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First-order piezoresistance coefficients in silicon crystals
Sensors and Actuators A 118 (2005) 33–43
First-order piezoresistance coefficients in silicon crystals S.I. Kozlovskiy∗ , I.I. Boiko V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, 41, Prospect Nauki, 03028 Kyiv, Ukraine Received 29 April 2004; received in revised form 16 June 2004; accepted 12 July 2004 Available online 25 August 2004
Abstract The first-order piezoresistance coefficients (π11 , π12 ) and (π11 , π12 , π44 ) are calculated for silicon crystals with electron and hole conductivities, respectively. Presented here analytical expressions for the piezoresistance coefficients are obtained for the first time with the account of a dependence of relaxation time τ of charged carriers on elastic strain. Our calculation of the piezoresistance coefficients in p-type silicon was carried out within the framework of the two-band anisotropic model taking into account the conventional dispersion law of light and heavy holes. For silicon crystals with electron conductivity it was obtained a good agreement between the calculated and experimental data in a wide range of impurity concentration (1016 –1020 cm−3 ). For crystals with the hole conductivity the satisfactory quantitative agreement was obtained for low and medium levels of the impurity concentrations (≤3 × 1018 cm−3 ). © 2004 Elsevier B.V. All rights reserved. Keywords: Piezoresistance; Silicon; Elastic strain
1. Introduction The piezoresistance effect in silicon and germanium was experimentally discovered, half century ago by Smith [1]. Large magnitudes of the effect in combination with perfect mechanical properties of silicon, and the possibilities of modern integral technology have made the silicon to be a basic material for the different integral electromechanical sensors. The need for effective sensors has initiated the numerous theoretical and experimental investigations for mechanism of piezoresistance effect and measurements of the piezoresistance tensor components. The first fundamental theoretical examinations of the piezoresistance have been carried out in references [1–5]. The explanation and interpretation of the large piezoresistance effect in n-type silicon was proposed on the basis of multivalley model of the conduction band. Here the elastic strain removes the degeneracy of the valleys occupied in the unstrained state by equal numbers of electrons. ∗ Corresponding author. Tel.: +380 44 265 83 38; fax: +380 44 235 10 80. E-mail address: [email protected] (S.I. Kozlovskiy).
0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.07.003
In p-type silicon large piezoresistance effect was associated with the splitting of light- and heavy-hole bands in the centre of the Brillouin’s zone. Elastic strain can also change the shape of isoenergetic surfaces in these bands. The first- and second-order piezoresistance coefficients were calculated and measured for practical needs on the basis of the conventional models (see references [6–18]). It should be noted that analytical calculations of the piezoresistance coefficients, performed up today, have not covered all cases important for practice. So, widely used analytical expressions for the piezoresistance coefficients in nSi (see references [2,6,16,17]) were obtained at neglect of dependence of the electron relaxation time on strain. In the fundamental papers [3–5] the isoenergetic surfaces of heavy and light holes in p-type silicon were approximated, at certain intermediate stage, by spheres. The subsequent investigation shows that such approximation can become a source of considerable discrepancy between experimental data and results of theoretical calculations (see, e.g., references [19,20]). One has to pay attention to references [8,16], where the strain-induced resistance of p-type silicon has been calculated for large strains on the basis of the model of ellipsoidal isoenergetic surfaces of holes (the same approximation was
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S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
used in earlier articles [21,22] as well). But such model is not acceptable outside of the area of low temperatures and large strains. Consequently, outcomes of these works do not give a possibility to calculate the first-order piezoresistance coefficients. In reference [18] the shear piezoresistance coefficient in p-type silicon π44 was calculated on basis of two-band model modified by influence of the third spin–orbit split-off band. Calculations performed neglected the dependences of the relaxation time on strain and on energy of holes. In reference [7] the first- and second-order piezoresistance coefficients for p-type silicon were calculated numerically on the basis of three-band model proposed earlier in reference [19]. The accepted model is valid for low impurity concentrations and temperatures. At temperature 77 K the mentioned model is suitable at hole concentrations below 6 × 1018 cm−3 . In present article, we calculate the first-order piezoresistance coefficients for n- and p-type silicon crystals within a quite wide range of impurity concentrations and temperatures. In the course of these calculations we took into account the dependence of the free carrier relaxation time on elastic strain, considering this point as very important. The basic attention was paid to cases unexplored sufficiently before. 2. Statement of the problem Calculation of the piezoresistance coefficients is performed on the following assumptions: 1. Conductivity of considered crystals is strictly monopolar. 2. The total number of charge carriers in the crystal does not depend on an elastic strain. 3. Variation of conductivity of strained crystal is provoked by dependence of carrier energy ε on strain. The elastic strain changes two values depending on energy: the equilibrium distribution function f0 (ε) and the relaxation time τ(ε). 4. The relaxation time τ is isotropic. Its dependence on strain is determined by the dependence of the carrier energy on strain only. 5. The piezoresistance effect is considered in the first order of the elastic stress. 6. The theoretical consideration is based on the standard dispersion laws of the free carriers. The many-valley [1,2] and two-bands models [3–5] will be used for n- and ptype silicon crystals, respectively. In τ-approximation components σ ij of conductivity tensor can be written (see, e.g., references [5,23]): ∂f0 (ε) e2 σij = − 3 τ(ε)ϑi ϑj d 3 k i, j ∈ {x, y, z}, (1) 4π ∂ε X)/∂k ˆ ¯ −1 ∂ε(k, here e is the electron charge, ϑi = h i is i-th component of the group velocity of charge carriers. For strained crystal the carrier energy ε depends on the and also on the magnitude and direction of the wave vectork,
ˆ that strained forces represented by the elastic stress tensorX; ˆ is ε = ε(k, X). Now we assume that energy of free carriers in strained crystal receives some additional ε , which is a function of ˆ the wave vector k and the stress tensorX: + ε(k, X). ˆ ε = ε(k) (2) X) ˆ is small in comparison with If the magnitude ε(k, the middle kinetic energy of the free carriers, the linear addition to components of tensor of conductivity is [4,5] ∂f0 (ε) e2 τ(ε) ϑi ϑj d 3 k σij = − 3 4π ∂ε e2 ∂ ∂f0 (ε) =− 3 ε τ(ε) ϑi ϑj ϑi ϑj d 3 k 4π ∂ε ∂ε 3 1 ∂f0 (ε) ∂ε ∂ε + τ(ε) ϑj + ϑi d k . (3) h ¯ ∂ε ∂ki ∂kj For unstrained crystal the conductivity is represented by the Eq. (1) at ε = ε0 (k). We use the first order piezoresistance coefficients determined by the relation (see references [6,23]) ˆ 1 ∂σij (X) πijkl = − , i, j, k, l ∈ {x, y, z}. (4) ˆ = 0) ∂Xkl X σij (X =0 As far as in the unstrained state conductivity of silicon crystal ˆ = 0) = δij σ0 (δij is the Kronecker delta). is isotropic σij (X When stress is applied conductivity tensor of silicon crystal could be written as ˆ = σ0 (δij − πijkl Xkl ). (5) σij (X) kl
For investigated here cases we use only linear approximation ˆ Than for components σij (X). π11 ≡ πxxxx = −
1 σxx − σ0 ; Xxx σ0
π12 = πxxyy = πyyxx = −
1 σyy − σ0 ; Xxx σ0
π44 ≡ 2πxyxy = −
1 σxy Xxy σ0
For uniaxial stress applied along the unit vector w we can write: Xij = −Xwi wj , where wi(j) are the directional cosines of the vector w relative to the principal crystallographic axes. The further consideration we carry out separately for nand p-type silicon crystals.
3. Piezoresistance of n-type silicon The conduction band of silicon contains six equivalent valleys [1,2]. In k-space isoenergetic surfaces of the valleys
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
Fig. 1. Schematic diagram of the valleys in k-space for n-type silicon.
are ellipsoids with principal axes oriented along the fourfold crystallographic axes (see Fig. 1). Each of the isoenergetic surfaces is characterized by longitudinal m and transverse m⊥ effective masses. The valleys are rather prolate (m = L m⊥ , L = 4.8). In the unstrained state the conductivity of unstrained silicon crystal is isotropic owing to the symmetry of disposition of the valleys; thus, the conductivity tensor degenerates into scalar. We investigate the situation when the main piezoresistance effect is defined by the strain-induced displacement of the different valleys in energy space. The different energy displacement of different valleys makes them non-equivalent because some parts of electron transfer from upper to lower valleys (the electron-transfer mechanism [1,2]). In this case the anisotropy of the separate valley plays the main role. As result the strain becomes the reason of anisotropy of the crystal. The shape of the isoenergetic surfaces of the valleys assumed to be not affected by strain. In reality, form of ellipsoids varies also when a stress is applied. But we consider this effect as negligible for values of the coefficients π11 and π12 . Our arguments are based on a small ratio of the piezoresistance coefficients: π44 /π11 . The value of the shear piezoresistance coefficient π44 is totally determined by variation of shape of ellipsoids; the value π11 –mainly by the electrontransfer mechanism. So our assumption of unchanged form of ellipsoids is well grounded (see also reference [17]). Note, that according to experimental results [24] the relative change of electron masses in valleys due to the uniaxial stress X = 245 MPa along [1 0 0]-axis is rather small (about 0.1%). In this section, we calculate the most interesting, for practice, longitudinal π11 and transverse π12 piezoresistance coefficients. Satisfactory calculation of shear piezoresistance coefficient π12 in n-type silicon was carried out in reference [11]. Here we consider the conductivity of the n-type silicon crystal subjected to the uniaxial compression (tension). We direct the Ox, Oy, Oz axes along the [1 0 0], [0 1 0], [0 0 1] crystallographic directions, respectively. For silicon crystal subjected to stress along the [1 0 0] direction the non-zero
35
Fig. 2. Schematic diagram of bottoms shift of electron valleys under uniaxial stress along the crystallographic direction [1 0 0].
