1107
47. Silicon on Mechanically Flexible Substrates
for Large-Area Electronics
Silicon on Mec
Low-temperature thin-film semiconductors and
dielectrics are critical requirements for large-area
electronics, including displays and imagers.
Despite the presence of structural disorder,
these materials show promising electronic
transport properties that are vital for devices
such as thin-film transistors (TFTs). This chapter
presents an overview of material and transport
properties pertinent to large-area electronics
on mechanically flexible substrates. We begin
with a summary of process challenges for lowtemperature fabrication of a-Si:H TFTs on plastic
substrates, followed by a description of transport
properties of amorphous semiconducting films,
along with their influence on TFT characteristics.
The TFTs must maintain electrical integrity under
mechanical stress, induced by bending of the
a-Si:H TFTs on Flexible Substrates ......... 1108
47.2 Field-Effect Transport
in Amorphous Films ............................. 1108
47.2.1 Localized and Extended States .... 1109
47.2.2 Density of States (DOS) ............... 1110
47.2.3 Effective Carrier Mobility ............ 1110
47.3 Electronic Transport
Under Mechanical Stress....................... 1113
47.3.1 Thin-Film Strain Gauges............. 1114
47.3.2 Strained Amorphous-Silicon
Transistors................................ 1116
References .................................................. 1118
flexible substrates. Bending-induced changes are
not limited to alteration of device dimensions and
involve modulation of electronic transport of the
active semiconducting layer.
ditions. The instability of TFTs stems from structural
and interfacial disorder in the semiconductor and insulator layers, and interfaces. This leads to a shift
in the threshold voltage ∆VT under prolonged bias
stress in a-Si:H [47.9] and organic [47.10] TFTs.
To deal with the shift in device parameters, pixel
circuits and driving schemes that can provide compensation have been proposed and are well under
investigation [47.11]. The lifetime of the OLED, on
the other hand, can be enhanced by efficient thin-film
encapsulation layers with water-vapor permeation rates
< 10−6 g/m2 /day at 25 ◦ C and less than 40% relative
humidity [47.12], which are imperative for active matrix organic light-emitting diode (AMOLED) displays
on plastic substrates.
This chapter presents an overview of material and
transport properties pertinent to mechanically flexible amorphous-silicon electronics. We begin with
an overview of low-temperature fabrication of highperformance a-Si:H transistors and associated processing challenges. The transport properties of amorphous
semiconducting films are then presented, along with
their influence on TFT characteristics. We also examine the impact of external mechanical stress,
Part E 47
There is growing interest in low-temperature thinfilm semiconductors and dielectrics for a variety
of applications such as organic light-emitting diode
(OLED) displays and lighting modules [47.1, 2], solar cells [47.3], and digital imagers [47.4, 5]. Interest
in these materials is driven by the promise of lowcost roll-to-roll manufacturing [47.6]. They enable
high-performance electronic devices including thin-film
transistors (TFTs), OLEDs, and p-i-n photodetectors
on large-area mechanically flexible plastic substrates.
OLEDs are becoming increasingly attractive as emission devices due to their fast response, high conversion
efficiency, wide viewing angle, and compatibility with
plastic substrates [47.7]. In high-information-content
displays, the OLED must be integrated with a TFT
circuit that provides stable drive currents [47.8].
Here, low-temperature a-Si:H TFTs or (solutionprocessed/vacuum-deposited) organic TFTs can be used
to realize these circuits.
In addition to advances in material deposition
and integration technologies, significant research efforts are being undertaken to improve the lifetime
of the TFT and the OLED in view of instabilities
induced by bias stress and exposure to ambient con-
47.1
1108
Part E
Novel Materials and Selected Applications
induced by bending of the flexible substrate, on the
characteristics of thin-film devices, including strain
gauges and TFTs. The shifts are not only limited to
changes in device dimensions but also to a modulation of electronic transport of the active semiconducting
layer.
47.1 a-Si:H TFTs on Flexible Substrates
Compared to conventional (rigid) glass substrates,
mechanically flexible substrates are attractive because of
their reduced fragility and weight [47.13]. Polyethylene
terephthalate (PET) [47.14] and polyimide [47.15] are
examples of such substrates. Most plastic substrates have
a low glass-transition temperature Tg (< 250 ◦ C). Consequently, the device processing temperatures must be
reduced without compromising the electronic properties
of the deposited materials. Despite the adverse effects of
reducing the processing temperature, high-performance
low-temperature (< 150 ◦ C) processes have been developed for fabrication of a-Si:H TFTs on flexible
substrates [47.14–16], and even lower temperature processes (75 ◦ C) are under investigation [47.17].
One of the key challenges in low-temperature fabrication is to achieve a high-quality a-SiOx [47.17] or
a-SiNx :H [47.16] gate and/or passivation dielectric with
low defect density, leakage, and interface state density. Nitrogen rich (x = 1.6 to 1.7) a-SiNx :H films with
high resistivity (ρ > 1015 Ω-cm) and breakdown field
(> 5 MV/cm) have been deposited using conventional
13.56 MHz plasma-enhanced chemical vapor deposition
(PECVD) at 120 ◦ C [47.16].
The fabrication challenges are not limited to the electrical quality of the deposited materials. Most plastic
substrates have a large coefficient of thermal expansion
(CTE) (> 50 ppm/◦ C), which is much higher than that
of the TFT layers (≈ 4 ppm/◦ C) [47.18]. As the deposited films go through a temperature drop ∆T upon
cooling from the process temperature [47.2], thermal
stress σth = Yf ∆C∆T develops in the film [47.19]. Here,
∆C is the difference between CTEs of the substrate
and the film, and Yf denotes the biaxial elastic modulus of the deposited film. The thermal stress bends
the composite structure, giving rise to handling problems. Thus, the substrate needs to be held flat in a frame
or glued to a rigid substrate [47.2]. In addition, since
most layers are patterned during processing, the thermal
stress exhibits a local pattern-dependent distribution,
which leads to problems in alignment of consecutive
masks. Lowering the process temperature and choosing a substrate with a smaller CTE, e.g. polyethylene
naphthalate (PEN) with a CTE of 13 ppm/◦ C [47.18],
will help alleviate the thermal stress, and associated
alignment issues. It is important to note that thin
rigid (steel, aluminum, or glass) substrates are also
mechanically flexible and are being considered for device fabrication [47.20]. Although these substrates can
withstand higher temperatures, handling issues still remain and depend on the desired degree of mechanical
flexibility.
