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Linear and Non-Linear Piezoresistance Coefficients in Cubic Semiconductors. I. Theoretical Formulations
S. Durand, C. Tellier
To cite this version:
S. Durand, C. Tellier. Linear and Non-Linear Piezoresistance Coefficients in Cubic Semiconduc- tors. I. Theoretical Formulations. Journal de Physique III, EDP Sciences, 1996, 6 (2), pp.237-266.
�10.1051/jp3:1996121�. �jpa-00249453�
J. Phys. III France 6 (1996) 237-266 FEBRUARY1996, PAGE 237
Linear and Non-Linear Piezoresistance Coefficients in Cubic Semiconductors.
I. Theoretical Formulations
S. Durand and C-It- Tellier (*)
Laboratoire de Chronom6trie, #lectronique et P16zo61ectricit6, ENSMM, 26 Chemin de I'#pitaphe, 25030 Besanqon Cedex, France
(Received 21 April 1995, revised 12 October1995, accepted 15 November 1995)
PACS.06.70.Dn Sensing and detecting devices
PACS.72.20.Fr Low-field transport and mobility; piezoresistance PACS.81.60.Cp Semiconductors and insulators
Abstract. This paper constitutes the first part of a work devoted to applications of piezore-
sistance effects in germanium and silicon semiconductors. In this part, emphasis is placed on a
formal explanation of non-linear effects. We propose a brief phenomenological description based
on the multi-valleys model of semiconductors before to adopt a macroscopic tensorial model from which general analytical expressions for primed non-linear piezoresistance coefficients
are
derived. Graphical representations of linear and non-linear piezoresistance coefficients allow us to characterize the influence of the two angles of cut and of directions of alignment The second part will primarily deal with specific applications for piezoresistive sensors.
Rdsumd, Cette publication constitue la premiAre partie d'un travail consacr6 aux applica-
tions des effets p16zor6sistifs darts les semiconducteurs germanium et silicium. Cette partie traite essentiellement de la mod61isation des effets non-lin6aires. AprAs une description ph6nom6nolo- gique h partir du modAle de bande des semiconducteurs nous d6veloppons un modAle tensoriel
macroscopique et nous proposons des 6quations gdndrales analytiques exprimant les coefficients
piezor6sistifs non-lin6aires dans des repAres tourn6s. Des repr6sentations graphiques des varia- tions des coefficients p16zor6sistifs lin6aires et non-lin6aires permettent une prd-caract6risation
de l'influence des angles de coupes et des directions d'alignement avant l'6tude d'applications sp6cifiques qui feront l'objet de la deuxiAme partie.
1, Introduction
During the two last decades a number of works [1-7] have been devoted to the application of piezoresistance effect to various mechanical sensors [1,3-8]. Actually we observe a revival of piezoresistive sensors essentially due to the development of microelectronics and microme-
chanics. Effectively several attempts [9-17] have been made to apply the photolithographic
process to the micromachining of mechanical devices [9-14,17] which constitute the sensing ele- ment of piezoresistive silicon sensors. The fabrication technique always involves an anisotropic wet-etching process [9-17] of silicon crystals. More recently experimental works [15-17] on
(*) Author for correspondence (Fax: (33) 81 88 57 14)
© Les Editions de Physique 1996
238 JOURNAL DE PHYSIQUE III NO?
the etching of (hk0) and (hhi) silicon planes have been undertaken. They lead to a b~tter
understanding of the anisotropic process.
From a practical point of view, it may be interesting to extent the micromachining to silicon
planes other than classical (100) or (110) wafers in order to design new piezoresistive silicon
sensors. Effectively up to no,v, number of published works deal with gauges diffused on (100)
silicon or (110) silicon [8] diaphragms [1,3-6] even if some authors [ii ha,>e suggested to use four- terminals sensors. Thus there is a need for a better understanding of the orientation depeiiden~e
of firstly. the linear piezoresistance coefficients and secondly, the non-linear piezoresistaiice coefficients. The final purpose is to determine specific orientations which offer interesting metrological performances for gauge and also for four-terminals sensors. The aim of this paper is to contribute to this study. Hence in part I. theoretical results related to linear and non-linear
piezoresistance coefficients are presented. lve start with a phenomenological description uf the non-linear effect in N-type germanium before to ~vork ~A<ith a classical macroscopic tensorial model. Emphasis is placed on analytical expressions for primed non-linear coefficients ,,.hich allo,,. us to follo,v the changes in coefficients with orientation. Thus at this point it is con;enient to gi,>e the state of the art for semiconducting crystals ,,<hich belong to class m3m namel;
germanium and silicon crystals.
