4) to calculate
the intrinsic stress in ceramics before cracking:
σ = (E/1 − ν)α(Tm − Troom ).
(4)
To be in the condition to use equation (4), we consider
AlN nanostructures built from evaporated Al material under
nitrogen atmosphere.
For a spherical nanoparticle with a size equal to 10 nm,
we get an internal stress around ∼100 MPa for Al and ∼6 GPa
for AlN. In a previous paper [11], we have indicated, for sizes
below ∼15 nm, the link between the yield stress (hardness)
of material and the intrinsic residual stress. As also noted
by Meyers [14], the stress first increases before decreasing as
the grain size increases. The grain size corresponding to the
maximum stress value depends on the strain rate.
Furthermore, Lucas et al [29] have studied the nitrogen
diffusion effect on the hardness of aluminium coating. Their
results are interesting; they noted an increase in hardness
with nitrogen incorporation into aluminium coating. Nitrogen
forms with aluminium precipitates of aluminium nitride into
the coating. These AlN precipitates act like a barrier to
dislocation motions: cutting or bowing mechanisms. In the
aluminium coating, the stress required to bend dislocations
around an AlN precipitate of diameter 2 nm, called the bowing
stress, is around 7.8 GPa [29]. This value can be compared
with the internal stress value (∼6 GPa) which is quite similar.
The transition between the two dislocations motions occurs
when the shear stress equals the bowing stress. This transition
occurs at L ≈ 18 nm [29]. The maximal strength is obtained
when the shear stress is equal to the bowing stress. This
transition can explain the inverse Hall–Petch behaviour where
dislocation motion seems to play a very important role.
Below ∼15–18 nm, there is no relaxation process, and the
stress/hardness increases with the size. Above ∼15–18 nm,
the dislocations decrease the hardness of the material. For
Zhao et al [30], the transition between the inverse Hall–Petch
relation and the Hall–Petch relation occurs between 15 and
30 nm, which is quite the same as ours.
As a harder material is more creep resistant, this
calculation confirms the superior creep resistance behaviour
of AlN towards Al. Furthermore, a harder material exhibits
a higher fatigue endurance limit [31]; therefore, AlN is
particularly suitable as bridge material in MEMS-RF.
Figure 2. Energy bandgap versus the diameter of the aluminium
nitride (AlN) nanoparticle, nanocylinder (h = 100 nm) and versus
the thickness of the nanofilm. The bulk energy bandgap is indicated
at 6.2 eV. This figure highlights the blue shift with size reduction.
where Eg,∞ is the energy bandgap of the bulk semiconductor.
It can be easily understood by writing the energy bandgap as a
function of the enthalpy and entropy between the conduction
and valence electronic bands: Eg = Hcv − T Scv .
In figure 2, the energy bandgap of AlN has been plotted
versus the size of free-standing nanostructures. A blue shift
appears with size reduction for all considered nanostructures.
For a given size, the blue shift magnitude varies with the
shape of the nanostructure, according to the following relation:
Eg (nanoparticle) > Eg (nanowire) > Eg (nanofilm).
6. Conclusions
The following properties of aluminium nitride: melting
temperature, creep temperature, intrinsic residual stress and
energy bandgap, have been investigated at the nanoscale. Not
only the size but also the shape of the nanostructure has an
influence on the material properties. From a thermodynamical
analysis, the higher creep resistance of AlN towards Al has
been demonstrated for all sizes and shapes of nanostructures.
The free-standing AlN nanostructures, with an energy bandgap
of >6.2 eV, are particularly well adapted for deep UV
nano/micro-applications. With the same shape parameter,
αshape , it allows one to calculate the melting temperature,
creep temperature, intrinsic residual stress and energy bandgap
for a given material at the nanoscale. Nevertheless, some
other effects such as segregation and composition can slightly
modify the surface properties such as surface tension and then
the related nano-properties of AlN. These effects will be the
subject another work.
5. Energy bandgap of AlN nanostructures
It has been detailed in [32] that with the same αshape parameter
(defined in section 1), we can describe the size effect on the
energy bandgap of semiconductors, Eg , with the following
equation:
Eg /Eg,∞ = 1 + αshape /L,
(5)
Acknowledgment
G Guisbiers thanks the ANR PNANO M&NEMS project for
financial support.
3
J. Phys. D: Appl. Phys. 41 (2008) 172001
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