Recent work on TFT fabrication on flexible substrates is starting to demonstrate successful handling
of material processing issues. For instance, TFTs fabricated on plastic substrate at 150 ◦ C have been reported
to have parameter values very similar to the high-temperature counterparts in terms of low reverse and high on
current (with switching ratio of > 108 ), high field-effect
mobility > 0.8 cm2 /Vs, threshold voltage of ≈ 1–2 V,
and low subthreshold slope < 0.3 V/dec [47.16].
Part E 47.2
47.2 Field-Effect Transport in Amorphous Films
The electronic properties of a semiconducting film are
primarily determined by its structural disorder and grain
size, which strongly depend on deposition conditions
and postprocessing treatment [47.21].
The structure of disordered semiconductors such
as a-Si:H lacks the long-range periodicity of crystalline silicon. However, the atoms of the materials bond
to their neighbors according to the 8-N rule [47.22]
for both amorphous and crystalline silicon, leading to
short-range structural order. The extent to which this
short-range order is conserved is dependent on processing conditions and leads to different degrees of
disorder. This range can be as small as a few nanometers in amorphous semiconductors or as large as a few
tens of microns in polycrystalline material. Nanocrystalline (nc) and microcrystalline (µc) materials reside
Silicon on Mechanically Flexible Substrates for Large-Area Electronics
between these extremes depending on grain size and
stoichiometry.
The observation of an absorption gap and thermally
activated conductivity in amorphous semiconductors
stems from the short-range order [47.24] and resembles
the band-gap properties of an ideal crystalline semiconductor, which has no states in the gap. The edges of the
gap are well defined and represent discontinuities in the
density of states (DOS). In contrast, the lack of longrange order in the amorphous semiconductor leads to
the presence of states in the gap, and thus, tailing of the
band edges.
47.2.1 Localized and Extended States
In the ideal crystalline semiconductor, the band states
are all extended with a corresponding band mobility
µband . In the case of an amorphous semiconductor, however, the structural disorder of the material leads to
the presence of localized states for which the mobillog g(E)
Extended
states
Localized
states
a)
EV
log µ
EF
EC
E
ity vanishes at T = 0. Anderson [47.25] demonstrated
that, for a sufficiently large degree of disorder, all states
in the band are localized. In his work, the disorder in
the material is modeled by a random but uniform distribution U0 for the energy of the wells at lattice sites.
He found that all of the states of the energy band become localized when U0 /B > 5, where B is the band
width and is given by B = 2Z I [47.22]. Here, Z is
the coordination number, and I the overlap energy for
the states in neighboring wells, which decreases exponentially with distance based on the tight-binding
approximation.
For a small degree of disorder, Anderson’s criteria
may not hold for states with large overlap energies (i. e.
the extended states). Mott and Davis [47.22] have stipulated that, for a given value of U0 , the states can be
localized or extended, depending on their overlap energy. Consequently, for some materials there are distinct
energies which separate the localized states from the extended ones and can be determined from Anderson’s
criteria U0 ≈ 5B. These energies are referred to as mobility edges (or mobility shoulders) due to the significant
change in the mobility expected at these edges. Figure 47.1 illustrates the DOS g(E) for the mobility gap of
a-Si:H indicating the extended and localized states. The
mobility shoulder E C of the conduction band separates
the localized states from the extended ones. A similar
band structure and mobility shoulder is present for the
valence band. This graph is compiled from the work
of Cohen et al. [47.23], who first demonstrated that the
change in mobility at these edges can be of the order of
103 –104 , signifying the different transport mechanisms
on the two sides of the edge.
In the extended states, the carriers have a finite mobility, and thus take part in metallic-type conduction.
This prevails at room temperature for a-Si:H and takes
the form σ0 = 0.06q 2 /~r(E) [47.22], where q is the elementary charge and r(E) the mean distance between the
wells that contribute to states with energy E. In the conduction band, r(E) will be close to the lattice constant,
which yields σ0 = 350 Ω−1 cm−1 . For Fermi energies
E F < E C , the conductivity depends on the Fermi energy
and reads
σ = σ0 exp(E F /kB T ) ,
b)
EV
EC
E
Fig. 47.1a,b Sketch of (a) DOS in valence and conduction
bands, and (b) hole and electron mobilities (after [47.23])
(47.1)
where kB is Boltzmann’s constant and T the temperature, predicting Arrhenius behavior for the conductivity.
Here, E F is defined as negative for the traps with respect
to the conduction band edge E C . This change in conductivity is due to a change in the concentration of carriers,
excited to the extended states n band with a band mobility
1109
Part E 47.2
Mobility
shoulders
47.2 Field-Effect Transport in Amorphous Films
1110
Part E
Novel Materials and Selected Applications
µband such that
σ = qµband n band ,
(47.2)
where
n band = Nb exp(E F /kB T ) ,
(47.3)
with Nb = σ0 /qµband . From the given value for σ0 , the
band mobility can be found as µband ≈ 12 cm2 /Vs for
a-Si:H.
In the localized states, the conduction is only possible by hopping of the carriers to the neighboring
localized states [47.26]. This method of conduction prevails at low temperatures (T < 200 K) and for highly
disordered materials.
The picture of the mobility gap can be completed
by adding defect states in the middle of the gap, as indicated by the dashed lines in Fig. 47.1a. These states
are attributed to the broken or unsaturated bonds such
as dangling bonds in the network structure. The density of dangling bonds is reduced to 1015 –1016 cm−3
in a-Si:H by hydrogen atoms, which passivate some of
these unsaturated bonds [47.24, 27].
47.2.2 Density of States (DOS)
The density of states (DOS) g(E), including the density of band tail and defect states, as schematically
represented in Fig. 47.1a, determines the transport properties of the disordered semiconductor. The DOS in
a-Si:H has been extensively studied by using different experimental techniques such as field-effect
measurements [47.28, 29], photoconductivity measurements [47.30], deep-level transient capacitance
spectroscopy (DLTS) [47.31], and capacitance–voltage–
frequency (C–V – f ) characteristics [47.32, 33]. It is
found that an exponential distribution of the deep and
tail states [47.34] can efficiently describe the distribution of the localized states and hold for a wide range
of materials with rapidly changing DOS. In the case of
a-Si:H, the localized states in the upper half of the mobility gap, closer to E C , behave as acceptor-like states,
while the states in the lower part of the gap, closer to
E V , behave as donor-like states. Acceptor-like states are
neutral when they are empty, and negatively charged after capturing an electron, whereas the donor-like states
are positively charged when they are empty and neutral
after capturing an electron.