For cubic semiconductor crystals references axes (xi .z2.~3) correspond to the cry.~tallo- graphic directions (100). A phenomenological description of the piezoresistive effect ,;hich
starts with the band structure and the constant energy< surface for electrons has been proposed by Mason [2] for slightly doped N-type silicon. This "microscopic'" model provides analytical expressions for the first order piezoresistance coefficients ~(i and ~(~ ,vhich in;olve changes iii energy levels of valleys. The success of this rather simple phenomenological model is partl;
due to the fact that theoretical predictions agree ~vell with experimental data. In particular in
a first approximation the theoretical ~ii /~21 ratio satisfies to the follo~A>ing relation:
~ii/~21 " -2
This ratio is of the same order of magnitude that the experimental ratio since a value close to -1.92 is generally cited in literature [2, 7]. For N-type silicon calculations are rather easy.
because the six constant energy ellipsoids lie along the reference axes ~i x2 and ~3. Then, the contributions to the total current density, Ji or J2. of conduction electrons in the ,-alleys are simply expressed in terms of the principal effecti,>e masses of the inverse mass tensor.
In germanium there are eight minima Ll of the conduction band lying somewhere along the
(111) directions in the Brillouin zone. The constant energy surface in their neighbourhoods
may be represented by formulae like:
fi2 ~~ j2 jh j2 ~h )2
l§'(k) = W(ko) + + ~ + ~ II
~ Rlil ~12 ~13
~A.here /hk is the ,<ector from the wave vector k to the centre, ko. Of the minimum pro;ided
~A.e have chosen principal axes of constant energy ellipsoids as special local axes in k space.
In this condition, m(i, m]~ and mj~ are the three principal effective masses of the inverse mass tensor ~vhich for germanium are usually designated as longitudinal and transverse masses
llljj " Rl~l, m1 7R(2 ~13 ~~~
Because prolate ellipsoids lie along directions which differ from the reference axes, it is les~
easy for N-type germanium to perform calculations than for N-tj<pe silicon. Effecti,>ely, in this
case. ,ve deal ~A.ith equivalent piezoresistance coefficients ~[iii and ~[~ii related to rotated
N°2 NON-LINEAR PIEZOItESISTANCE EFFECTS 239
axes xi, xi, xl with xl axis lying parallel to a (111) direction. Analytical expression for
~[iii has been previously derived by Mason et al. [18] but to our kno,vledge, no calculations have been performed when a current density J, (0, J(,0) and an electric field E (0, E(, 0) are applied simultaneously with an uniaxial stress T(i. Thus it may be of interest to develop such calculations in order to appreciate the inherent non-linearity in the relative change of electrical resistivity induced by stress.
In a macroscopic point of view, the piezoresistivity effect is described by tensors which
connect the Cartesian components of the electric field E to the components of the current
density J and the applied stresses, Tki [19]. In a tensorial description non-linearity is generally
accounted for second order piezoresistance coefficients related to a 6th rank tensor. For sensor
applications a maximum piezoresistance effect is generally required but in many cases the maximum sensitivity occurs along directions other than the reference axes. Thus we are usually concerned with primed first order, ~(~~~, and second order, ~(~~j~~, piezoresistance coefficients.
A great number of works [1, 2, 7,18, 20,24] have been devoted to the calculation of primed
~(~~j which determine the sensitivity of a semiconductor single gauge. Papers which deal ~vith primed second-order coefficients are less numerous. Only Kanda and Matsuda [21,22] proposed expressions for the ~(~~j~~ which unfortunately are limited to some special geometrical config-
urations. But due to the promising applications of four terminals piezoresistive sensors [24-26],
there is an increasing need of analytical equations for the primed second-order piezoresistance coefficients. Effectively, it seems essential to estimate simultaneously the sensitivity (associated
with the ~(~~~) and the non-linearity of sensors for various special orientations of wafers and for various directions of alignment of gauges and of four-terminals sensors.
2. A Multi-Valley Formulation of Non-Linear Piezoresistance Coefficients
2,I. EFFECT OF AN UNIAXIAL [III] STRESS ON CONDUCTIVITY. Let us be concerned
~vith a N-type germanium, plate of orientation (111). Under the influence of an electric field E (0, E2,0) the current flows parallel to the xi axis which coincides with the [i12] direction
(Fig. la). This plate is affected by the application of an uniaxial [111] tensile stress T(i.