components of the strain tensor eˆ are (see, e.g., reference [22]) exx = s11 X,
eyy = ezz = s12 X,
(6)
here X is the magnitude of stress, the values s11 and s12 are elastic compliance constants. For tension and compression the stress X is considered as negative and positive value, respectively. Let us number the valleys as it is shown in Fig. 1. For the assumed uniaxial compression energy of two bottoms of the valleys (1,4) reduces with the magnitude Ex as it is shown in Fig. 2. At the same time the energies of other four valleys increase with the magnitudes Ey (valleys 2,5) and Ez (valleys 3,6), respectively [17]. The energy shift Ei of the valley belonging to Oi axis (here i ∈ {x, y, z}) could be expressed trough the strain components and the deformation potential constants Ξd (dilation) and Ξu (shear) [6,17]: Ei = Ξd Sp(ˆe) + Ξu eii ,
(7)
here Sp(ˆe) = exx + eyy + ezz . As far as the energy shifts of the valleys do not depend on the wave vector, the difference between energies of the valley bottoms is En = Ex − Ey = Ex − Ez = Ξu (s11 − s12 )X. (8) In the unstrained state the equilibrium distribution function of electrons in the v-th valley is f0 (v) (ε) =
f0 (ε) . 6
(9)
The equilibrium distribution function f0 (ε) is normalized on the electron density n: f0 (ε)d 3 k = 4π3 n. (10) In the linear approximation the requirement of the electron density conservation can be written so:
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S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
6
(v)
v=1
=
6
(v) ∂f0 (ε) (v) 3 d k ε ∂ε
(v) f0 (ε) +
r=1
=
i, j ∈ {x, y, z}
1 ∂f0 (ε) (v) 3 (v) f0 (ε) + d k ε 6 ∂ε
6 v=1
=
Taking into account the condition (10) we obtain from Eqs. (15) and (16)
e2 n ∂ 1/2 (v) ε τ(ε) , σij = −δij (v) ε(v) ε−1/2 ∂ε 6mj
f0 (ε + ε(v) )d 3 k
6
f0 (ε)d 3 k. (v)
2 (2m + m⊥ ) σ0 = e2 n 9 m m⊥ (11)
v=1
here ε(v) is the energy shift of the v-th valley (v = 1, . . ., 6, see Fig. 1) induced by the strain (see Eq. (2)). It follows from Eq. (11): 6
ε(v) = 0.
(12)
v=1
The energy displacement of the valleys ε(v) can be found from Eqs. (8) and (12) (see Fig. 2) ε(1) = ε(4) = ε
(2)
= ε
(3)
−2En ; 3
= ε
(5)
= ε
(6)
En = , 3
(13)
if energy shifts of the valleys do not depend on the wave vector the variation of conductivity of v-th valley, given earlier by the by Eq. (3), is simplified now: (v) ∆σij
e2 ε(v) =− 3 4π 6
∂f0 (ε) ∂ τ(ε) ϑi ϑj d 3 k. ∂ε ∂ε
(14)
For the ellipsoids oriented along axes of the fourfold symmetry the diagonal components of the conductivity tensor do not vanish only. Using Eqs. (1), (14) and the ellipsoid disper (here m = (¯h2 /2)(km sion lawε0 (k) ˆ −1 is the second rank ˆ −1 k) tensor of reciprocal effective mass) that was applied to each of the six ellipsoids, we obtain (see Appendix A):
2 2m m2 e ⊥ ∂f0 (ε) (v) ˆ σii (X ε3/2 τ(ε) = 0) = − dε (15) 3 (v) 2 ∂ε 9π h ¯ mi
(v)
σii = δij
e2
2m m2⊥
ε(v) (v) 6π2h ¯ 3 mi ∂f0 (ε) × ε1/2 τ(ε) dε ∂ε
(v)
(17)
ε
−1/2
∂ 3/2 ε τ(ε) . ∂ε
(18)
The angular brackets in Eqs. (17) and (18) designate the Fermi average of arbitrary function of energy: ∞ f0 (ε)A(ε)ε1/2 dε A(ε) = 0 ∞ . (19) 1/2 dε 0 f0 (ε)ε The strain induced variation of conductivity tensor components can be found by summation over all valleys:
2e2 n (v) −1/2 ∂ 1/2 σij = σij = δij En ε ε τ(ε) , 9Mj ∂ε v (20) here Mx −1 = m −1 − m⊥ −1 = (1 − L)/m , Mz −1 = −1/2Mx . Thus, −1/2 ∂ 1/2 ε τ(ε) σij m En ∂ε ε . = δij ∂ σ0 Mj (2L + 1) ε−1/2 ∂ε ε3/2 τ(ε)
My −1 =
(21)
For further calculations we use a common model for energy dependence of relaxation time τ() (see references [17,23,25,26]): ε r , (22) τ(ε) = τ0 kB T here τ 0 is the constant, which does not depend on the carrier energy, kB is the Boltzmann constant, T is the absolute temperature, r is the factor determined by basic mechanism of scattering. Using Eqs. (21) and (22) we obtain the following resulting expressions for the piezoresistance coefficients: σxx En 1 − L 1 + 2r εr−1 π11 = − = −2π12 = Xσ0 X 2L + 1 3 + 2r εr =
Ξu (s11 − s12 )(1 − L) Fr−1/2 (η) . kB T (2L + 1)(r + 3/2) Fr+1/2 (η)
(23)
For further discussion it is convenient to represent the expression as follows: (n)
π11 (r, η) = (v = 1, . . . , 6)
(16)
here mi is the electron effective mass of the v-th ellipsoid in the direction of the i-th principal axis.
Π11 (r)Fr−1/2 (η) . Fr+1/2 (η)
(24)
here (n)
Π11 (r) =
Ξu (s11 − s12 )(1 − L) kB T (2L + 1)(r + 3/2)
(25)
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
and Fr (η) =
1 ,(r+1)
∞
xr dx/[exp (x − η) + 1] is the Fermi
0 (n)
(n),(p)
integral [27], η = EF /kB T , EF is the Fermi energy of electrons (n) or holes (p). For high degeneration, when η 1, 1 Fr−1/2 (η) = (r + 3/2)Fr+1/2 (η) η
(26)
at absence of degeneration, that is at (−η) 1, Fr−1/2 (η) = 1. Fr+1/2 (η)
(27)
Note that neglect by the dependence of the relaxation time on strain leads to the loss of the factor 3/(2r + 3) in the Eq. (25). Thus, the analogous formulas in reference [2] can be considered as valid formally in the case r = 0 only; this is equivalent to total neglect of dependence of the relaxation time on energy. More comments concerning Eq. (24) will be made in Section 5.
4. Piezoresistance of p-type silicon crystals There are three groups of free carriers relevant to the three sub-bands in the hole band of silicon crystal. They are two sub-bands of heavy (HH) and light (LH) holes, and also one sub-band (SO) splitted by the spin–orbit interaction (see Fig. 3). The last band is not considered here as far as it is depopulated in practice [18]. Isoenergetic surfaces of the bands of light and heavy holes have their centres in the centre of Brillouin’s zone (k = 0). The large piezoresistance effect in p-type silicon is associated with destruction of symmetry of crystals subjected to
37
uniaxial strain. The strain induces anisotropic change in energy ε and the latter depends on the wave vector k [3–5] in distinction with the situation explored for n-type silicon. At small strains and high temperatures the majority of holes occupies the states with rather large wave vectors. Here isoenergetic surfaces of holes have shape of the warped spheres that significantly differ from ellipsoids. We use here the form [3–5] eˆ ) = E(v) (k) + ε(v) (k, eˆ ) E(v) (k, eˆ ), = −Ak2 ∓ Θk + ε(v) (k,
(28)
where Θk = B2 k4 + C2 (kx2 ky2 + kx2 kz2 + ky2 kz2 ) and A, B, C are the valence band constants. At calculation we use the following values of band constants (in units of h ¯ 2 /2m0 , where m0 is free electron mass) [28]: A = −4.28, B = −0.68, C2 = 24. The upper and lower signs in Eq (28) refer, respectively, to heavy (v = 1) and light (v = 2) holes. The energy shift eˆ )will be treated in Section 4.1. The form (28) does ε(v) (k, not take into account an influence of the spin–orbit split-off band on isoenergetic surfaces of light and heavy holes. To calculate piezoresistance coefficients it is necessary to know the conductivity of the crystal in the unstrained state (see Eq. (23)) as well. The conductivity of a v-th sub-band in unstrained state after integration over angles ϕ and θ in k-space can be presented in the following form [5]:
2 e2 p (v) −1/2 ∂ 3/2 σ0 = ε ε , (29) τ(ε) v 3 m(v) ∂ε 3 is total density of holes. The ex here p = f0 (ε)d 3 k/4π (v) pressions for m ¯ are given in Appendix B. Under application of stress the strain-induced change of conductivity of the v-th sub-band is [5]
Fig. 3. Dependence of holes energy on wave vector for p-type silicon calculated for common used dispersion law [5,24,25]. HH denotes the heavy-hole band, LH the light-hole band, and SO the spin–orbit split-off band.
38
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
(v) σij
∂ ∂f0 (ε) (v) (v) 3 ε τ(ε) ϑ i ϑj d k ∂ε ∂ε 2 ∂f0 (ε) (v) ∂2 E(v) 3 − τ(ε) ε d k . h ¯ ∂ε ∂ki ∂kj
e2 = 4π3
Table 1 (v) (v) (v) Numerical values of constants λm , Γxx , Γyy , Γxy calculated on the basis of Eqs. (B.1)–(B.4) collected in Appendix B
(v)
(30)
The relative change of conductivity of the crystal can be found by summation over sub-bands: (v) (1) (2) σij + σij σij v σij = (v) = (1) (2) σ0 σ0 + σ 0 v σ0 (1) (2) σij σij 1 = λ (1) + (2) . λ+1 σ σ 0
(31)
0
(1)
σ0
(2) σ0
λm =
,
λ = λ τ λm ,
λτ =
(2)
(1)
(2)
(1)
(2)
Γxx
Γxx
Γyy
Γyy
Γxy
Γxy
0.359
−0.561
−0.016
0.280
0.0080
0.156
0.247
such expression:
Γ (v) X ∂ 1/2 ij σij (v) = 2e2 p ε−1/2 ε τ(ε)v ∂ε m(v)
(35)
(v)
Here (see Appendix B) λ=
(1)
λm
τ01 ; τ02
m(2) m(1)
entered here coefficients Γij depend only on values of the band constants A, B, C. The expressions for coefficients (v) Γij are rather complicated. Therefore, we collect them in Appendix B. Using Eqs. (29)–(35) we obtain resulting expressions for longitudinal and transverse piezoresistance coefficients: π11 = −
(32)
For two-band model we have to use different forms for the relaxation intraband times instead of the unique form (22). Here we assume [5,7,23,25,26]: ε r τv (ε) = τ0v . (33) kB T In the case of scattering by acoustic phonons or by impurities the values τ0v are proportional to (mdv )−3/2 or (mdv )−1/2 , respectively [23,25,26,29], where mdv = −1 m0 (−A ∓ B2 + C2 /5) is the average effective mass of holes for densities of the states for the v-th sub-band [4,5,28]. The upper and lower signs refer, respectively, to heavy (v = 1) and light (v = 2) holes.
