Mechanical properties, elastic constants, lattice vibrations of Silicon (Si)

Si - Silicon

Mechanical properties, elastic constants, lattice vibrations

Basic Parameter
Elastic constants
Acoustic Wave Speeds
Phonon frequencies

Basic Parameter

Bulk moduluss 9.8·1011 dyn/cm2
Density 2.329 g/cm3
Hardness 7 on the Mohs scale
Surface microhardness (using Knoop's pyramid test) 1150 kg/mm2
Cleavage plane { 111 }

Elastic constants

C11 = 16.60·1011 dyn/cm2
C12 = 6.40·1011 dyn/cm2
C44 = 7.96·1011 dyn/cm2
Temperature dependences of elastic constants.
Dashed lines - (Mc. Skimin [1953]).
Circles - (Nikanorov et al. [1971]).

For 400 K < T < 873 K (in units of 1011 dyn/cm2)

C11 ≈16.38 - 1.28·10-3T;
C12 ≈5.92 - 0.48·10-3T
C44 ≈8.17 - 0.59·10-3T

For 300 K

Bulk modulus (compressibility-1) Bs = 9.8·1011 dyn/cm2
Shear modulus C' = 5.2·1011 dyn/cm2
[111] Young's modulus Y111 = 18.5·1011 dyn/cm2
[110] Young's modulus Y110 = 17.0·1011 dyn/cm2
[100] Young's modulus Y100 = 13.0·1011 dyn/cm2
[100] Poisson ratio σ100 = 0.28
[111] Poisson ratio σ111 = 0.26
Si, Ge. Young's modulus vs. direction in the (001) plane.
Wortman & Evans (1965)
Si, Ge. Poisson ratio vs. direction in the (001) plane.
Wortman & Evans (1965)

Acoustic Wave Speeds

Wave propagation Direction Wave character Expression for wave speed Wave speed (in units of 105 cm/s)
[100] VL (C11/ρ)1/2 8.43
VT (C44/ρ)1/2 5.84
[100] Vl [(C11+Cl2+2C44)/2ρ]1/2 9.13
Vt|| Vt||=VT=(C44/ρ)1/2 5.84
Vt⊥ [(C11-C12)/2ρ]1/2 4.67
[111] Vl' [(C11+2C12+4C44)/3ρ]1/2 9.36
Vt' [(C11-C12+C44)/3ρ]1/2 5.10

Phonon frequencies (in units of 1012 Hz)

phonon wavenumbers:   Remarks Referens
Si  νLTO(Γ25') 15.5 1012 Hz T=300K  
  νTA(X3) 4.5 1012 Hz    
  νLAO(X1) 12.3 1012 Hz    
  νTO(X4 ) 13.9 1012 Hz    
  νTA (L3) 3.45 1012 Hz    
  νLA(L2') 11.3 1012 Hz    
  νLO(L1) 12.6 1012 Hz    
  νTO(L3') 14.7 1012 Hz    
Si. Dispersion curves for acoustic and optical phonons.
Dolling (1963) and Tubino et al. (1972)