For germanium, the constant energy surface are prolate half-ellipsoids in number M
= 8. Then
in absence of stress the density of electrons, nj in a (111) valley may be written as:
'll ~ j'la 13)
where na is the usual density of states corresponding to an ellipsoid a (Eq. ii)).
3
no = 2 ~~~~$~~) ~ exp
(- ~) /~
(4)'B with
I
md " (T~~lT~(2T~13)~ ~~~
WC corresponds to the conduction band minimum and WF to the Fermi level. kB and h are
respectively the Boltzmann's and Planck's constants. In absence of stress the total density of conduction electrons is just
Suppose now that the ntravalley scattering does not ontribute to the iezoresistive effect
240 JOURNAL DE PHYSIQtIE III NO?
(hi A (II I)-plane
2' ..
I'
,
B
~~
_,"
c
plane3
a)
~' 3**
M
l~
~**
B
b) ~ 1**
x~
~i ~8 G (
x~
(
~
~
~
~
C) Xl
~
Fig. 1. The geometry of the model a) starting axes b). Definition of successive transformation for
(iii) ellipsoidal energy surfaces. c) Definition of the angle of alignment. ~ for a doubly rotated plate.
level in the strained lattice. Under this assumption the piezoresistive effect arises essentially because the electron energy levels of N-type germanium depend on the state of strain of the
crystal This effect was theoretically investigated [19,27j and changes in energy extrema were
expressed in terms of deformation potentials constants. In particular shifts in the four minima Li (ill) of the conduction band of germanium were theoretically calculated [27j for several systems of stress. For a compressive ill lj stress this model predicts that more electrons will be in the minimum Li ill I) at the expense of the three other minima accordingly to the following
N°2 NON-LINEAR PIEZORESISTANCE EFFECTS 241
calculated shifts:
ALI[lllj
= -14.49 eV x (T[i( x 10~~~m~IN
ALI [liij
= ALI[ill]
= ALI ill()
= -8.51 eV x (T[1(10~~~m~/N (7)
Since the electron density can now be conveniently written as:
~liiij " exP (-~~~~)~ ~~ (8)
the following relation holds
A(Li [ill] WF) = fliT[1 (10)
A(Li(lli] WF)
= -fl2T[1 ill)
At constant temperature we have only a redistribution of the conduction electrons amongst
the four minima Li (ill) so that [2]:
no = np it + 3npijj (12)
In the present configuration (Fig. I) where the external electrical field E[ (0,E[,0) is applied along the xi axis of the (ill) plate, we have calculate the contribution J[ (0, J(~,0) of each
ellipsoids a to the total electric current density ii. Each contribution may be written as:
~~o " ~~l(ill)/l~2~~ (~~)
where ~J[~ is the component of the mobility tensor in the rotated Cartesian system associated with the (ill) plate (Fig. la):
/1(~ = e(T) (14)
m~
We assume that the mobility tensor remains unaffected by the strain, consequently for the
[ill] ellipsoid, the mobility is just the transverse mobility:
/11 =
~~~~ (15)
ml
according to the local mobility tensor attached to the longitudinal and transverse directions of the ellipsoids:
1
j
II
/llj ~ ~l~) l16)
22
1
(
Let be x[*, x[*, x[* the local axes related to a [ill] ellipsoid. The illI] valley lies at an angle
~T = 70°30' of other (ill) valleys. Turning our attention to Figure 16, it clearly appears that
242 JOURNAL DE PHYSIQUE III N°2
a single rotation of
~T degree along the x[ axis ([l10j direction) transforms the illIi ellipsoid
into the iillj ellipsoid. ~laking use of the transformation matrix
lci
S~ 0~13~ ~~l ~l ~ ~~~~
0 0 1
with
C~ = cos ~, S~ = sin
~T
The mobility /1[~illIi related to electrons of the Ill(] valley becomes:
~J[~[lllj
= C(~Ji + S(pjj =
~Jjj
+ pi (18)
The calculations are rather more complicated for [111j and [III) valleys because we have to
take into account two successive transformations. Let us for example start with the [111j
ellipsoid. A first rotation of -~T degrees along the xj direction lying parallel to Bli (Fig. lb)
makes the [lllj direction in coincidence with the [iii) direction. So that applying the tensor transformation rule we obtain the following intermediate mobilities:
p(( = C(/tjj+S(/t1
/l11 ~~~)(~~~~~ ~~~~
/lI( /ll
which refer to the rotated axes xj~, xj~ and x(*.