=
(1) + Γ (2) ) b(s11 − s12 ) (λΓxx Fr−1/2 (η) xx kB T (1 + λ) (r + 3/2)Fr+1/2 (η)
π12 = − =
σxx σ0 X
(36)
σyy σ0 X
(1) + Γ (2) ) Fr−1/2 (η) b(s11 − s12 ) (λΓyy yy . kB T (1 + λ) (r + 3/2)Fr+1/2 (η)
(37)
(v)
The results of calculation of parameters λm and Γij are shown in Table 1. The presented values were obtained by numerical integration of Eqs. (A2.1)–(A2.4) with the help of Romberg method (see, e.g., reference [30]). 4.2. Uniaxial compression (tension) along the twofold axis [1 1 0]
4.1. Uniaxial compression (tension) along the fourfold axis [100]
In this case the strain tensor components are [22] Taking into account the Eq. (6) we present the strain induced energy shift of the v-th sub-band in the following small stress approximation: (kx2 exx + ky2 eyy + kz2 ezz ) Bb (v) ε (k, eˆ ) = ± √ 3 − Sp(ˆe) k2 2 Θk Bb(s11 − s12 )(3kx2 − k2 ) √ = ±X , 2 Θk
(34)
here b is the deformation potential of the valence band. It follows from this formula that the energy change eˆ ) does not depend on the absolute value of the wave ∆ε(v) (k, After integration of Eq. (30) over angles one gives vector k.
(s11 + s12 )X ; ezz = s12 X; 2 = S44 X/4; exz = eyz = 0.
exx = eyy = e¯ xy
(38)
For the v-th sub-band shift of energy of holes is eˆ ) = ±X ε(v) (k,
Bb(s11 − s12 )(k2 − 3kz2 ) + Dds44 kx ky √ 4 Θk (39)
here D2 = C2 + 3B2 , d is the deformation potential, which defines the band splitting under the shear strain. Using Eqs. (29)–(32) and Eq. (39) we obtain the analytic expression for the piezoresistance shear coefficient:
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
π44 = −2 = −2
σxy σ0 X ds44 (λΓxy1 + Γxy2 ) Fr−1/2 (η) kB T 1+λ (r + 3/2)Fr+1/2 (η)
Table 2 Calculated and experimental values of the first-order piezoresistance coefficients π11 and π12 (10−10 Pa−1 ) for n-type low doped silicon at 300 K
(40)
We have obtained the Eqs. (36), (37) and (40) under high energy approximation proposed in references [3–5] for dispersion laws of heavy and light holes. This is a case when the majority of holes is located far from the point k = 0. Influence of elastic strain is considered as small perturbation of the dispersion law, when kinetic energy of holes is much higher than the strain-induced change of energy ε(v) . Obviously this situation takes place at small strains and at high temperatures. It can be shown that the assumed above approximation is valid for non-degenerated holes if kinetic energy of holes satisfies to the following inequality (see reference [4] for more details): E(2) (k)ε (v) (k, eˆ ) √ kB T max (41) . Θk It should be noted that the right part of inequality (41) does not depend on the absolute value of the wave vector and depends only on angles ϕ and θ. Therefore, a problem to evaluate the maximum of the right part in inequality (41) is rather simplified. For valence band of the constants used here we can present the condition (41) for non-degenerate holes as the following relation between temperature (in Kelvin degrees) and stress (in Pa): T 1.3 × 10−6 |X|
(42)
At high degeneration (η 1) the inequality (41) should be replaced by EF 1.1 × 10−10 |X| (p)
39
(43)
(p)
here EF is the Fermi energy of holes (in eV). 5. Discussion 5.1. Silicon crystals with n-type conductivity Obtained above expressions for piezoresistance coefficients π11 and π12 (see Eq. (23)) differ from the appropriate expressions in the references [2,6,12,16,17], by the factor (2r/3+1)−1 . This factor takes into account a direct influence of strain on the electron relaxation time. The coincidence of our and other analytical expressions takes place in the case r = 0, which corresponds to dominant scattering by optical phonons. It is known [23,25,26] that the probability of scattering by these phonons strongly depends on temperature. It becomes important for temperatures comparable to the Debye temperature θ D . For silicon that is enough high: θ D = 620 K [31]. Therefore, in our calculations provided at temperatures T ≤ 350 K we have to consider the contribution
Experimental data [1] Experimental data [12] Experimental data [6] Calculated data obtained with formulas of reference [2] Calculated data obtained with formula (23) r = −0.5
π11
π12
−10.2 −10.7 −8.4 −7.7
5.3 ≈5.3 4.3 3.86
−11.56
5.78
of optical phonons in scattering as very small in comparison with that of acoustic phonons or impurities. Now we make comparison of experimental and calculated dependencies, which represent relation of piezoresistance coefficient π11 with temperature and impurity concentration. These dependencies were calculated for the following assumed numerical data: the deformation potential Ξu = 8.5 eV, [6,16], the elastic compliance constants (Pa−1 ): s11 = 0.768 × 10−11 , s12 = −0.214 × 10−11 , s44 = 1.26 × 10−11 [22,28]. The experimental dependencies were taken from reference [12]. The experimental and calculated values of the piezoresistance coefficients for the low impurity concentrations are represented by Table 2. As small impurity concentration (here and in Section 5.2) we mean concentrations and temperatures for which the Fermi integrals ratios in Eqs. (23), (36), (37) and (40) are close to unity. The compared curves are shown in Fig. 4. The curves 1, 4 and 3, 6 in Fig. 4 were both calculated on the basis of the formula (23). The curves 1 and 4 relate to the case r = −1/2. The curves 3 and 6 relate to the case r = 0 in component (25) and r = −1/2 in the Fermi integrals (this case corresponds to results in references [2,6,12,16,17]. In the course of our calculations we have also taken into account a diminution of ratio of the longitudinal and transverse mobilities following with growth of the impurity concentration (see [6,12]). The curves 1, 3 and 4, 6 in Fig. 4 were calculated at L = 4.8 and 3.4 accordingly. Apparently, in Fig. 4, the calculated data, obtained with above-mentioned dependence of relaxation time on strain, much better agree with the experimental data than the calculated data obtained with neglect of strain dependence of the relaxation time. 5.2. Silicon crystals with p-type conductivity The analytical expressions for the piezoresistance coefficients (Eqs. (36), (37) and (40)) could be rewritten in the following form: (p)
πlm (r, η) = (p)
Πlm (r)Fr−1/2 (η) , Fr+1/2 (η)
lm ∈ {11, 12, 44}
(44)
here Πlm (r) is the first-order piezoresistance coefficients for pure samples [6,13,16].
40
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
Fig. 4. Calculated (curves 1, 3, 4, 6) and experimental [12] (curves 2, 5) dependencies of the piezoresistance coefficient π11 in n-type silicon plotted as function of temperature for different impurity concentrations. Presented values were obtained with the account (curves 1, 4) and in neglect (curves 3, 6) of dependence of electron relaxation time on elastic strain.
In difference from appropriate expressions in references [6,13,16], we have obtained the analytical expression for (p) factor Πlm (r), which takes into account direct influence of strain on the hole relaxation time and reflects also the bands anisotropy of heavy and light holes. It should be noticed that results of calculations of the de(p) pendence of the value Π44 (r) on r performed on the basis of analytical formula (40) coincides with accuracy ±8% with appropriated results obtained in reference [7] by numerical calculations. The expression for the shear piezoresistance coefficient π44 in reference [18] may be received formally from obtained Eq. (40) if it is assumed r = 0 (that corresponds to dominant scattering by optical phonons). But it is difficult to imagine optical phonons as dominant in temperature range 173–373 K as far as these temperatures are significantly lower of the Debye temperature. The experimental and calculated values of the piezoresistance coefficients for the low impurity concentrations are represented in Table 3. In our calculation the following values of the deformation potential constants were accepted (eV): b = −2.2, d = −5.1 [28]. For scattering of holes by acoustic phonons we use such values [23,25,26,29]: r = −1/2, λτ = (md2 /md1 )3/2 . Hence, here λτ = 0.166. Table 3 Calculated and experimental values of the first-order piezoresistance coefficients π11 , π12 and π44 (10−10 Pa−1 ) in p-type low doped silicon at 300 K Experimental data [1] Experimental data [6] Calculated data obtained with formulas in reference [5] Calculated data obtained with Eqs. (36), (37) and (40)
π11
π12
π44
0.7 −0.0 1.15
−0.1 0.2 −0.58
13.8 11.9 6.31
−0.388
0.194
11.9
In Table 3 alongside with experimental data and results of our calculations we included as well results of calculation performed on the basis of formulas in reference [5]. There at the intermediate stage of calculation the spherical approximation for dispersion law was used. Experimental and our calculated dependences (latter at r = −1/2) of the shear piezoresistance coefficient π44 on temperature are represented in Fig. 5 for different impurity concentrations. As it follows from Fig. 5 a good agreement of calculated and experimental data appears for relatively low impurity concentrations. At impurity concentrations N ≥ 5 × 1019 cm−3 the quantitative agreement is violated. The calculated value of shear piezoresistance coefficient π44 is less than the experimentally observed values. We can see here only qualitative agreement between calculated and experimental data. There is no simple explanation of this fact. For the case of extreme degeneration the Fermi integrals ratio Fr−1/2 (η)/Fr+1/2 (η) in Eq. (40) reduces to the simple factor (r + 3/2)/η (see Eq. (27)). Then for the case r = 0, considered in fact in reference [18], the reduced factor becomes 1.5 times more than for the case r = −1/2 accepted in our paper. So, it appears that calculated in reference [18] value of the shear piezoresistance coefficient π44 may occasionally correspond to experiment data at room temperatures better than our value. One could suppose that this discrepancy is explained by appreciable contribution of impurity scattering. However, the simple calculation for Eq. (40) at r = 3/2 (that formally corresponds to impurity scattering) gives the lower numerical value π44 , than calculation performed at r = −1/2. We can assume that the numerical discrepancies at high impurity are caused by assumption, accepted in course of calculations, about isotropic scattering of holes; then the matrix element of the hole transfer from one state in another does not
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
41
Fig. 5. Calculated (curves 2, 4, 6) and experimental [13] (curves 1, 3, 5) dependencies of the piezoresistance coefficient π44 for p-type silicon on temperature at different impurity concentrations.
depend on scattering angle. However, in reality the scattering by ionized impurities is anisotropic in distinction with scattering by acoustic phonons (see, for example, [23,25,26,32]. Therefore, obtained here piezoresistance coefficients at the case of dominating scattering by ionized impurities can appear unfair. Another possible reason of the discrepancy could be variation of the isoenergetic surfaces of light and heavy holes due to contribution of the splitting by spin–orbit sub-band [5,18,33]; we have neglected in our calculations. In any case, we think that calculation of the piezoresistance coefficients at the high impurity concentration requires a special separate investigation.