In a-Si:H, the number of donor-like states closer to
the valence band is much higher than the number of
acceptor-like states. As a result, following the neutrality
condition, the position of the Fermi energy E i in an
intrinsic a-Si:H sample in the dark lies closer to E C
due to the asymmetrical DOS distribution [47.27]. This
results in a much stronger electron conduction in a-Si:H,
which signifies the role of the conduction band tail in
dispersive electron transport. The density of states for
the conduction band tail g(E) is written as
g(E) = Nt /kB Tt exp(E/kB Tt ) ,
where Nt is the total acceptor-like states in the conduction band tail, and Tt the associated slope of the
exponential state distribution.
log (n)
47.2.3 Effective Carrier Mobility
Nt
Part E 47.2
Tail
states
N0
Reference
concentration
Due to the high density of tail and deep states, only
a small number of carriers are thermally excited to
the extended states and contribute to conduction as described in (47.2). The total carrier density n is the sum
of the excited and trapped carriers such that
n = n band + n t .
Transport
Band
Deep
states
Ea0
EF0
(47.4)
EC
EF
Fig. 47.2 Density of trapped carriers as a function of the
Fermi energy, showing the reference concentration N0 and
transport (conduction) band
(47.5)
To obtain the trapped carrier density n t as a function of
the Fermi energy E F , we integrate the product of the
DOS and the probability of occupation of a state f (E)
over the mobility gap
0
nt =
g(E) f (E) dE .
EV
(47.6)
Silicon on Mechanically Flexible Substrates for Large-Area Electronics
In equilibrium, the probability of occupation can be
described by the Fermi–Dirac function,
f (E) =
1
.
1 + exp [(E − E F )/kB T ]
(47.7)
Equation (47.6) can be numerically solved and approximated along the lines given by Shaw and Hack [47.37]
as
n t (E F ) = Nt exp(E F /kB Tt )u(Tt /T, E F ) .
(47.8)
The underlying assumption is that E F moves no closer
than a few kB T to the mobility edge. This is true because of the high density of tail states, which tends
to pin the movement of the Fermi energy. Here,
u(Tt /T, E F ) represents the changes in the trappedcarrier density with normalized ambient temperature
Tt /T . For T Tt , u(Tt /T, E F ) is often approximated
by [sin(πT/Tt )/πT/Tt ] [47.38] with a value close to 1.
The ratio Tt /T , referred to as the dispersion parameter, characterizes the dispersive transport of electrons in
the conduction band tail, and can be obtained from the
time dependence of the electron drift mobility in timeof-flight experiments [47.39]. The presence of a high
trapped-carrier density in amorphous semiconductors
leads to an effective trapped-carrier mobility that is lower
than the band mobility. Street [47.27] has defined the
drift mobility µD of the carriers as the band mobility reduced by the fraction of time that the carrier is
trapped,
τband
µD = µband
,
τband + τtrap
Ea (eV)
0.3
µFE (cm2/Vs)
1
PTV
VGS
a-Si:H
0.2
20
10
5
Pentacene PQT-12
a-Si:H
0
0
10
20
PQT-12
–2
円VGT 円 (V)
30
VGT
– 40
– 10
–20
10 – 4
VGS
–10
–20
10
–10
–5
–8
3
4
5
6
–5
7
8
9
1000/T (1/K)
Fig. 47.3 Temperature dependence of µFE for a-Si:H,
PQT-12 [47.35], pentacene and PTV [47.36]. Inset shows
the bias dependence of the activation energy E a of µFE in
these materials
Part E 47.2
–20
PTV
(47.10)
(47.11)
– 30
Pentacene
VGS
10 – 6
where g is the conductance of the film, and qn is the
field-induced charge in the semiconducting film. The
terms g and n are averaged over the device active area
and are related to the number of band carriers n band and
trapped carriers n t . The µFE is conventionally retrieved
from measurement of the transistor current (IDS,lin ) in
the linear regime (VDS = 0.1 V), by using the following:
L
∂IDS,lin
µFE =
,
WCi VDS ∂VGS
1111
where W and L are the channel width and length, respectively, Ci the gate capacitance, and VGS and VDS are
the gate–source and drain–source biases, respectively.
According to (47.3) and (47.8), the densities of
the band and trapped carriers increase differently with
increasing E F . Consequently, when n band and n t are
averaged over the volume of the semiconducting film
to obtain the µFE described by (47.10), the mobility
becomes a function of device parameters (e.g. layer
thicknesses and bias conditions).
The bias dependence is also evident in temperaturedependence measurements of mobility and conductivity
in a-Si:H and organic TFTs. Figure 47.3 illustrates
the temperature dependence of µFE for a-Si:H, pentacene and polythienylene vinylene (PTV) [47.36],
and poly[5,5’-bis(3-alkyl-2-thienyl)-2,2’-bithiophene)]
(PQT-12) [47.35,40] TFTs at different gate biases. Here,
the activation energy E a turns out to be bias-dependent
(inset of Fig. 47.3). More importantly, an anomaly arises
in which the µFE and E a become dependent on the gate
capacitance Ci , which is solely a geometrical capaci-
(47.9)
where τband and τtrap are the times that carriers
spend in the extended and localized states, respectively.
Carriers accumulated by the field effect in TFTs also
demonstrate similar transport properties. The field-effect
mobility µFE in TFTs is conventionally defined as
µFE = g/qn ,
47.2 Field-Effect Transport in Amorphous Films
1112
Part E
Novel Materials and Selected Applications
tance. A higher Ci implies higher carrier accumulation
and consequently higher µFE for the same bias.
To remove this anomaly, we need to identify the
effective carrier mobility such that µFE = µeff × f (φ),
where µeff is the effective physical mobility and f (φ)
describes the device attributes such as bias and geometry.
We reconsider the densities of trapped and free carriers according to (47.3) and (47.8). Since both densities
vary exponentially with the Fermi energy, the density of
band carriers in terms of trapped carriers can be written
as
T /T
n band = θn t t
,
(47.12)
where θ = Nb /(Nt u)Tt /T . Equation (47.12) has been
found to hold empirically for a wide range of disordered semiconductor systems and operating conditions,
including organic semiconductors [47.41, 42].