A second rotation of fl = 60° along the [iii) direction (xi axis) transforms the zj~, xj*,
~j~ axes into the xi, z[. ~[ axes. A similar treatment may be applied to the [[iii valley-, consequently for the two valleys the effective transverse mobility may be expressed as:
/1[~[illj
= /1[~[lllj
= C)(S(pjj + C(~Ji) + S)/11 (20)
where Cp = cos fl and Sp = sin fl.
After some calculation, we finally obtain:
~tl~ll14
=
~tjj
+ jJti 121)
The total electric current density may be written as:
~~ ~ e (njlll)~l + R[lli)(/l~2(~~~j ~ ~#~2'~~~j)) (22) Making use of equations (10), (12), iii) and (21) yields to the final equation for the electrical
conductivity of the strained plate:
~~~ l ~~eA
~~"
~
~~~ ~~ ~
~~l~~~~
It is obvious that for zero stress (A - 0) equation (23) leads to the isotropic conductivity
ao (18j:
°° " ~)° [~ll + 2~ij (?~)
N°2 NON-LINEAR PIEZORESISTANCE EFFECTS 243
2.2. FIRST, SECOND AND THIRD ORDER COEFFICIENTS. The geometrical configuration corresponds to a gauge whose output is the fractional change, AR/R, in resistance. Neglecting
dimensions changes with respect to resistivity change we can derive
~~
m
~~)=
~° l (25)~ P0 22 ~22
from equations (23) and (24). We finally find a fractional change in resistivity
~~
~~
i~~~ ~)~ ~~i~~A
~~~~
where the term A is directly proportional to the applied stress T[i.
Assuming a weak state of strain we can expand eA in ascending powers of A. Limiting the expansion to the first term and substituting A from equation (10) we obtain the approximate equation:
AP ~l l~ll fll + fl2
~j j~~~
po ~~ 4(~jj + 2~i) kBT ~~
Thus we easily identify the equivalent piezoresistance coefficient 7r[~ii as
~~~~~ 4(~~+)~i) ~i~/~ ~~~~
Let us recall that Mason [8] calculated the piezoresistance coefficient for rotated axes:
ir[~~~ =
~" ~l fli + fl2
2 l~l1 + 2~1i k~T 129)
Consequently these two equivalent piezoresistance coefficients satisfy the following simple re-
lation:
7~~lll ~~7~~211 (~~)
Thus provided the uniaxial stress T[i was applied along the longitudinal axis of an ellipsoid, N-type silicon and germanium behave quite identically: the piezoresistive effect verifies law
(30) which refers to Cartesian axes connected with the ellipsoid axes.
Now retaining second and third order terms in the expansion of equation (26) we can write:
~~)~
7~~211~~l ~ 7~~21111~~~ ~ 7~~2111111~~~ (~~)0 22
where 7r[~iiii and 7r[~iiiiii constitute respectively the non-linear second and third order piezoresistance coefficients:
j~i ~tii fli + fl~ ~
7~~21111 132)
j4(/l(( + 2/l1)j~ ~~~
~~2111111 "
~~~ ~~~))~ ~~~~ ~~~~~~~
(fll + l2)~
~ ~ ~ll + 2~i ))~ kBT (33)
244 JOURNAL DE PHYSIQUE III N°2
Table1. Theoretical formulations of non-linear piezoresistance coejficients. In the above equations A
= (fli + fl2)/kBT. The theoretical values for coejficients are as follows-. N-Type
Si 7r[lint =17.8 x10~~~m~N~~, 7r[~iiii
=
-1.92 x10~~~m~N~~ N-type Ge: 7r[lint
=
19.9 x 10~~~m~N~~~ 7r[~iiii
= 2.78 x 10~~~m~N~~
N-Type Silicon N-Type Germanium
,
i (pjj pi)(pjj 2pi)
~~ i (pjj pi)jpjj 4pi)
7riiiiii 3 i~ + ~~i)2 ? ii~ + ~~i)i~ ~~
i lpi( ~ti)lP( 8Jtjj~ti + 4~t[)
~~ i l~tji Jti)lJt( 32p[ 40pjjpi)
~~~~~~~~~ ~
9 (~Jjj + 2~Ji)~ 24 [2(/1jj + ~Ji ))~
'~
7r121ii
i
Ill)
i~ll/A~ See E~. 132)7r12111111 1~~~
~ll~+/ill
~~~~A~ See Eq. 133)Table I gives the theoretical expressions for linear and non-linear piezoresistance coefficient~
related to slightly doped N-type silicon and gernianium. For N-type silicon, stresses and
current are applied along crystallographic axes. For N-type germanium the geometry is just
that defined in this section (Fig. la). Using published values [2, 24j for coefficients 7rii and n12
(silicon) or calculated values for primed coefficients 7r[1 and 7r[1 (germanium) it is possible to estimate the factor A appearing in equations listed in Table I. Then we can evaluate for the non-linear coefficients the values listed in the caption of Table I.