ϑ i ϑj =
(A.2) −1
−1 = (m(v) ) where Mxx xx −1 Mzz
−1 (m(v) zz )
−1
−1 = (m(v) ) sin θ cos ϕ; Myy yy
sin θ
(v) mii
= cos θ etc. Here are, principal sin ϕ; values of tensor of reciprocal effective masses for the vellipsoid. 1/2
(v)
σij = −
e2 (2m m2⊥ ) 2π3h ¯3
Iij
ε3/2
∂f0 (v) (ε) τ(ε)dε ∂ε
(A.3)
1/2
(v)
e2 (2m m2⊥ ) Iij ε(v) 2π3h ¯3 (v) ∂f0 (ε) 3/2 ∂ × ε τ(ε) dε, ∂ε ∂ε
σij = −
6. Conclusion Direct dependence of electron relaxation time on strain is important point for calculations of the longitudinal and transverse piezoresistance coefficients in n-type silicon crystals. For p-type silicon crystals a satisfactory quantitative agreement between calculated and experimental [6,13] data was obtained for low and medium levels of impurity concentrations.
2ε , Mij (θ, ϕ)
heref0 (v) (ε) = f0 (ε)/6,
Iij =
π
sin θ dθ
0
(A.4) 2π
dϕ(1/Mij ) =
0
(v)
δij (4π/3mij ). Eq. (16) could be obtained by integration of Eq. (A.4) by parts.
Appendix B At our calculations we have used such set of symbols [5] at i, j ∈ {x, y, z}and v = 1, v = 2:
Appendix A To perform calculations in Eqs. (1) and (14) we use the −1 = (¯h2 /2)(k m obvious dispersion lawε (k) k). It is convenient to replace there the integration over three components of wave vector on integration over energy , polar angle θ and azimuth angle ϕ. Then we have 1/2 √
d 3 k = (2m m2⊥ )
ε dε sin θ dθ dϕ/¯h3
(A.1)
m
(v)
1 = (v) 3γ
2π
π dϕ
0
(m∗v )3/2 sin θ dθ
(B.1)
0
(v)
(v) Γij
=
γij
γ (v)
(B.2)
42
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
1 = 2
(v) γij
2π
π
0
(v)
∆Eij
dϕ 0
(v) 2
(v) 2
× 3(Λi ) (Λj ) m∗v − 2Bij
(v)
(m∗v )3/2 sin θ dθ (B.3)
2π γ (v) =
π dϕ
0
2
∗ 5/2 (Λ(v) sin θ dθ x ) (mv )
(B.4)
0
The following symbols are used in the above expressions: m∗v = [−A ∓ M]−1
M = B2 + C2 (x2 z2 + x2 y2 + y2 z2 )
(B.5) (B.6)
(v)
Λi = i(−A ∓ Ri )
(B.7)
(v)
(B.8)
(v)
(B.9)
∆Eii = ±B(3x2 − 1)/M ∆Eij = ±Dxy/4M (v) = ±xy(2B2 + C2 − 2Rx Ry )/M Bxy
(v) Bii
2 (B + Ri ) = −A ∓ Ri + 2i M
y = sin ϕ sin θ, Ri =
[2B2
x = cos ϕ sin θ,
z = cos θ,
+ C2 (1 − i2 )] 2M
(B.10)
(B.11)
The upper and lower signs in Eqs. (B.5), (B.7)–(B.11) correspond to heavy (v = 1) and light (v = 2) holes, respectively. In expression for π44 (see Eq. (B.9)) we have omitted the term, which does not contribute to the piezoresistance coefficient.
References [1] C.S. Smith, Piezoresistance effect in germanium and silicon, Phys. Rev. 94 (1954) 42–49. [2] C. Herring, E. Vogt, Transport and deformation-potential theory for many-valley semiconductors with anisotropic scattering, Phys. Rev. 101 (1956) 944–961. [3] G.E. Pikus, G.L. Bir, Effect of deformation on the hole energy spectrum of germanium and silicon, Sov. Phys.-Solid State. 1 (1960) 1502–1517. [4] G.L. Bir, G.E. Pikus, Effect of deformation on electrical properties of the p-type germanium and silicon, Sov. Phys.-Solid State. 1 (1959) 1828–1840 (in Russian). [5] G.E. Pikus, G.L. Bir, Symmetry and Strain Induced Effects in Semiconductors, Wiley, New York, 1974.
[6] K. Matsuda, K. Suzuki, K. Yamamura, Y. Kanda, Nonlinear piezoresistance effects in silicon, J. Appl. Phys. 73 (1993) 1838–1847. [7] J.T. Lenkkeri, Nonlinear effects in the piezoresistivity of p-type silicon, Phys. Stat. Sol. (b) 136 (1986) 373–385. [8] K. Suzuki, H. Hasegawa, Y. Kanda, Origin of the linear and nonlinear piezoresistance effects in p-type silicon, Jpn. J. Appl. Phys. 23 (1984) L871–L874. [9] K. Matsuda, Y. Kanda, K. Yamamura, K. Suzuki, Second-order piezoresistance of p-type silicon, Jpn. J. Appl. Phys. (1990) 29. [10] K. Matsuda, Y. Kanda, K. Suzuki, Second-order piezoresistance of n-type silicon, Jpn. J. Appl. Phys. 28 (1989) L1676–L1677. [11] Y. Kanda, K. Suzuki, Origin of the shear piezoresistance coefficient of n-type silicon, Phys. Rev. B 43 (1991) 6754–6756. [12] O.N. Tufte, E.L. Stelzer, Piezoresistive properties of heavily doped n-type silicon, Jpn. J. Appl. Phys. 29 (1990) L1941–L1942. [13] O.N. Tufte, E.L. Stelzer, Piezoresistive properties of silicon diffused layers, J. Appl. Phys. 34 (1963) 313–318. [14] Y.Kanda. Graphical representation of the piezoresistance coefficients in silicon. IEEE Trans. Electron Devices, ED-29 (1982) 64-70. [15] Y. Kanda, Graphical representation of the piezoresistance coefficients in silicon-shear coefficients in plane, Jap. J. Appl. Phys. 26 (1987) 1031–1033. [16] Y. Kanda, Piezoresistance effect of silicon, Sens. Actuators A 28 (1991) 83–91. [17] R.W. Keyes, The effects of elastic deformation on the electrical conductivity of semiconductors, in: F. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 11, Academic Press, New York, London, 1960, pp. 149–221. [18] T. Toriyama, S. Sugiyama, Analysis of piezoresistance in p-type silicon for mechanical sensors, J. Microelectromech. Syst. 11 (2002) 598–604. [19] C.K. Kim, M. Cardona, S. Rodriguez, Effect of free carriers on the elastic constants of p-type silicon and germanium, Phys. Rev. B 13 (1976) 5429–5441. [20] N.T. Gorbachuk, V.V. Mitin, Yu.A. Thorik, Yu.M. Shwarz, On the definition of the strain potential constans of p-Ge type semiconductors using the temperature dependence of piezoresistance, Sov. Phys. Semicond. 15 (1981) 549–652 (in Russian). [21] H. Hasegawa, Theory of cyclotron resonance in strained silicon crystals, Phys. Rev. 129 (1963) 1029–1040. [22] J.C. Hensel, H. Hasegawa, G. Feher, Cyclotron resonance experiments in uniaxially stressed silicon: valence band inverse mass parameters and deformation potentials, Phys. Rev. 129 (1963) 1041–1062. [23] K. Seeger, Semiconductor Physics, Springer-Verlag, Wien, New York, 1973. [24] J.C. Hensel, H. Hasegawa, M. Nakayama, Cyclotron resonance in uniaxially stressed silicon. II. Nature of covalent bond, Phys. Rev. 138 (1965) 238–255. [25] R.A.Smith, Semiconductors, Cambridge, 1959. [26] B.K. Ridly, Quantum Processes in Semiconductor, Clarendon Press, Oxford, 1982. [27] J.S. Blakemore, Semiconductor statistics, Pergamon Press, Inc., New York, 1962. [28] P.Y. Yu, M. Cardona, Fundamentals of semiconductors, in: Physics and Material Properties, Springer, 2002. [29] M. Ashe, J. von Boerzeszkowski, On the temperature dependence of hole mobility in silicon, Phys. Stat. Sol. 37 (1970) 433–438. [30] N.N. Kalitkin, Numerical Methods, Nauka, Moscow, 1978 (in Russian). [31] R.L. Sproull, Modern physics, in: The Quantum Physics of Atoms, Solids, Nuclei, Wiley, Inc., New York, London, 1963. [32] I.I. Boiko, To theory of electron mobility in semiconductors, Solid State Phys. 1 (4) (1959) 574–652 (in Russian). [33] J.F. Creemer, P.J. French, The piezojunction effect in bipolar transistors at moderate stress levels: a theoretical and experimental study, Sens. Actuators A 82 (2000) 181–185.
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Biographies S.I. Kozlovskiy (1952) graduated from the Polytechnic Institute in Kiev (1975), postgraduated at the Institute of Semiconductor Physics of the Ukrainian Academy of Sciences. PhD (1982), Doctor of Sciences (1995). He is working as a researcher at the Institute of Semiconductor Physics of the Ukrainian Academy of Sciences since 1975 (now is leading researcher). His present interest involves physics and mathematical modelling of integrated silicon pressure and acoustic sensors. Author of 60 published scientific articles and one monograph.
43
I.I. Boiko was born in Ukraine in 1935. In 1958 he graduated from the Physical department of Moscow State University. Since 1958 till now is working in Kiev in National Academy of Sciences of Ukraine. Specialist in theoretical physics and mathematical modelling. PhD (1964), Doctor of Sciences (1974), full professor (1991), active member of New York Academy of Sciences (1995), Chief scientific researcher of Ukrainian National Academy of Sciences (1992). Author of more than 120 scientific articles and five monographs.