According to (47.2) and (47.12), the conductivity σ(n) as a function of carrier concentration can be
written as
σ(n) = qµeff N0 (n/N0 )Tt /T ,
(47.13)
where µeff is the effective mobility defined at a reference
concentration as
σ(n = N0 )
Nb N0 Tt /T
µeff ≡
= µband
.
qN0
N0 Nt u
(47.14)
Consequently, the conductivity at any carrier concentration is given by µeff and Tt /T . Here, µeff represents
the effective carrier mobility at concentration N0 and
Tt /T describes the change in conductivity with carrier
concentration.
The significance of this representation becomes clear
when we use it to obtain the current–voltage characteristics of a TFT (Fig. 47.4). Using the gradual channel
Part E 47.2
VS
VD
Contact layers
0
x
L
y
Gate
Carrier
accumulation
Semiconductor
VG
Fig. 47.4 Simplified schematic of an inverted staggered
TFT
approximation, we can write
W
IDS,lin = VDS
L
δ
σ(y) dy ,
(47.15)
0
where y denotes the location across the channel and δ the
channel depth. Using (47.13) for σ and changing the integral parameter from y to n, we find after mathematical
manipulation [47.43]
W
IDS,lin = µeff ζ (Ci VGT )α−1 VDS ,
(47.16)
L
where VGT = VGS − Vthr , Vthr the threshold voltage,
α = 2Tt /T the saturation current–voltage characteristics
power parameter, and
(2kB Tt N0 )1−α/2
.
(47.17)
α−1
Here, ζ is just a function of Tt /T and accounts for the
carrier distribution across the film. Revisiting the definition of the field-effect mobility as given by (47.11), but
this time employing a more meaningful representation
of the current, i. e. (47.16), we find
µFE
= ζ (α − 1)(Ci VGT )α−2 .
(47.18)
f (φ) ≡
µeff
Representation of µeff according to (47.14) is valid
only when there is an exponential relationship between
the carrier concentration n and the Fermi energy E F
as given by (47.8). This is true for Fermi energy locations below the transport band edge and above the
energies of the deep states (the region shown by the
solid slope in Fig. 47.2). Since the deep states mostly
contribute to the threshold voltage Vthr and are filled
before the device turns on, the mobility definition of
(47.14) is valid for the above-threshold regime. Consequently, the value of the reference concentration N0
must be selected such that the Fermi energy EF0 associated with N0 resides well above the deep states.
This requires the charge accumulated in the channel
Q channel to be higher than the charge Ci Vthr needed to
turn on the device. If the carrier concentration at the
semiconductor interface is N0 , Q channel = Ci V0 , where
V0 = (2kB Tt N0 )1/2 /Ci . Thus, for Q channel > Ci Vthr , we
2 /(2k T ), indicating
conclude V0 > Vthr or N0 > Ci2 Vthr
B t
the lower limit for N0 . Using typical values, Vthr = 2 V
and Ci = 20 nF/cm2 , we have N0 > 6 × 1016 cm−3 . For
instance, with N0 = 1017 cm−3 , we have for the bias
(1−α/2)
dependence factor ζ = 4.1 × 10−16 α
/(α − 1).
We now revisit the temperature dependence of µeff
by recasting (47.14) in the following manner
ζ=
µeff = µeff0 exp(−E a0 /kB T ) ,
(47.19)
Silicon on Mechanically Flexible Substrates for Large-Area Electronics
47.3 Electronic Transport Under Mechanical Stress
1113
Table 47.1 Extracted transport parameters (µeff , Tt , and E a0 ) for a selection of disordered semiconductors at
N0 = 1017 cm−3
Semiconductor
µeff at 300 K
(cm2 /Vs)
Tt
(K)
Ea0
(eV)
Reference
a-Si:H
PQT-12
Pentacene
PTV
0.98
3.9 × 10−2
9.0 × 10−4
1.1 × 10−5
350
326
385
380
0.23
0.24
0.31
0.42
Our measurements
[47.35]
[47.36]
[47.36]
where
Nb
µeff0 ≡ µband
and
N0
Nt u
= −E F0 .
E a0 ≡ kB Tt ln
N0
(47.20)
Equation (47.19) predicts Arrhenius behavior for the
effective mobility with an activation energy of E a0 =
−E F0 (Fig. 47.2). The activation energy of µeff is not
bias-dependent and corresponds to the energy needed
for carriers to thermalize from E F0 to the mobility edge.
We now examine the relation between the two activation energies E a and E a0 . To do so, we look at the
temperature dependence of µFE and µeff , viz.,
∂µeff
∂ f (φ)
∂µFE
=
+
µFE ∂T
µeff ∂T
f (φ)∂T
2Tt VGT E a0
,
+
ln
=−
kB T 2 T 2 V0 (47.21)
which yields
E a = E a0 − 2kB Tt ln |VGT /V0 | .
(47.22)
Equation (47.22) describes the bias dependence observed for E a in the inset of Fig. 47.3.
where γ is the effective overlap parameter for electronic states in the band tail. Baranovskii et al. [47.46]
have generalized this concept of transport band beyond the exponential DOS assumption and to a broader
range of disordered materials with Gaussian or similar rapidly changing distributions. The generalized
band concept can also accommodate the percolationbased hopping transport described by Matters [47.36]
for amorphous organic semiconductors, which
pre-
dicts a hopping band that is just kB Tt ln BC /3π 3
higher than that predicted by Monroe [47.26]. Here,
BC ≈ 2.8 is the critical number for percolation in
three-dimensional amorphous systems. Table 47.1 summarizes the values for µeff at room temperature Tt , and
E a0 at N0 = 1017 cm−3 , determined from the results
presented for different disordered materials reported in
literature.
47.3 Electronic Transport Under Mechanical Stress
Mechanical stress deforms the structure of the thin film
leading to modulation in carrier mobility and density
of states, and consequently, modulation of resistance.
The change in resistance of a solid with elastic strain
or stress is commonly referred to as the elastoresistance or piezoresistance effect, respectively [47.47]. The
Part E 47.3
Amorphous Organic Semiconductors
The concept of effective mobility and transport band
may be generalized to accommodate different amorphous organic semiconductors despite differences in the
underlying transport mechanism. This generalization
follows from the relation n band = θn Tt /T (47.12) for
organic semiconductors for a wide range of temperatures and carrier concentrations [47.41]. For this relation
to hold, the distribution of trapped and band carriers
must be exponential. Although evidence of a Gaussian
trap distribution has been reported for organic materials [47.44], the Gaussian distribution is effectively seen
as an equivalent exponential distribution due to the small
variation in the Fermi energy because of the large tail
state distribution. In addition, Shapiro and Adler [47.45]
have demonstrated that a transport band is present in
which hopping conduction dominates irrespective of
the position of Fermi energy. Similar to the mobility
edge, the trapped carriers are thermalized to the hopping band [47.26]. Relative to E F0 , the hopping band is
located at
γ 3 2T 3
(47.23)
E 0hopping = kB Tt ln
,
N0 3Tt
1114
Part E
Novel Materials and Selected Applications
magnitude of the change is a function of the electronic
properties of the material, the dimensions of the solid,
and the direction of current flow.