~ie observe some departures between these values and the data reported by Kanda et fil.
(Tab. II, [7,21,22j). The origin of these deviations is not very clear: experimental values are deduced from resistance measurements performed on specific sensors [21j. In particular for Kanda and co-~vorkers [7,21j the N-L-P-C- were determined by combination oflongitudinal and transverse piezoresistance effects and by assuming that some non-linearities can be neglected.
Mastuda et al. [21j estimated that the determination error can reach 60% for some N-L-P-C-
Unfortunately it remains very difficult to extract the value of intrinsic N-L-P-C- from practical
sensor data. In this condition, we have chosen to retain these published data to evaluate the
influence of orientation on N-L-P-C-
3. A Tensorial Formulation of Non-Linear Piezoresistance Coefficients
3.I. l~AcRoscoPic CONSTITUTIVE EQUATIONS. In this section we consider a reference
system whose Cartesian axes correspond to (100) crystallographic axes and we start ».ith an
equilibrium fundamental state in which the semiconductor is submitted to zero stress and electrical current. From the previous section it is clear that for the semiconductor under stress the cartesian components, E~, of the electric field, E, are function of two types of independent
variables namely, the Cartesian components, Jj, of the electrical current density, J and the stress components, Tki-Assuming small current densities and stresses at constant temperature,
N°2 NON-LINEAR PIEZOItESISTANCE EFFECTS 245
Table II. Independent linear and non-linear piezoresistance coejficients and experimental values. A and N are respectively for assumed zero and not measured. Here we use experi-
mental values previously published by Kanda f?/. Superscript (*) indicates that Kanda's values
are multiplied by a factor 2. (In the theoretical formulation of non-linear effects Kanda distin-
guishes the product T~jTki from the product TkiT~j as soon as subscripts kl differ from subscripts
11).
Linear piezoresistance coefficients 7ropQ lx 10~°m~IN) N-type Silicon P-type Silicon N-type Germanium
7rii -102.2 6.6 -4.7
7r12 53.4 -1.1 -5
7r44 -13.6 -138.1 -137.9
Non linear piezoresistance coefficients 7roPQ(x 10~°m~IN)
2riii
7ri12 -70* A N
7r122 -36 A N
2r144 N -57 N
2r166 A 98 N
2r123 N A N
2r441 N 88* N
1r456 N -102~ N
1r661 -10* -44* N
we can expand E~ in McLaurin series about the reference state:
dE~ = ~~
dJj +
~~
°Jj °Tki dTki
+ ~~~) dJjdJm +
~~~
dTkidT~o + 2
~~~ JjdTki)
2 JJ Jm °Tki°T~o °Jj°Tki
+
~~~ dJj dJmdJp + ~~~
dTkidT~odTqr
6 0Jj0Jm0Jp 0Tki°T~o°Tqr
~3~ ~3~
+3 ~ dJjdJmdTki + 3 ~ JjdTkidT~o)
°Jj°Jm°Tii 0Jj°Tki0T~o (34)
In this equation the derivative ~~, ~~, ~~~
are components of tensors of rank
0Jj 0Tki 0Jj0Tki"'
2, 3, 4 For cubic semiconductors which possess a centre of symmetry, components of all
tensors of odd rank must vanish. Then limiting the expansion to tensor of maximum even rank
6, equation (34) reduces to:
dE~ = ~)dJj +
~
j~(
dJjdTki+ ~~
j/j
dJjdTkidT~o (35)j J ki 2
j ki T~o
if we retain only terms involving increments in stress.