First-order piezoresistance coefficients in silicon crystals S.I. Kozlovskiy∗ , I.I. Boiko V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, 41, Prospect Nauki, 03028 Kyiv, Ukraine Received 29 April 2004; received in revised form 16 June 2004; accepted 12 July 2004 Available online 25 August 2004
Abstract The first-order piezoresistance coefficients (π11 , π12 ) and (π11 , π12 , π44 ) are calculated for silicon crystals with electron and hole conductivities, respectively. Presented here analytical expressions for the piezoresistance coefficients are obtained for the first time with the account of a dependence of relaxation time τ of charged carriers on elastic strain. Our calculation of the piezoresistance coefficients in p-type silicon was carried out within the framework of the two-band anisotropic model taking into account the conventional dispersion law of light and heavy holes. For silicon crystals with electron conductivity it was obtained a good agreement between the calculated and experimental data in a wide range of impurity concentration (1016 –1020 cm−3 ). For crystals with the hole conductivity the satisfactory quantitative agreement was obtained for low and medium levels of the impurity concentrations (≤3 × 1018 cm−3 ). © 2004 Elsevier B.V. All rights reserved. Keywords: Piezoresistance; Silicon; Elastic strain
1. Introduction The piezoresistance effect in silicon and germanium was experimentally discovered, half century ago by Smith [1]. Large magnitudes of the effect in combination with perfect mechanical properties of silicon, and the possibilities of modern integral technology have made the silicon to be a basic material for the different integral electromechanical sensors. The need for effective sensors has initiated the numerous theoretical and experimental investigations for mechanism of piezoresistance effect and measurements of the piezoresistance tensor components. The first fundamental theoretical examinations of the piezoresistance have been carried out in references [1–5]. The explanation and interpretation of the large piezoresistance effect in n-type silicon was proposed on the basis of multivalley model of the conduction band. Here the elastic strain removes the degeneracy of the valleys occupied in the unstrained state by equal numbers of electrons. ∗ Corresponding author. Tel.: +380 44 265 83 38; fax: +380 44 235 10 80. E-mail address: [email protected] (S.I. Kozlovskiy).
0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.07.003
In p-type silicon large piezoresistance effect was associated with the splitting of light- and heavy-hole bands in the centre of the Brillouin’s zone. Elastic strain can also change the shape of isoenergetic surfaces in these bands. The first- and second-order piezoresistance coefficients were calculated and measured for practical needs on the basis of the conventional models (see references [6–18]). It should be noted that analytical calculations of the piezoresistance coefficients, performed up today, have not covered all cases important for practice. So, widely used analytical expressions for the piezoresistance coefficients in nSi (see references [2,6,16,17]) were obtained at neglect of dependence of the electron relaxation time on strain. In the fundamental papers [3–5] the isoenergetic surfaces of heavy and light holes in p-type silicon were approximated, at certain intermediate stage, by spheres. The subsequent investigation shows that such approximation can become a source of considerable discrepancy between experimental data and results of theoretical calculations (see, e.g., references [19,20]). One has to pay attention to references [8,16], where the strain-induced resistance of p-type silicon has been calculated for large strains on the basis of the model of ellipsoidal isoenergetic surfaces of holes (the same approximation was
34
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
used in earlier articles [21,22] as well). But such model is not acceptable outside of the area of low temperatures and large strains. Consequently, outcomes of these works do not give a possibility to calculate the first-order piezoresistance coefficients. In reference [18] the shear piezoresistance coefficient in p-type silicon π44 was calculated on basis of two-band model modified by influence of the third spin–orbit split-off band. Calculations performed neglected the dependences of the relaxation time on strain and on energy of holes. In reference [7] the first- and second-order piezoresistance coefficients for p-type silicon were calculated numerically on the basis of three-band model proposed earlier in reference [19]. The accepted model is valid for low impurity concentrations and temperatures. At temperature 77 K the mentioned model is suitable at hole concentrations below 6 × 1018 cm−3 . In present article, we calculate the first-order piezoresistance coefficients for n- and p-type silicon crystals within a quite wide range of impurity concentrations and temperatures. In the course of these calculations we took into account the dependence of the free carrier relaxation time on elastic strain, considering this point as very important. The basic attention was paid to cases unexplored sufficiently before. 2. Statement of the problem Calculation of the piezoresistance coefficients is performed on the following assumptions: 1. Conductivity of considered crystals is strictly monopolar. 2. The total number of charge carriers in the crystal does not depend on an elastic strain. 3. Variation of conductivity of strained crystal is provoked by dependence of carrier energy ε on strain. The elastic strain changes two values depending on energy: the equilibrium distribution function f0 (ε) and the relaxation time τ(ε). 4. The relaxation time τ is isotropic. Its dependence on strain is determined by the dependence of the carrier energy on strain only. 5. The piezoresistance effect is considered in the first order of the elastic stress. 6. The theoretical consideration is based on the standard dispersion laws of the free carriers. The many-valley [1,2] and two-bands models [3–5] will be used for n- and ptype silicon crystals, respectively. In τ-approximation components σ ij of conductivity tensor can be written (see, e.g., references [5,23]): ∂f0 (ε) e2 σij = − 3 τ(ε)ϑi ϑj d 3 k i, j ∈ {x, y, z}, (1) 4π ∂ε X)/∂k ˆ ¯ −1 ∂ε(k, here e is the electron charge, ϑi = h i is i-th component of the group velocity of charge carriers. For strained crystal the carrier energy ε depends on the and also on the magnitude and direction of the wave vectork,
ˆ that strained forces represented by the elastic stress tensorX; ˆ is ε = ε(k, X). Now we assume that energy of free carriers in strained crystal receives some additional ε , which is a function of ˆ the wave vector k and the stress tensorX: + ε(k, X). ˆ ε = ε(k) (2) X) ˆ is small in comparison with If the magnitude ε(k, the middle kinetic energy of the free carriers, the linear addition to components of tensor of conductivity is [4,5] ∂f0 (ε) e2 τ(ε) ϑi ϑj d 3 k σij = − 3 4π ∂ε e2 ∂ ∂f0 (ε) =− 3 ε τ(ε) ϑi ϑj ϑi ϑj d 3 k 4π ∂ε ∂ε 3 1 ∂f0 (ε) ∂ε ∂ε + τ(ε) ϑj + ϑi d k . (3) h ¯ ∂ε ∂ki ∂kj For unstrained crystal the conductivity is represented by the Eq. (1) at ε = ε0 (k). We use the first order piezoresistance coefficients determined by the relation (see references [6,23]) ˆ 1 ∂σij (X) πijkl = − , i, j, k, l ∈ {x, y, z}. (4) ˆ = 0) ∂Xkl X σij (X =0 As far as in the unstrained state conductivity of silicon crystal ˆ = 0) = δij σ0 (δij is the Kronecker delta). is isotropic σij (X When stress is applied conductivity tensor of silicon crystal could be written as ˆ = σ0 (δij − πijkl Xkl ). (5) σij (X) kl
For investigated here cases we use only linear approximation ˆ Than for components σij (X). π11 ≡ πxxxx = −
1 σxx − σ0 ; Xxx σ0
π12 = πxxyy = πyyxx = −
1 σyy − σ0 ; Xxx σ0
π44 ≡ 2πxyxy = −
1 σxy Xxy σ0
For uniaxial stress applied along the unit vector w we can write: Xij = −Xwi wj , where wi(j) are the directional cosines of the vector w relative to the principal crystallographic axes. The further consideration we carry out separately for nand p-type silicon crystals.
3. Piezoresistance of n-type silicon The conduction band of silicon contains six equivalent valleys [1,2]. In k-space isoenergetic surfaces of the valleys
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
Fig. 1. Schematic diagram of the valleys in k-space for n-type silicon.
are ellipsoids with principal axes oriented along the fourfold crystallographic axes (see Fig. 1). Each of the isoenergetic surfaces is characterized by longitudinal m and transverse m⊥ effective masses. The valleys are rather prolate (m = L m⊥ , L = 4.8). In the unstrained state the conductivity of unstrained silicon crystal is isotropic owing to the symmetry of disposition of the valleys; thus, the conductivity tensor degenerates into scalar. We investigate the situation when the main piezoresistance effect is defined by the strain-induced displacement of the different valleys in energy space. The different energy displacement of different valleys makes them non-equivalent because some parts of electron transfer from upper to lower valleys (the electron-transfer mechanism [1,2]). In this case the anisotropy of the separate valley plays the main role. As result the strain becomes the reason of anisotropy of the crystal. The shape of the isoenergetic surfaces of the valleys assumed to be not affected by strain. In reality, form of ellipsoids varies also when a stress is applied. But we consider this effect as negligible for values of the coefficients π11 and π12 . Our arguments are based on a small ratio of the piezoresistance coefficients: π44 /π11 . The value of the shear piezoresistance coefficient π44 is totally determined by variation of shape of ellipsoids; the value π11 –mainly by the electrontransfer mechanism. So our assumption of unchanged form of ellipsoids is well grounded (see also reference [17]). Note, that according to experimental results [24] the relative change of electron masses in valleys due to the uniaxial stress X = 245 MPa along [1 0 0]-axis is rather small (about 0.1%). In this section, we calculate the most interesting, for practice, longitudinal π11 and transverse π12 piezoresistance coefficients. Satisfactory calculation of shear piezoresistance coefficient π12 in n-type silicon was carried out in reference [11]. Here we consider the conductivity of the n-type silicon crystal subjected to the uniaxial compression (tension). We direct the Ox, Oy, Oz axes along the [1 0 0], [0 1 0], [0 0 1] crystallographic directions, respectively. For silicon crystal subjected to stress along the [1 0 0] direction the non-zero
35
Fig. 2. Schematic diagram of bottoms shift of electron valleys under uniaxial stress along the crystallographic direction [1 0 0].