Structural order in the material influences the elastoresistance effect. In the case of crystalline silicon,
anisotropic scattering of electrons in the n-type material leads to a strong orientation dependence of the
elastoresistive behavior [47.48]. In polycrystalline silicon, the crystallite size and orientation, and material
texture play a critical role in determining the magnitude
of the effect [47.49, 50]. In amorphous semiconductors,
the random network behaves like an isotropic medium,
and the anisotropy found in the crystalline material is
less visible. However, the elastoresistance coefficients
still depend on the relative orientation of the current and
applied strain [47.51].
In sputtered amorphous silicon, Welber and Brodsky [47.52] have reported a decrease in the absorption
gap with hydrostatic
pressure with
a coefficient of about
−1 meV/kbar −10−11 eV/Pa observed from the shift
in the absorption edge. Weinstein [47.53] has also reported similar results drawn from photoluminescence
experiments. The change in the optical gap is similar in
sign and magnitude to that measured for crystalline silicon [47.54]. Lazarus [47.55], however, has reported an
exponential increase in the resistivity of a-Si:H with increasing hydrostatic pressure at room temperature. The
increase in resistivity is ascribed to either a decrease in
the number of carriers or a reduction of the mobility with
compressive strain [47.51].
In this section, we investigate the impact of mechanical stress on electrical properties of thin-film devices,
insight into which is critical for design of mechanically
flexible electronics. We begin with metallic and semiconductor thin films for strain-gauge applications and
continue with a-Si:H TFTs.
Substrate
Fig. 47.5 Schematic of a thin-film strain gauge
gauge, respectively. The last term on the right-hand side
of (47.24) reflects the strain-induced change in resistivity of the sample, whereas the first three terms refer to
geometrical changes only.
Usually, the gauge length is oriented in a direction
where strain is largest to achieve the highest gauge factor.
This is referred to as the longitudinal orientation, where
the current flows parallel to the strain. Similarly, the
transverse orientation is defined when the gauge length
is oriented in a direction perpendicular to the maximum
strain (Fig. 47.6). The modulation of resistivity in the different orientations with the strain and correspondingly
stress can be summarized by the following expression:
∆ρi
= γij j = πij τ j with i = 1, 2, 3
ρ
and j = 1, 2, ..., 6 , (47.25)
where γij and πij are the elements of the compact matrix
of the elastoresistance and piezoresistance coefficients,
Curr
ent f
Part E 47.3
en
Curr
Strain gauges are thin-film transducer elements embedded on a substrate Fig. 47.5. The strain in the substrate
leads to a change in the geometry, and therefore, the resistance of the gauge, which is detected and measured
by external circuitry. The sensitivity of the resistance R
to strain is referred to as the gauge factor k, which can
be written as
low
t
w
t flo
47.3.1 Thin-Film Strain Gauges
∆R ∆L ∆W ∆t ∆ρ
=
−
−
+
,
k≡
R
L
W
t
ρ
Gauge
length
Thin film
L
W
L
W
Transverse
Longitudinal
3
2
1
τ1 = τext
ε1 = εext
(47.24)
where ρ is the resistivity of the material, and W, L,
and t are the width, length, and thickness of the strain
Fig. 47.6 Longitudinal and transverse wires under uniaxial
stress and strain components in the 1-direction
Silicon on Mechanically Flexible Substrates for Large-Area Electronics
respectively, and j and τ j denote the strain and stress
components, respectively. (Here, we have used compact
notation for these tensors [47.47].)
In metal gauges, the sensitivity of ρ to strain is
assumed to be negligible, leading to the well-known
longitudinal gauge factor of kl = 1 − 2ν, where ν is the
Poisson’s ratio. However, Arlt [47.56] has shown that the
term ∆ρ/ρ also includes geometrical attributes. This is
due to the change in the volume of the wire and the resulting change in carrier density. According to Arlt [47.56],
we have for the longitudinal gauge factor
∆(γµ)l
kl = 2 −
,
γµ
∆V/V (× 10 –3)
1.0
2.0
Tensile, longitudinal
Mo gauge
I = 100 µA
V = 0.69 V
∆ (mm)
1.66
0.8
1.33
0.6
1.0
0.66
0.4
Load
0.33
(47.26)
Unload
3ts ∆
(1 − x/L 0 ) ,
2L 20
(47.27)
with the positive and negative signs denoting tensile and
compressive configurations, respectively.
Load
x
1
∆
0
Compressive
configuration
50
100
150
200
250
300
Time (s)
Fig. 47.8 Normalized change in output voltage of a longi-
tudinal Mo strain gauge
Figure 47.8 illustrates the result of the beamdeflection experiment on a 150-nm-thick molybdenum
strain gauge with 30 turns, line width of 20 µm, and
length of 1100 µm. The strain gauges are biased with
a constant current of 100 µA and the voltage drop
across its terminals is measured. The gauges are subjected to a sequence of loading/unloading steps in tensile
configuration to eliminate systematic errors associated
with slowly varying transients. By averaging the value
of the voltage modulation, we retrieve a gauge factor
of approximately kl ≈ 2.06 for the longitudinal strain
gauges.
The gauge factor for semiconductors is much
higher due to the higher elastoresistance coefficients.
Dössel [47.57] has related the gauge factor to the elastoresistance coefficients as follows:
kl = γ1 − (νs + ν h )γ2 and kt = −νs γ1 + (1 − νh )γ2 ,
ν h = νf
3
2
0
Substrate
Fig. 47.7 Schematic of a beam-deflection experiment
1 − νs
,
1 − νf
(47.28)
and γ1 = 2 − ∆(γµ)l /γµ and γ2 = ∆(γµ)t /γµ are the
elastoresistance coefficients, and νs and νf denote the
Poisson’s ratios for the substrate and the film, respectively.
