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a. The piezoresistive effect in single crystalline β-SiC/3C-SiC

Since 3C-SiC can be epitaxially grown on large Si wafers, the cost of 3C-SiC devices can be significantly reduced. 3C-SiC-on-Si platforms are also highly compatible with MEMS processes [14].

A comprehensive comparison of the piezoresistive effect in β-SiC can be found in [111]. Shor et al. reported the first study on the piezoresistive effect in n-type single crystalline 3C-SiC grown on a Si substrate by APCVD [93]. In the work, a negative GF of a 10¹⁶-10¹⁷cm⁻³doped 3C-SiC was found to be -31.8. Several growth

Table2.3:Listofgaugefactors(GF)ofSiCreportedintheliterature PolytypeType/DopantGrowthCarrierconcentrationGF OrientationStress Roomtemp.Hightemp. Single4H-SiC[96]n/N-1.5×1019 20.8-(0001)uniaxial Single6H-SiC[97]n/N-3.8×1018 -29.4-17(250◦ C)(0001)uniaxial Single6H-SiC[98]n/N-2×1019 -22-11(250◦ C)(0001)uniaxial Single6H-SiC[98]p/Al-2×10192712(250◦C)(0001)uniaxial Single3C-SiC[93]n/NAPCVD1018 -31.8-18(450◦ C)[100]uniaxial Single3C-SiC[99]n/NHMCVD1018 -27-[100]uniaxial Single3C-SiC[100]n/NAPCVD--18-7(400◦ C)[100]biaxial Single3C-SiC[101]n/NLPCVD0.4−2×1019 -24.8-11(450◦ C)[100]biaxial Single3C-SiC[102]n/NAPCVDhighlydoped-16-12.5(400◦ C)[100]uniaxial Single3C-SiC[103]p/AlLPCVD5×1018 30.3-[110]uniaxial Single3C-SiC[104]p/AlLPCVD1.3−10×1018 20-30-[110]uniaxial Single3C-SiC[105]p/AlLPCVDhighlydoped2825(300◦ C)[110]uniaxial Poly3C-SiC[106]n/NLPCVDlowdoped-10--biaxial Poly3C-SiC[100]n/NLPCVD--2.1--biaxial Poly3C-SiC[107]p/BoLPCVD1018 −1020 107(200◦ C)-uniaxial NanocrystalineSiC[108]p/AlLPCVD2×1018 14.5--uniaxial AmorphousSiC[109,110]n/NPECVD-49--uniaxial n/NSputtering-31--uniaxial N=Nitrogen Al=Aluminum Bo=Boron

processes had been carried out to develop 3C-SiC, including a selective deposition of cubic silicon carbide on SOI substrates [112], low temperature LPCVD [113] and hot mesh chemical vapour deposition [99]. The GF of the SEG on Si wafer was found to be -18 [112] while Yasui reported a GF of -27 for n-type 3C-SiC [99]. To eliminate the current leakage between SiC and Si layers, 3C-SiC thin films were transferred to or grown on insulating substrates, such as an unintentionally doped 3C-SiC transferred to SiO₂/Si with a GF of -18 [100] and 3C-SiC fabricated by a bonding-free method with a GF of -17.8 [114]. In general, at temperatures above 300^◦C, the GFs of n-type β-SiC were found to be less dependent on the temperature change [93].

Phan et al. reported GFs of p-type 3C-SiC of 25 to 28 at the temperature range of 27^◦C to 300^◦C using an in situ measurement method [105]. It was concluded that a combination of the piezoresistive and thermoresistive effects possibly increases the GF of p-type 3C-SiC to approximately 20% in high temperatures.

b. The piezoresistive effect in single crystalline α-SiC

10 6 2 -2 -6

-10A LM Γ A H K Γ A LM ΓA HK Γ

a) b)

Wave vector (a.u.) Wave vector (a.u.)

Energy (eV)

4H-SiC 6H-SiC

Figure 2.6: Energy band structures of a) 4H-SiC and b) 6H-SiC. Courtesy of Kaeckell et al. [115].

Figure2.6shows the band structures of 4H- and 6H- SiC calculated using the density functional theory (DFT) and the local density approximation (LDA). Subsequently, the solution for the electron-electron interaction can be derived [115,116]. Commer-cial bulk α-SiC wafers have been made widely available with mass productions from CREE Inc. [117]. The primary advantage of using commercial bulk α-SiC wafers is

that the thermal mismatch between layers is completely eliminated since the func-tioning layer and substrate are made from the same material. Following are reports on the piezoresistive effect in α-SiC, including 4H-SiC and 6H-SiC.

The piezoresistive effect in 4H-SiC

Figure 2.7: a) SiC piezoresistors placed on top of cantilever beam, b) Cross-sectional view of layers, and c) Measured longitudinal and transverse gauge factor

of n-type 4H piezoresistors. Reprinted with permission from [96].