components of the strain tensor eˆ are (see, e.g., reference [22]) exx = s11 X,
eyy = ezz = s12 X,
(6)
here X is the magnitude of stress, the values s11 and s12 are elastic compliance constants. For tension and compression the stress X is considered as negative and positive value, respectively. Let us number the valleys as it is shown in Fig. 1. For the assumed uniaxial compression energy of two bottoms of the valleys (1,4) reduces with the magnitude Ex as it is shown in Fig. 2. At the same time the energies of other four valleys increase with the magnitudes Ey (valleys 2,5) and Ez (valleys 3,6), respectively [17]. The energy shift Ei of the valley belonging to Oi axis (here i ∈ {x, y, z}) could be expressed trough the strain components and the deformation potential constants Ξd (dilation) and Ξu (shear) [6,17]: Ei = Ξd Sp(ˆe) + Ξu eii ,
(7)
here Sp(ˆe) = exx + eyy + ezz . As far as the energy shifts of the valleys do not depend on the wave vector, the difference between energies of the valley bottoms is En = Ex − Ey = Ex − Ez = Ξu (s11 − s12 )X. (8) In the unstrained state the equilibrium distribution function of electrons in the v-th valley is f0 (v) (ε) =
f0 (ε) . 6
(9)
The equilibrium distribution function f0 (ε) is normalized on the electron density n: f0 (ε)d 3 k = 4π3 n. (10) In the linear approximation the requirement of the electron density conservation can be written so:
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S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
6
(v)
v=1
=
6
(v) ∂f0 (ε) (v) 3 d k ε ∂ε
(v) f0 (ε) +
r=1
=
i, j ∈ {x, y, z}
1 ∂f0 (ε) (v) 3 (v) f0 (ε) + d k ε 6 ∂ε
6 v=1
=
Taking into account the condition (10) we obtain from Eqs. (15) and (16)
e2 n ∂ 1/2 (v) ε τ(ε) , σij = −δij (v) ε(v) ε−1/2 ∂ε 6mj
f0 (ε + ε(v) )d 3 k
6
f0 (ε)d 3 k. (v)
2 (2m + m⊥ ) σ0 = e2 n 9 m m⊥ (11)
v=1
here ε(v) is the energy shift of the v-th valley (v = 1, . . ., 6, see Fig. 1) induced by the strain (see Eq. (2)). It follows from Eq. (11): 6
ε(v) = 0.
(12)
v=1
The energy displacement of the valleys ε(v) can be found from Eqs. (8) and (12) (see Fig. 2) ε(1) = ε(4) = ε
(2)
= ε
(3)
−2En ; 3
= ε
(5)
= ε
(6)
En = , 3
(13)
if energy shifts of the valleys do not depend on the wave vector the variation of conductivity of v-th valley, given earlier by the by Eq. (3), is simplified now: (v) ∆σij
e2 ε(v) =− 3 4π 6
∂f0 (ε) ∂ τ(ε) ϑi ϑj d 3 k. ∂ε ∂ε
(14)
For the ellipsoids oriented along axes of the fourfold symmetry the diagonal components of the conductivity tensor do not vanish only. Using Eqs. (1), (14) and the ellipsoid disper (here m = (¯h2 /2)(km sion lawε0 (k) ˆ −1 is the second rank ˆ −1 k) tensor of reciprocal effective mass) that was applied to each of the six ellipsoids, we obtain (see Appendix A):
2 2m m2 e ⊥ ∂f0 (ε) (v) ˆ σii (X ε3/2 τ(ε) = 0) = − dε (15) 3 (v) 2 ∂ε 9π h ¯ mi
(v)
σii = δij
e2
2m m2⊥
ε(v) (v) 6π2h ¯ 3 mi ∂f0 (ε) × ε1/2 τ(ε) dε ∂ε
(v)
(17)
ε
−1/2
∂ 3/2 ε τ(ε) . ∂ε
(18)
The angular brackets in Eqs. (17) and (18) designate the Fermi average of arbitrary function of energy: ∞ f0 (ε)A(ε)ε1/2 dε A(ε) = 0 ∞ . (19) 1/2 dε 0 f0 (ε)ε The strain induced variation of conductivity tensor components can be found by summation over all valleys:
2e2 n (v) −1/2 ∂ 1/2 σij = σij = δij En ε ε τ(ε) , 9Mj ∂ε v (20) here Mx −1 = m −1 − m⊥ −1 = (1 − L)/m , Mz −1 = −1/2Mx . Thus, −1/2 ∂ 1/2 ε τ(ε) σij m En ∂ε ε . = δij ∂ σ0 Mj (2L + 1) ε−1/2 ∂ε ε3/2 τ(ε)
My −1 =
(21)
For further calculations we use a common model for energy dependence of relaxation time τ() (see references [17,23,25,26]): ε r , (22) τ(ε) = τ0 kB T here τ 0 is the constant, which does not depend on the carrier energy, kB is the Boltzmann constant, T is the absolute temperature, r is the factor determined by basic mechanism of scattering. Using Eqs. (21) and (22) we obtain the following resulting expressions for the piezoresistance coefficients: σxx En 1 − L 1 + 2r εr−1 π11 = − = −2π12 = Xσ0 X 2L + 1 3 + 2r εr =
Ξu (s11 − s12 )(1 − L) Fr−1/2 (η) . kB T (2L + 1)(r + 3/2) Fr+1/2 (η)
(23)
For further discussion it is convenient to represent the expression as follows: (n)
π11 (r, η) = (v = 1, . . . , 6)
(16)
here mi is the electron effective mass of the v-th ellipsoid in the direction of the i-th principal axis.
Π11 (r)Fr−1/2 (η) . Fr+1/2 (η)
(24)
here (n)
Π11 (r) =
Ξu (s11 − s12 )(1 − L) kB T (2L + 1)(r + 3/2)
(25)
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
and Fr (η) =
1 ,(r+1)
∞
xr dx/[exp (x − η) + 1] is the Fermi
0 (n)
(n),(p)
integral [27], η = EF /kB T , EF is the Fermi energy of electrons (n) or holes (p). For high degeneration, when η 1, 1 Fr−1/2 (η) = (r + 3/2)Fr+1/2 (η) η
(26)
at absence of degeneration, that is at (−η) 1, Fr−1/2 (η) = 1. Fr+1/2 (η)
(27)
Note that neglect by the dependence of the relaxation time on strain leads to the loss of the factor 3/(2r + 3) in the Eq. (25). Thus, the analogous formulas in reference [2] can be considered as valid formally in the case r = 0 only; this is equivalent to total neglect of dependence of the relaxation time on energy. More comments concerning Eq. (24) will be made in Section 5.
4. Piezoresistance of p-type silicon crystals There are three groups of free carriers relevant to the three sub-bands in the hole band of silicon crystal. They are two sub-bands of heavy (HH) and light (LH) holes, and also one sub-band (SO) splitted by the spin–orbit interaction (see Fig. 3). The last band is not considered here as far as it is depopulated in practice [18]. Isoenergetic surfaces of the bands of light and heavy holes have their centres in the centre of Brillouin’s zone (k = 0). The large piezoresistance effect in p-type silicon is associated with destruction of symmetry of crystals subjected to
37
uniaxial strain. The strain induces anisotropic change in energy ε and the latter depends on the wave vector k [3–5] in distinction with the situation explored for n-type silicon. At small strains and high temperatures the majority of holes occupies the states with rather large wave vectors. Here isoenergetic surfaces of holes have shape of the warped spheres that significantly differ from ellipsoids. We use here the form [3–5] eˆ ) = E(v) (k) + ε(v) (k, eˆ ) E(v) (k, eˆ ), = −Ak2 ∓ Θk + ε(v) (k,
(28)
where Θk = B2 k4 + C2 (kx2 ky2 + kx2 kz2 + ky2 kz2 ) and A, B, C are the valence band constants. At calculation we use the following values of band constants (in units of h ¯ 2 /2m0 , where m0 is free electron mass) [28]: A = −4.28, B = −0.68, C2 = 24. The upper and lower signs in Eq (28) refer, respectively, to heavy (v = 1) and light (v = 2) holes. The energy shift eˆ )will be treated in Section 4.1. The form (28) does ε(v) (k, not take into account an influence of the spin–orbit split-off band on isoenergetic surfaces of light and heavy holes. To calculate piezoresistance coefficients it is necessary to know the conductivity of the crystal in the unstrained state (see Eq. (23)) as well. The conductivity of a v-th sub-band in unstrained state after integration over angles ϕ and θ in k-space can be presented in the following form [5]:
2 e2 p (v) −1/2 ∂ 3/2 σ0 = ε ε , (29) τ(ε) v 3 m(v) ∂ε 3 is total density of holes. The ex here p = f0 (ε)d 3 k/4π (v) pressions for m ¯ are given in Appendix B. Under application of stress the strain-induced change of conductivity of the v-th sub-band is [5]
Fig. 3. Dependence of holes energy on wave vector for p-type silicon calculated for common used dispersion law [5,24,25]. HH denotes the heavy-hole band, LH the light-hole band, and SO the spin–orbit split-off band.
38
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
(v) σij
∂ ∂f0 (ε) (v) (v) 3 ε τ(ε) ϑ i ϑj d k ∂ε ∂ε 2 ∂f0 (ε) (v) ∂2 E(v) 3 − τ(ε) ε d k . h ¯ ∂ε ∂ki ∂kj
e2 = 4π3
Table 1 (v) (v) (v) Numerical values of constants λm , Γxx , Γyy , Γxy calculated on the basis of Eqs. (B.1)–(B.4) collected in Appendix B
(v)
(30)
The relative change of conductivity of the crystal can be found by summation over sub-bands: (v) (1) (2) σij + σij σij v σij = (v) = (1) (2) σ0 σ0 + σ 0 v σ0 (1) (2) σij σij 1 = λ (1) + (2) . λ+1 σ σ 0
(31)
0
(1)
σ0
(2) σ0
λm =
,
λ = λ τ λm ,
λτ =
(2)
(1)
(2)
(1)
(2)
Γxx
Γxx
Γyy
Γyy
Γxy
Γxy
0.359
−0.561
−0.016
0.280
0.0080
0.156
0.247
such expression:
Γ (v) X ∂ 1/2 ij σij (v) = 2e2 p ε−1/2 ε τ(ε)v ∂ε m(v)
(35)
(v)
Here (see Appendix B) λ=
(1)
λm
τ01 ; τ02
m(2) m(1)
entered here coefficients Γij depend only on values of the band constants A, B, C. The expressions for coefficients (v) Γij are rather complicated. Therefore, we collect them in Appendix B. Using Eqs. (29)–(35) we obtain resulting expressions for longitudinal and transverse piezoresistance coefficients: π11 = −
(32)
For two-band model we have to use different forms for the relaxation intraband times instead of the unique form (22). Here we assume [5,7,23,25,26]: ε r τv (ε) = τ0v . (33) kB T In the case of scattering by acoustic phonons or by impurities the values τ0v are proportional to (mdv )−3/2 or (mdv )−1/2 , respectively [23,25,26,29], where mdv = −1 m0 (−A ∓ B2 + C2 /5) is the average effective mass of holes for densities of the states for the v-th sub-band [4,5,28]. The upper and lower signs refer, respectively, to heavy (v = 1) and light (v = 2) holes.