The same beam-deflection experiment can also be
performed on n+ µc-Si:H strain gauges. Figure 47.9
displays the results of tensile and compressive tests on
metal and semiconductor gauges, indicating a higher
gauge factor. The values are kl = −17.0 and kt = −3.41
Part E 47.3
Films
0.0
where
Tensile
configuration
ts
1115
0.2
where γ denotes the number of free electrons per atom
and µ the electron mobility. Since the last term on
the right-hand side of (47.26) is relatively small for
metallic gauges, the gauge factor mainly stems from
geometrical changes; it is close to 2 and temperature
independent [47.57].
These results can be validated using beam-deflection
(cantilever) experiments performed on strain gauges integrated on glass and silicon substrates. Figure 47.7
illustrates the schematic of a beam-deflection system.
One end of the sample is clamped (x = 0) and the free
end is deflected by a displacement ∆. Rajanna and Mohan [47.58] used this method in measurements of both
tensile and compressive configurations by simply placing the sample with the films on the top or bottom,
respectively (see Fig. 47.7). The strain at location x along
a sample with a length L 0 reads
(x) = ±
47.3 Electronic Transport Under Mechanical Stress
1116
Part E
Novel Materials and Selected Applications
∆V/V (× 10 –3)
10
∆ID /ID (× 10 –3)
IMo = 100 µA
In+ = 100 nA
Longitudinal
n+
8
Longitudinal
Mo
Transverse
n+
2
–2
–2
Transverse
–4
–6
–6
–8
Compressive
–10
–6
–4
–2
Tensile
0
Compressive
2
4
6
Strain (× 10 –4)
Fig. 47.9 Change in voltage of longitudinal and transverse
n+ µc-Si:H and longitudinal Mo gauges under tensile and
compressive strains
for longitudinal and transverse semiconductor gauges,
respectively. Assuming νs = νf = 0.23, we find that γ1 =
−22 and γ2 = −10.9 for n+ µc-Si:H films.
The longitudinal gauge factor obtained for the n+ µcSi:H gauges corroborate the results of Germer [47.59]
for phosphorus-doped microcrystalline thin-film samples with a doping density of ≈ 1020 cm−3 . For the
transverse gauge factor, Germer has observed a small
(negative or positive) gauge factor for doping densities in the range 5 × 1019 –1020 . This shows that the
∆ID /ID(%)
0
Tensile, longitudinal
W/L = 100 µm/25 µm
VG = 20 V
VD = 0.5 V
ID = 0.45 µA
Load
Part E 47.3
–2
–3
Shear
2
0
–4
Longitudinal
4
0
–1
W/L = 400 µm/400 µm
VG = 20 V
VD = 0.5 V
6
6
4
8
2 1.66
∆ (mm)
1.33
1
0.66
–8
–6
–4
–2
Tensile
0
2
4
6
Strain (×10 –4)
Fig. 47.11 Normalized change in TFT current as a function
of tensile and compressive strain
transverse gauge factor is highly sensitive to the doping
density and other process conditions, which explains the
slight difference between our values and that reported
by Germer.
47.3.2 Strained Amorphous-Silicon
Transistors
Figure 47.10 illustrates measurement results for the transient drain current of a longitudinal TFT subject to
a sequence of tensile loading/unloading steps. We see
immediate changes in current superimposed on the intrinsic transient response of the TFT. The measured
change in current ∆ID decreases with decreasing displacement. Correspondingly, we define the sensitivity
S (ID ) =
1 ∂∆ID
|=0 ,
ID ∂
(47.29)
0.33
Strain direction
Unload
–4
–5
0
50
100
150
200
250
300
Time (s)
Fig. 47.10 Change in TFT current in deflection experiments
Longitudinal
Transverse
Shear
Fig. 47.12 Longitudinal, transverse, and shear TFTs
Silicon on Mechanically Flexible Substrates for Large-Area Electronics
S⑀(ID)
W/L = 400 µm/400 µm
30
Linear
Saturation
25
20
Longitudinal
15
10
Shear
5
Transverse
0
–5
5
10
15
20
VGS (V)
Fig. 47.13 Bias dependence of the TFT current sensitivity
Table 47.2 Values for SH (ID ), S (µeff ), and S (Vthr ) for
different orientations
Parameter
Longitudinal
Transverse
Shear
SH (ID ) = ∆ID /ID S (µeff ) = ∆µeff /µeff S (Vthr ) = ∆Vthr /Vthr 12.1
11
5
−1.1
−1.1
4.5
4.5
4.0
4.7
1117
In addition to TFT orientation, we observe that the
gate bias alters the magnitude of strain-induced change
in current ∆ID . To examine the impact of bias, deflection experiments were performed for different values
of the gate bias VGS in the range 4–20 V in 1 V steps.
Figure 47.13 illustrates the measured S (ID ) as a function of VGS for TFTs of different orientations. Solid
symbols denote measurement data for the linear regime
(VDS = 0.5 V) while the open symbols are those for the
saturation regime where the gate and drain terminals
are shorted. Interestingly, the modulation in the current
shifts toward positive values as the gate bias decreases.
This is true for TFTs of all orientations, and independent of whether the devices were integrated on glass or
silicon substrates.
However, the sensitivity S (ID,t ) for the transverse
TFT undergoes a sign change. At high biases, the
S (ID,t ) is generally small and negative (i. e. ∆ID is
positive for tensile strain). As the gate bias decreases,
the S (ID,t ) virtually vanishes at approximately 7 V, and
subsequently increases to a sizable positive value (i. e.
∆ID is negative for tensile strain) at lower voltages
(VGS < 7 V).
From the results of Fig. 47.13, we identify two
distinct components underlying the strain-induced modulation of current: the high-bias (VGS > 7 V) SH (ID ) and
low-bias (VGS < 7 V) SL (ID ) components such that
S (ID ) = SH (ID ) + SL (ID ) .
(47.30)
At high biases, S (ID ) of longitudinal, transverse, and
shear TFTs gradually approach constant values. As seen
from Table 47.2, the extracted values for SH (ID ) are
strongly orientation dependent, suggesting the presence
of strain-induced modulation in carrier mobility, whose
sensitivity we define as S (µeff ) = ∆µeff /µeff . The
mobility change in the longitudinal orientation is higher
than that in the transverse orientation.
Superimposed on the high-bias component is the
low-bias component SL (ID ) which manifests itself
as a bias-dependent positive shift in S (ID ). This
component can be attributed to the modulation in
threshold voltage. Correspondingly, we define the
threshold-voltage sensitivity as S (Vthr ) = ∆Vthr /Vthr .