There are very limited studies and applications of 4H-SiC utilising the piezoresistive effect due to the wafer availability and difficulties in the metallisation for good Ohmic contact. Akiyama et al. fabricated n-type 4H-SiC piezoresistors from a 4^◦ off-cut 4H-SiC wafer with a sheet resistance of 0.02 Ωcm and a 295 µm-thick substrate [96]. The layer structure is illustrated in Fig. 2.7(b) with three epitaxial layers with different doping concentrations. The piezoresistors were formed by patterning the top n-type epitaxial 1 µm-thick layer with a doping concentration N_d = 1.5× 10¹⁹ cm⁻³. The subsequent layer was 1 µm-thick p-type with a doping concentration N_a= 5.4× 10¹⁴ cm⁻³. An n-type 2 µm-thick buffer layer had a doping concentration N_d= 2.1× 10¹⁹ cm⁻³. The electrical isolation of the piezoresistors was generated by the p–n junction

between two top layers. The results showed the highest GF of 20.8 in a single element piezoresistor in transverse orientation and -10 for the longitudinal orientation with a clustered piezoresistor.

The piezoresistive effect in 6H-SiC

Γ

M K

=

M K

M M

M

M M

M M

K M Γ

L

M M

Figure 2.8: Six conduction band minima at M points in the first Brillouin zone (six semi-ellipsoids and equivalent three full ellipsoids). Reprinted with permission

from [118].

The piezoresistive effect in n-type 6H-SiC is attributed to the electron transfer and mobility shift in the temperature range of from 300K to 773K, with an impurity concentration of from 2 to 3.3×10¹⁹cm⁻³[118,119]. Figure2.9compares theoretical and experimental results of the GFs in 6H-SiC. Three important GFs including the longitudinal, transverse and shear GFs were calculated using band parameters

G_l = 1 + 2νgauge−3D1+ D2

24k_BT

m1− m2

m₁+ m₂(1 + ν_sub)f (2.36)

G_t= 1 + 2ν_gauge+3D₁+ D₂ 24k_BT

m₁− m2

m₁+ m₂(1 + ν_sub)f (2.37) G_s=−3D₁+ D₂

24k_BT

m₁− m2

m₁+ m₂(1 + ν_sub)f (2.38)

f ≡ Fs−1/2/F_s−1/2 (2.39) where νgaugeand νsubare Poisson’s ratios of the gauge and the substrate, and 06C61 is the gauge shape factor [120]. In n-type 6H-SiC, the energy extrema are located in between M and L along ≡U axis of the Brillouin zone [116]. Therefore, a semi-ellipsoidal six-valley model was given to represent the band energy surface, as shown in Fig. 2.8. At an energy level below 12 meV, the constant energy surface contains double-well-like minima, when the energy level higher than 12 meV, the minima became ellipsoidal shapes with a centre at M point [116]. In the temperature range from 298K to 773K, the corresponding energy level 3/2k_BT is in a range from 38.5 to 100 meV (far above 12 meV) [118]. The ellipsoidal approximation represents the energy surface of the extrema at M points and three anisotropic mass components m₁, m₂ and m₃, with the C_v2 symmetry at M points in the wurtzite crystals.

273 373 473 573 673 773

Temperature (K)

273 323 373 423 473 523 573

Temperature (K)

Figure 2.9: Comparison of the GFs reported by Shor, Okojie and Toriyama. a) longitudinal b) transverse. Reprinted with permission from [118].

The piezoresistive effect of a low doped n-type 6H-SiC grown homo-epitaxially on p-type 6H-SiC were characterized by Shor et al. [97]. At room temperature, the longitudinal GFs of n-type 6H-SiC, with the doping concentrations of 1.8×10¹⁷cm⁻³ and 3.3×10¹⁸cm⁻³, were found to be -35 and -29.4, respectively. These GFs decreased by roughly 43% at 250^◦C. At room temperature, the transverse GF was found to be 20 then dropped to 12 at 250^◦C. The piezoresistive effect in n-type 6H-SiC was found to be decreased with increasing temperature and doping level, which was in good agreement with the multi-valley theory. For example, increasing doping level from

1.8 to 3×10¹⁷ cm⁻³ led to the reduction of the GF by 20%, while an increasing temperature from 25 to 250^◦C reduced GF by approximately 40%.

The piezoresistive effect in highly doped (i.e. 2×10¹⁹ cm⁻³) n-type and p-type 6H-SiC were investigated by Okojie et al. [98]. The GF of n-type 6H-6H-SiC was found to be 22 at room temperature, followed by a decrease of 52% at 250^◦C, while the GF of p-type 6H-SiC was 27 at room temperature, then decreased 55% at 250^◦C. The experiments of Shor and Okojie agreed that the magnitude of the GFs in n- and p-type 6H-SiC decreases when increasing the carrier concentration and temperature.