=
(1) + Γ (2) ) b(s11 − s12 ) (λΓxx Fr−1/2 (η) xx kB T (1 + λ) (r + 3/2)Fr+1/2 (η)
π12 = − =
σxx σ0 X
(36)
σyy σ0 X
(1) + Γ (2) ) Fr−1/2 (η) b(s11 − s12 ) (λΓyy yy . kB T (1 + λ) (r + 3/2)Fr+1/2 (η)
(37)
(v)
The results of calculation of parameters λm and Γij are shown in Table 1. The presented values were obtained by numerical integration of Eqs. (A2.1)–(A2.4) with the help of Romberg method (see, e.g., reference [30]). 4.2. Uniaxial compression (tension) along the twofold axis [1 1 0]
4.1. Uniaxial compression (tension) along the fourfold axis [100]
In this case the strain tensor components are [22] Taking into account the Eq. (6) we present the strain induced energy shift of the v-th sub-band in the following small stress approximation: (kx2 exx + ky2 eyy + kz2 ezz ) Bb (v) ε (k, eˆ ) = ± √ 3 − Sp(ˆe) k2 2 Θk Bb(s11 − s12 )(3kx2 − k2 ) √ = ±X , 2 Θk
(34)
here b is the deformation potential of the valence band. It follows from this formula that the energy change eˆ ) does not depend on the absolute value of the wave ∆ε(v) (k, After integration of Eq. (30) over angles one gives vector k.
(s11 + s12 )X ; ezz = s12 X; 2 = S44 X/4; exz = eyz = 0.
exx = eyy = e¯ xy
(38)
For the v-th sub-band shift of energy of holes is eˆ ) = ±X ε(v) (k,
Bb(s11 − s12 )(k2 − 3kz2 ) + Dds44 kx ky √ 4 Θk (39)
here D2 = C2 + 3B2 , d is the deformation potential, which defines the band splitting under the shear strain. Using Eqs. (29)–(32) and Eq. (39) we obtain the analytic expression for the piezoresistance shear coefficient:
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
π44 = −2 = −2
σxy σ0 X ds44 (λΓxy1 + Γxy2 ) Fr−1/2 (η) kB T 1+λ (r + 3/2)Fr+1/2 (η)
Table 2 Calculated and experimental values of the first-order piezoresistance coefficients π11 and π12 (10−10 Pa−1 ) for n-type low doped silicon at 300 K
(40)
We have obtained the Eqs. (36), (37) and (40) under high energy approximation proposed in references [3–5] for dispersion laws of heavy and light holes. This is a case when the majority of holes is located far from the point k = 0. Influence of elastic strain is considered as small perturbation of the dispersion law, when kinetic energy of holes is much higher than the strain-induced change of energy ε(v) . Obviously this situation takes place at small strains and at high temperatures. It can be shown that the assumed above approximation is valid for non-degenerated holes if kinetic energy of holes satisfies to the following inequality (see reference [4] for more details): E(2) (k)ε (v) (k, eˆ ) √ kB T max (41) . Θk It should be noted that the right part of inequality (41) does not depend on the absolute value of the wave vector and depends only on angles ϕ and θ. Therefore, a problem to evaluate the maximum of the right part in inequality (41) is rather simplified. For valence band of the constants used here we can present the condition (41) for non-degenerate holes as the following relation between temperature (in Kelvin degrees) and stress (in Pa): T 1.3 × 10−6 |X|
(42)
At high degeneration (η 1) the inequality (41) should be replaced by EF 1.1 × 10−10 |X| (p)
39
(43)
(p)
here EF is the Fermi energy of holes (in eV). 5. Discussion 5.1. Silicon crystals with n-type conductivity Obtained above expressions for piezoresistance coefficients π11 and π12 (see Eq. (23)) differ from the appropriate expressions in the references [2,6,12,16,17], by the factor (2r/3+1)−1 . This factor takes into account a direct influence of strain on the electron relaxation time. The coincidence of our and other analytical expressions takes place in the case r = 0, which corresponds to dominant scattering by optical phonons. It is known [23,25,26] that the probability of scattering by these phonons strongly depends on temperature. It becomes important for temperatures comparable to the Debye temperature θ D . For silicon that is enough high: θ D = 620 K [31]. Therefore, in our calculations provided at temperatures T ≤ 350 K we have to consider the contribution
Experimental data [1] Experimental data [12] Experimental data [6] Calculated data obtained with formulas of reference [2] Calculated data obtained with formula (23) r = −0.5
π11
π12
−10.2 −10.7 −8.4 −7.7
5.3 ≈5.3 4.3 3.86
−11.56
5.78
of optical phonons in scattering as very small in comparison with that of acoustic phonons or impurities. Now we make comparison of experimental and calculated dependencies, which represent relation of piezoresistance coefficient π11 with temperature and impurity concentration. These dependencies were calculated for the following assumed numerical data: the deformation potential Ξu = 8.5 eV, [6,16], the elastic compliance constants (Pa−1 ): s11 = 0.768 × 10−11 , s12 = −0.214 × 10−11 , s44 = 1.26 × 10−11 [22,28]. The experimental dependencies were taken from reference [12]. The experimental and calculated values of the piezoresistance coefficients for the low impurity concentrations are represented by Table 2. As small impurity concentration (here and in Section 5.2) we mean concentrations and temperatures for which the Fermi integrals ratios in Eqs. (23), (36), (37) and (40) are close to unity. The compared curves are shown in Fig. 4. The curves 1, 4 and 3, 6 in Fig. 4 were both calculated on the basis of the formula (23). The curves 1 and 4 relate to the case r = −1/2. The curves 3 and 6 relate to the case r = 0 in component (25) and r = −1/2 in the Fermi integrals (this case corresponds to results in references [2,6,12,16,17]. In the course of our calculations we have also taken into account a diminution of ratio of the longitudinal and transverse mobilities following with growth of the impurity concentration (see [6,12]). The curves 1, 3 and 4, 6 in Fig. 4 were calculated at L = 4.8 and 3.4 accordingly. Apparently, in Fig. 4, the calculated data, obtained with above-mentioned dependence of relaxation time on strain, much better agree with the experimental data than the calculated data obtained with neglect of strain dependence of the relaxation time. 5.2. Silicon crystals with p-type conductivity The analytical expressions for the piezoresistance coefficients (Eqs. (36), (37) and (40)) could be rewritten in the following form: (p)
πlm (r, η) = (p)
Πlm (r)Fr−1/2 (η) , Fr+1/2 (η)
lm ∈ {11, 12, 44}
(44)
here Πlm (r) is the first-order piezoresistance coefficients for pure samples [6,13,16].
40
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
Fig. 4. Calculated (curves 1, 3, 4, 6) and experimental [12] (curves 2, 5) dependencies of the piezoresistance coefficient π11 in n-type silicon plotted as function of temperature for different impurity concentrations. Presented values were obtained with the account (curves 1, 4) and in neglect (curves 3, 6) of dependence of electron relaxation time on elastic strain.
In difference from appropriate expressions in references [6,13,16], we have obtained the analytical expression for (p) factor Πlm (r), which takes into account direct influence of strain on the hole relaxation time and reflects also the bands anisotropy of heavy and light holes. It should be noticed that results of calculations of the de(p) pendence of the value Π44 (r) on r performed on the basis of analytical formula (40) coincides with accuracy ±8% with appropriated results obtained in reference [7] by numerical calculations. The expression for the shear piezoresistance coefficient π44 in reference [18] may be received formally from obtained Eq. (40) if it is assumed r = 0 (that corresponds to dominant scattering by optical phonons). But it is difficult to imagine optical phonons as dominant in temperature range 173–373 K as far as these temperatures are significantly lower of the Debye temperature. The experimental and calculated values of the piezoresistance coefficients for the low impurity concentrations are represented in Table 3. In our calculation the following values of the deformation potential constants were accepted (eV): b = −2.2, d = −5.1 [28]. For scattering of holes by acoustic phonons we use such values [23,25,26,29]: r = −1/2, λτ = (md2 /md1 )3/2 . Hence, here λτ = 0.166. Table 3 Calculated and experimental values of the first-order piezoresistance coefficients π11 , π12 and π44 (10−10 Pa−1 ) in p-type low doped silicon at 300 K Experimental data [1] Experimental data [6] Calculated data obtained with formulas in reference [5] Calculated data obtained with Eqs. (36), (37) and (40)
π11
π12
π44
0.7 −0.0 1.15
−0.1 0.2 −0.58
13.8 11.9 6.31
−0.388
0.194
11.9
In Table 3 alongside with experimental data and results of our calculations we included as well results of calculation performed on the basis of formulas in reference [5]. There at the intermediate stage of calculation the spherical approximation for dispersion law was used. Experimental and our calculated dependences (latter at r = −1/2) of the shear piezoresistance coefficient π44 on temperature are represented in Fig. 5 for different impurity concentrations. As it follows from Fig. 5 a good agreement of calculated and experimental data appears for relatively low impurity concentrations. At impurity concentrations N ≥ 5 × 1019 cm−3 the quantitative agreement is violated. The calculated value of shear piezoresistance coefficient π44 is less than the experimentally observed values. We can see here only qualitative agreement between calculated and experimental data. There is no simple explanation of this fact. For the case of extreme degeneration the Fermi integrals ratio Fr−1/2 (η)/Fr+1/2 (η) in Eq. (40) reduces to the simple factor (r + 3/2)/η (see Eq. (27)). Then for the case r = 0, considered in fact in reference [18], the reduced factor becomes 1.5 times more than for the case r = −1/2 accepted in our paper. So, it appears that calculated in reference [18] value of the shear piezoresistance coefficient π44 may occasionally correspond to experiment data at room temperatures better than our value. One could suppose that this discrepancy is explained by appreciable contribution of impurity scattering. However, the simple calculation for Eq. (40) at r = 3/2 (that formally corresponds to impurity scattering) gives the lower numerical value π44 , than calculation performed at r = −1/2. We can assume that the numerical discrepancies at high impurity are caused by assumption, accepted in course of calculations, about isotropic scattering of holes; then the matrix element of the hole transfer from one state in another does not
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
41
Fig. 5. Calculated (curves 2, 4, 6) and experimental [13] (curves 1, 3, 5) dependencies of the piezoresistance coefficient π44 for p-type silicon on temperature at different impurity concentrations.
depend on scattering angle. However, in reality the scattering by ionized impurities is anisotropic in distinction with scattering by acoustic phonons (see, for example, [23,25,26,32]. Therefore, obtained here piezoresistance coefficients at the case of dominating scattering by ionized impurities can appear unfair. Another possible reason of the discrepancy could be variation of the isoenergetic surfaces of light and heavy holes due to contribution of the splitting by spin–orbit sub-band [5,18,33]; we have neglected in our calculations. In any case, we think that calculation of the piezoresistance coefficients at the high impurity concentration requires a special separate investigation.