The change in Vthr can be attributed to a strain-induced
change in the density of deep states, which is orientation independent [47.51]. The modulation in threshold
voltage leads to a significant change in current at low biases, and can be quantitatively explained by looking at
the current–voltage relation in the linear regime. From
(47.16) and using partial differentiation with respect to
Part E 47.3
where ID is the unstrained value of current and 0
a reference strain value.
Figure 47.11 depicts measurement results for the
change in drain current of the longitudinal, shear, and
transverse TFTs (Fig. 47.12) under tensile and compressive strains. The TFTs have W/L = 400 µm/400 µm and
are biased in the linear regime with constant VGS = 20 V
and VDS = 0.5 V. As seen in the figure, the results for
compressive and tensile strain are similar but opposite
in sign. For the longitudinal TFT, the current increases
with tensile strain with a sensitivity S (ID,l ) = 12.5.
In contrast, for the transverse TFT, this is small and
negative S (ID,t ) = −1.1, clearly signifying an orientation dependence. The value of S (ID,s ) = 4.5 for the
shear TFT can be explained from the linear superposition of the effects of longitudinal and transverse strain
components.
The measured value for S (ID,l ) is close to the
value of 15 ± 3 reported by Spear and Heintze [47.51]
for intrinsic a-Si:H at room temperature. Gleskova
et al. [47.60] have found a higher longitudinal sensitivity
S (ID,l ) = 26. For the transverse sensitivity, Spear and
Heintze have reported a positive value of S (ID,t ) = 7
for intrinsic a-Si:H samples.
47.3 Electronic Transport Under Mechanical Stress
1118
Part E
Novel Materials and Selected Applications
strain, we can write S (IDS,lin ) as
αVthr
S (IDS,lin ) = SG + S (µeff ) − S (Vthr )
VGT
VGT α
− − 1
,
+ S (α) α ln V0 2 α − 1
(47.31)
where V0 = (2kB Tt N0
)1/2 /C
i,
and
Ei =
SG = S (W ) − S (L) + (α − 1)S (Ci )
Here, SG includes the effect of change in device dimensions. The other terms on the right-hand side of
(47.31), in order from the left, describe the dependence
of S (IDS,lin ) on S (µeff ), S (Vthr ), and S (α), respectively, which represent strain-induced modulation of the
different TFT parameters (µeff , Vthr , and α).
As seen in (47.31), the modulation in mobility
S (µeff ) is directly reflected in the change of current.
In contrast, the strain-induced change in threshold voltage ∆Vthr /Vthr is scaled, and by a factor of αVthr /VGT .
This is particularly visible at low biases and its effect
decreases with increasing gate bias VGT . Thus, at high
biases, the impact of threshold-voltage modulation is
minimal, which yields
SH (ID ) = SG + S (µeff ) .
(47.32)
In contrast, the low-bias component can be written as
SL (ID ) = −S (Vthr )
αVthr
.
VGT
modulation of α. This term contains a bias-dependent
scaling factor of α ln |VGT /V0 |, which does not correlate
with the observed bias dependence of S (ID ) seen in
Fig. 47.13. This implies that modulation in Vthr and µeff
are the dominant contributors to the observed changes
in current.
The activation energy for the temperature dependence of SL (ID ) and SH (ID ) is defined as
(47.33)
It is important to note that the last term on the right-hand
side of (47.31) represents the impact of strain-induced
∂Si (ID )
with i = H or L .
∂(1/kB T )
(47.34)
The values for E H and E L are found to be 140 meV
and 0.58 eV, respectively. The much lower activation energy (140 meV) at high biases indicates that the SH (ID )
stems from the shallow states in the conduction-band
tail. Again, this corroborates our previous findings that
the sensitivity of the current at high biases is associated with the mobility modulation that is principally
determined by the tail states. Spear and Heintze [47.51]
have found an activation energy of 0.52 eV for intrinsic and doped a-Si:H layers. This corroborates with
our low-voltage sensitivity data. The high activation
energy for SL (ID ) identifies the role of deep states
in the gap, which is in agreement with our previous
finding that S (ID ) ∝ S (Vthr ) at low biases. Here, the
strain is believed to modify the energy of the deep
states [47.51].
From (47.32) and (47.33), the values of S (µeff ) and
S (Vthr ) for different orientations can be determined (Table 47.2). The values can be incorporated in a compact
model for the different TFT orientations, which can be
used for computer-aided design (CAD) of mechanically
flexible TFT circuits [47.61].
References
Part E 47
47.1
47.2
47.3
47.4
47.5
47.6
47.7
S. R. Forrest: Nature 428, 911–918 (2004)
S. Wagner, H. Gleskova, J. C. Sturm, Z. Suo: In:
Technology and Applications of Amorphous Silicon,
ed. by R. A. Street (Springer, Berlin 2000) pp. 222–
251
S. E. Shaheen, R. Radspinner, N. Peyghambarian,
G. E. Jabbour: Appl. Phys. Lett. 79, 2996 (2001)
R. A. Street, M. Mulato, R. Lau, J. Ho, J. Graham,
Z. Popovic, J.Hor: Appl. Phys. Lett. 78, 4193–4195
(2001)
P. Servati, Y. Vygranenko, A. Nathan, S. Morrison,
A. Madan: J. Appl. Phys. 96, 7575–7582 (2004)
L. Collins: IEE Rev. 49(2), 42–45 (2003)
A. B. Chwang, M. A. Rothman, S. Y. Mao, R. H. Hewitt, M. S. Weaver, J. A. Silvernail, K. Rajan,
47.8
47.9
47.10
47.11
M. Hack, J. J. Brown, X. Chu, L. Moro, T. Krajewsky, N. Rutherford: Appl. Phys. Lett. 83(3), 413–415
(2003)
A. Nathan, A. Kumar, K. Sakariya, P. Servati,
S. Sambandan, D. Striakhilev: IEEE J. Solid-State
Circ. 39(9), 1477–1486 (2004)
M. J. Powell, C. van Berkel, A. R. Franklin,
S. C. Deane, W. I. Milne: Phys. Rev. B 45(8), 4160–
4170 (1992)
D. Knipp, R. A. Street, A. Volkel, J. Ho: J. Appl. Phys.
93, 347 (2003)