ϑ i ϑj =
(A.2) −1
−1 = (m(v) ) where Mxx xx −1 Mzz
−1 (m(v) zz )
−1
−1 = (m(v) ) sin θ cos ϕ; Myy yy
sin θ
(v) mii
= cos θ etc. Here are, principal sin ϕ; values of tensor of reciprocal effective masses for the vellipsoid. 1/2
(v)
σij = −
e2 (2m m2⊥ ) 2π3h ¯3
Iij
ε3/2
∂f0 (v) (ε) τ(ε)dε ∂ε
(A.3)
1/2
(v)
e2 (2m m2⊥ ) Iij ε(v) 2π3h ¯3 (v) ∂f0 (ε) 3/2 ∂ × ε τ(ε) dε, ∂ε ∂ε
σij = −
6. Conclusion Direct dependence of electron relaxation time on strain is important point for calculations of the longitudinal and transverse piezoresistance coefficients in n-type silicon crystals. For p-type silicon crystals a satisfactory quantitative agreement between calculated and experimental [6,13] data was obtained for low and medium levels of impurity concentrations.
2ε , Mij (θ, ϕ)
heref0 (v) (ε) = f0 (ε)/6,
Iij =
π
sin θ dθ
0
(A.4) 2π
dϕ(1/Mij ) =
0
(v)
δij (4π/3mij ). Eq. (16) could be obtained by integration of Eq. (A.4) by parts.
Appendix B At our calculations we have used such set of symbols [5] at i, j ∈ {x, y, z}and v = 1, v = 2:
Appendix A To perform calculations in Eqs. (1) and (14) we use the −1 = (¯h2 /2)(k m obvious dispersion lawε (k) k). It is convenient to replace there the integration over three components of wave vector on integration over energy , polar angle θ and azimuth angle ϕ. Then we have 1/2 √
d 3 k = (2m m2⊥ )
ε dε sin θ dθ dϕ/¯h3
(A.1)
m
(v)
1 = (v) 3γ
2π
π dϕ
0
(m∗v )3/2 sin θ dθ
(B.1)
0
(v)
(v) Γij
=
γij
γ (v)
(B.2)
42
S.I. Kozlovskiy, I.I. Boiko / Sensors and Actuators A 118 (2005) 33–43
1 = 2
(v) γij
2π
π
0
(v)
∆Eij
dϕ 0
(v) 2
(v) 2
× 3(Λi ) (Λj ) m∗v − 2Bij
(v)
(m∗v )3/2 sin θ dθ (B.3)
2π γ (v) =
π dϕ
0
2
∗ 5/2 (Λ(v) sin θ dθ x ) (mv )
(B.4)
0
The following symbols are used in the above expressions: m∗v = [−A ∓ M]−1
M = B2 + C2 (x2 z2 + x2 y2 + y2 z2 )
(B.5) (B.6)
(v)
Λi = i(−A ∓ Ri )
(B.7)
(v)
(B.8)
(v)
(B.9)
∆Eii = ±B(3x2 − 1)/M ∆Eij = ±Dxy/4M (v) = ±xy(2B2 + C2 − 2Rx Ry )/M Bxy
(v) Bii
2 (B + Ri ) = −A ∓ Ri + 2i M
y = sin ϕ sin θ, Ri =
[2B2
x = cos ϕ sin θ,
z = cos θ,
+ C2 (1 − i2 )] 2M
(B.10)
(B.11)
The upper and lower signs in Eqs. (B.5), (B.7)–(B.11) correspond to heavy (v = 1) and light (v = 2) holes, respectively. In expression for π44 (see Eq. (B.9)) we have omitted the term, which does not contribute to the piezoresistance coefficient.
References [1] C.S. Smith, Piezoresistance effect in germanium and silicon, Phys. Rev. 94 (1954) 42–49. [2] C. Herring, E. Vogt, Transport and deformation-potential theory for many-valley semiconductors with anisotropic scattering, Phys. Rev. 101 (1956) 944–961. [3] G.E. Pikus, G.L. Bir, Effect of deformation on the hole energy spectrum of germanium and silicon, Sov. Phys.-Solid State. 1 (1960) 1502–1517. [4] G.L. Bir, G.E. Pikus, Effect of deformation on electrical properties of the p-type germanium and silicon, Sov. Phys.-Solid State. 1 (1959) 1828–1840 (in Russian). [5] G.E. Pikus, G.L. Bir, Symmetry and Strain Induced Effects in Semiconductors, Wiley, New York, 1974.
[6] K. Matsuda, K. Suzuki, K. Yamamura, Y. Kanda, Nonlinear piezoresistance effects in silicon, J. Appl. Phys. 73 (1993) 1838–1847. [7] J.T. Lenkkeri, Nonlinear effects in the piezoresistivity of p-type silicon, Phys. Stat. Sol. (b) 136 (1986) 373–385. [8] K. Suzuki, H. Hasegawa, Y. Kanda, Origin of the linear and nonlinear piezoresistance effects in p-type silicon, Jpn. J. Appl. Phys. 23 (1984) L871–L874. [9] K. Matsuda, Y. Kanda, K. Yamamura, K. Suzuki, Second-order piezoresistance of p-type silicon, Jpn. J. Appl. Phys. (1990) 29. [10] K. Matsuda, Y. Kanda, K. Suzuki, Second-order piezoresistance of n-type silicon, Jpn. J. Appl. Phys. 28 (1989) L1676–L1677. [11] Y. Kanda, K. Suzuki, Origin of the shear piezoresistance coefficient of n-type silicon, Phys. Rev. B 43 (1991) 6754–6756. [12] O.N. Tufte, E.L. Stelzer, Piezoresistive properties of heavily doped n-type silicon, Jpn. J. Appl. Phys. 29 (1990) L1941–L1942. [13] O.N. Tufte, E.L. Stelzer, Piezoresistive properties of silicon diffused layers, J. Appl. Phys. 34 (1963) 313–318. [14] Y.Kanda. Graphical representation of the piezoresistance coefficients in silicon. IEEE Trans. Electron Devices, ED-29 (1982) 64-70. [15] Y. Kanda, Graphical representation of the piezoresistance coefficients in silicon-shear coefficients in plane, Jap. J. Appl. Phys. 26 (1987) 1031–1033. [16] Y. Kanda, Piezoresistance effect of silicon, Sens. Actuators A 28 (1991) 83–91. [17] R.W. Keyes, The effects of elastic deformation on the electrical conductivity of semiconductors, in: F. Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 11, Academic Press, New York, London, 1960, pp. 149–221. [18] T. Toriyama, S. Sugiyama, Analysis of piezoresistance in p-type silicon for mechanical sensors, J. Microelectromech. Syst. 11 (2002) 598–604. [19] C.K. Kim, M. Cardona, S. Rodriguez, Effect of free carriers on the elastic constants of p-type silicon and germanium, Phys. Rev. B 13 (1976) 5429–5441. [20] N.T. Gorbachuk, V.V. Mitin, Yu.A. Thorik, Yu.M. Shwarz, On the definition of the strain potential constans of p-Ge type semiconductors using the temperature dependence of piezoresistance, Sov. Phys. Semicond. 15 (1981) 549–652 (in Russian). [21] H. Hasegawa, Theory of cyclotron resonance in strained silicon crystals, Phys. Rev. 129 (1963) 1029–1040. [22] J.C. Hensel, H. Hasegawa, G. Feher, Cyclotron resonance experiments in uniaxially stressed silicon: valence band inverse mass parameters and deformation potentials, Phys. Rev. 129 (1963) 1041–1062. [23] K. Seeger, Semiconductor Physics, Springer-Verlag, Wien, New York, 1973. [24] J.C. Hensel, H. Hasegawa, M. Nakayama, Cyclotron resonance in uniaxially stressed silicon. II. Nature of covalent bond, Phys. Rev. 138 (1965) 238–255. [25] R.A.Smith, Semiconductors, Cambridge, 1959. [26] B.K. Ridly, Quantum Processes in Semiconductor, Clarendon Press, Oxford, 1982. [27] J.S. Blakemore, Semiconductor statistics, Pergamon Press, Inc., New York, 1962. [28] P.Y. Yu, M. Cardona, Fundamentals of semiconductors, in: Physics and Material Properties, Springer, 2002. [29] M. Ashe, J. von Boerzeszkowski, On the temperature dependence of hole mobility in silicon, Phys. Stat. Sol. 37 (1970) 433–438. [30] N.N. Kalitkin, Numerical Methods, Nauka, Moscow, 1978 (in Russian). [31] R.L. Sproull, Modern physics, in: The Quantum Physics of Atoms, Solids, Nuclei, Wiley, Inc., New York, London, 1963. [32] I.I. Boiko, To theory of electron mobility in semiconductors, Solid State Phys. 1 (4) (1959) 574–652 (in Russian). [33] J.F. Creemer, P.J. French, The piezojunction effect in bipolar transistors at moderate stress levels: a theoretical and experimental study, Sens. Actuators A 82 (2000) 181–185.
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Biographies S.I. Kozlovskiy (1952) graduated from the Polytechnic Institute in Kiev (1975), postgraduated at the Institute of Semiconductor Physics of the Ukrainian Academy of Sciences. PhD (1982), Doctor of Sciences (1995). He is working as a researcher at the Institute of Semiconductor Physics of the Ukrainian Academy of Sciences since 1975 (now is leading researcher). His present interest involves physics and mathematical modelling of integrated silicon pressure and acoustic sensors. Author of 60 published scientific articles and one monograph.
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I.I. Boiko was born in Ukraine in 1935. In 1958 he graduated from the Physical department of Moscow State University. Since 1958 till now is working in Kiev in National Academy of Sciences of Ukraine. Specialist in theoretical physics and mathematical modelling. PhD (1964), Doctor of Sciences (1974), full professor (1991), active member of New York Academy of Sciences (1995), Chief scientific researcher of Ukrainian National Academy of Sciences (1992). Author of more than 120 scientific articles and five monographs.