S. Alexander, P. Servati, G. R. Chaji, S. Ashtiani,
R. Huang, D. Striakhilev, K. Sakariya, A. Kumar,
A. Nathan, C. Church, J. Wzorek, P. Arsenault: J.
Soc. Info. Display 13(7), 587–595 (2005)
Silicon on Mechanically Flexible Substrates for Large-Area Electronics
47.12
47.13
47.14
47.15
47.16
47.17
47.18
47.19
47.20
47.21
47.22
47.23
47.24
47.25
47.26
47.27
47.28
47.29
47.30
47.31
47.32
47.33
47.34
47.36
47.37
47.38
47.39
47.40
47.41
47.42
47.43
47.44
47.45
47.46
47.47
47.48
47.49
47.50
47.51
47.52
47.53
47.54
47.55
47.56
47.57
47.58
47.59
47.60
47.61
M. J. C. M. Vissenberg, M. Matters: Phys. Rev. B
57(20), 12964–12967 (1998)
J. G. Shaw, M. Hack: J. Appl. Phys. 64(9), 4562–
4566 (1988)
C. Tanase, E. J. Meijer, P. W. M. Blom, D. M. de Leeuw:
Phys. Rev. Lett. 91(21), 216–601 (2003)
T. Tiedje, A. Rose: Solid State Comm. 37, 49–52
(1980)
B. S. Ong, Y. Wu, P. Liu, S. Gardner: J. Am. Chem.
Soc. 126, 3378–3379 (2004)
A. J. Campbell, M. S. Weaver, D. G. Lidzey,
D. D. C. Bradley: J. Appl. Phys. 84(12), 6737–6746
(1998)
D. Natali, M. Sampietro: J. Appl. Phys. 92(9), 5310–
5318 (2002)
P. Servati, D. Striakhilev, A. Nathan: IEEE Trans.
Electr. Devices 50(11), 2227–2235 (2003)
C. Tanase, E. J. Meijer, P. W. M. Blom, D. M. de Leeuw:
Organic Elect. 4, 33 (2003)
F. R. Shapiro, D. Adler: J. Non-Cryst. Solids 74(2–3),
189–194 (1985)
S. D. Baranovskii, T. Faber, F. Hensel, P. Thomas:
J. Phys.: Condens. Matter 9, 2699 (1997)
A. Nathan, H. Baltes: Microtransducer CAD
(Springer, Wien 1999)
C. Herring, E. Vogt: Phys. Rev. 101(3), 944–961 (1956)
V. A. Gridchin, V. M. Lubimsky, M. P. Sarina: Sens.
Actuators A 49(1-2), 67–72 (1995)
A. Dévényi, A. Belu, G. Korony: J. Non-Cryst. Solids
4, 380–390 (1970)
W. E. Spear, M. Heintze: Philos. Mag. B 54(5), 343–
358 (1986)
B. Welber, M. H. Brodsky: Phys. Rev. B 16(8), 3660–
3664 (1977)
B. A. Weinstein: Phys. Rev. B 23(2), 787–793 (1981)
R. Zallen, W. Paul: Phys. Rev. 155(3), 703–711 (1967)
D. Lazarus: Phys. Rev. B 24(4), 2282–2284 (1981)
G. Arlt: J. Appl. Phys. 49(7), 4273–4274 (1978)
O. Dössel: Sens. Actuators 6(3), 169–179 (1984)
K. Rajanna, S. Mohan: Phys. Status Solidi A 105(2),
K181–K184 (1988)
W. Germer: Sens. Actuators 7(2), 135–142 (1985)
H. Gleskova, S. Wagner, W. Soboyejo, Z. Suo:
J. Appl. Phys. 92(10), 6224–6229 (2002)
P. Servati, A. Nathan: Appl. Phys. Lett. 86(7),
033504 (2005)
1119
Part E 47
47.35
G. L. Graff, R. E. Williford, P. E. Burrows: J. Appl.
Phys. 96(4), 1840–1849 (2004)
J. A. Rogers, Z. Bao, A. Dodabalapur, A. Makhija:
IEEE Electr. Devices Lett. 21(3), 100–103 (2000)
C.-S. Yang, L. L. Smith, C. B. Arthur, G. N. Parsons:
J. Vac. Sci. Technol. B 18(2), 683–689 (2000)
H. Gleskova, S. Wagner, V. Gašparík, P. Kováč: Appl.
Surf. Sci. 175-176, 12–16 (2001)
D. Stryahilev, A. Sazonov, A. Nathan: J. Vac. Sci.
Technol. A 20(3), 1087–1090 (2002)
M. Meitine, A. Sazonov: Mat. Res. Soc. Symp. Proc.
769, H6.6.1 (2003)
W. A. MacDonald: J. Mater. Chem. 14, 4–10 (2004)
K. L. Chopra: Thin Film Phenomena (McGraw–Hill,
Toronto 1969)
E. Y. Ma, S. Wagner: Appl. Phys. Lett. 74, 2661–2662
(1999)
A. Madan, P. G. Le Comber, W. E. Spear: J. NonCryst. Solids 20, 239–257 (1976)
N. F. Mott, E. A. Davis: Electronic Processes in
Non-Crystalline Materials (Oxford University Press,
Oxford 1971)
M. H. Cohen, H. Fritzsche, S. R. Ovshinsky: Phys.
Rev. Lett. 22(20), 1066–1068 (1969)
C. Popescu, T. Stoica: In: Thin Film Resistive Sensors,
ed. by P. Ciureanu, S. Middelhoek (IOPP, New York
1992) pp. 37–112
P. W. Anderson: Phys. Rev. 109(5), 1492–1505 (1958)
D. Monroe: Phys. Rev. Lett. 54(2), 146–149 (1985)
R. A. Street: Hydrogenated Amorphous Silicon
(Cambridge University Press, New York 1992)
W. E. Spear, P. G. Le Comber: J. Non-Cryst. Solids
8-10, 727–738 (1972)
M. J. Powell: Philos. Mag. B 43(1), 93–103 (1981)
C.-Y. Huang, S. Guha, S. J. Hudgens: Phys. Rev. B
27(12), 7460–7465 (1983)
J. D. Cohen, D. V. Lang, J. P. Harbison: Phys. Rev.
Lett. 45(3), 197–200 (1980)
M. Hirose, T. Suzuki, G. H. Döhler: Appl. Phys. Lett.
34(3), 234–236 (1979)
P. Viktorovitch, G. Moddel: J. Appl. Phys. 51(9),
4847–4854 (1980)
M. Shur, M. Hack: J. Appl. Phys. 55(10), 3831–3842
(1984)
A. Salleo, T. W. Chen, A. R. Völkel, Y. Wu, P. Liu,
B. S. Ong, R. A. Street: Phys. Rev. B 70, 115–311 (2004)
References