Introduction to Condensed Matter Physics Elasticity M.P. Vaughan
Overview
Overview of elasticity Classical description of elasticity Speed of sound Strain Stress Young s modulus Shear modulus Poisson ratio The harmonic potential Continuum approximation Symmetry properties
Overview of elasticity Symmetry properties The strain tensor Elastic constants Crystal systems Cubic system Waves in elastic media Strain in elastic media
Classical description of elasticity
Elastic properties - the speed of sound We have already seen that, for small q, the group velocity of acoustic phonons is fairly constant. This corresponds to the speed of sound in the material. In terms of sound waves, longitudinal modes longitudinal waves transverse modes shear waves The speed of sound is related to the elastic properties of the medium.
Normal strain Normal strain is defined as the extension of a material divided by its original length (i.e. the fractional change in length). Thus, if a rod of length L is stretched to L + L, then the normal strain is L L In the ith Cartesian direction, this may be written i u x i i..
Shear strain
Shear strain With reference to the previous image, the shear strain (in 2D) is given as the change in angle at corner A. That is, xy u y x The strain tensor may then be written as u y x. xx yx zx xy yy zy xz yz zz.
Normal stress The normal stress s is the force F per unit area applied normally to the crosssectional area A. The normal stress s is defined by s F A.
Shear stress The shear stress t is the force F per unit area applied coplanar to the cross-sectional area A. The shear stress t is defined in the same way as normal stress via t F A.
Cauchy stress tensor Note s is used to denote normal stress as well as shear stress in the above.
Cauchy stress tensor The components of the Cauchy stress tensor are s s s s xx yx zx s s s xy yy zy s s s xz yz zz. The stress vectors T shown in the previous image are obtained from s via T = sn, where n is a normal vector.
Young s modulus Young s modulus is defined as the ratio of the tensile stress applied to a material and the tensile strain that it induces. Thus, if a tensile force F is applied along a rod of length L and cross-sectional area A, then Young s modulus Y is given by Y s FL AL, where L/L is the fractional change in length.
Shear modulus The shear modulus G is defined as the ratio of shear stress to shear strain. Hence s xy G xy.
Poisson ratio The Poisson ration is defined as the negative ratio of transverse to the axial strain.
Poisson ratio In the image transversestrain L' L, axial strain L L, so L' n L L L L'. L
Bulk modulus The bulk modulus is be described as a materials resistance to compression. In terms of the pressure P and volume V, the bulk modulus B is defined as B V dp dv.
The speed of sound Longitudinal waves v L B 4 3 C 1/ 2 Y1 n 1 n 1 2n 1/ 2, where is the density of the material and C is a material constant. Shear waves v S C 1/ 2.
The Harmonic Potential
The harmonic potential Previously, we found that the harmonic potential could be written in the form U 1 2 n' n u T n D nn' u n', where the potential energy terms are incorporated into the entity D nn.
The continuum approximation It can be shown that U may also be rendered in the form U 1 4 n' n T u u D u u. n' n nn' n' n We now consider the continuum limit, in which the material is taken to approximate a classical elastic medium. In line with this assumption, we take u R' ur R' Rur. rr
The continuum approximation This yields U 1 2 u n, nst xs x R t R u R E, where E stn is a fourth rank tensor given by stn E stn 1 2 R R s D n R R. t
Symmetry properties of E stn Note that due to the commutativity of scalar multiplication R s D Since we also have n D RR R D R R. n t E stn is unaffected by interchange of s and t, or and n. t n R D R, n s
Symmetry properties of E stn Due to these symmetries, for a given pair st or n there are six possible values of E stn for the other pair xx, yy, zz, yz, zx, xy. This gives 6 6 = 36 independent numbers required to specify E stn.
Rotational properties of E stn The potential energy is unaffected by a rigid rotation. Generally, such a rotation by an infinitesimal angle dw would yield u R dwn R, where n is the unit vector normal to the rotation.
The strain tensor It can be shown that this implies that U can only depend on the derivatives in the symmetrical combination s 1 2 u x s u x s. This quantity is known as the strain tensor.
The elastic constants Transforming to an integration, the potential energy may then be written U 1 3 s cstn tn d 2 R, nst where the elastic stiffness constants c stn are given by c s tn R s D t R n R D st R n r, 1 Rs DnRt R Dsn Rt, 8 and is the volume of the primitive cell. R
The elastic constants Incorporating the symmetry relations of c stn, we find that the number of independent elastic constants is, in general, reduced to 21. This number may be further reduced by the symmetries of the particular crystal system.
Crystal systems Lattice system Independent elastic constants Triclinic 21 Monoclinic 13 Orthorhombic 9 Tetragonal (1) 7 Tetragonal (2) 6 Rhombohedral (1) 7 Rhombohedral (2) 6 Hexagonal 5 Cubic 3 N.B. The extra categories (1) and (2) are associated with different point groups.
Classical elasticity theory In classical elasticity theory, we use the notation e e n n n 2, n, n, n. We also reduce the subscript on e to e, for = 1,...,6 where xx 1, yy 1, zz 1, yz 4, zx 5, xy 6.
Classical elasticity theory The integral for the potential energy then becomes U 1 3 e Ce d 2 r, where C c snt and the double subscripts s and tn are replaced by single subscripts and respectively in the same way as previously given.
Classical elasticity theory This classical expression has wave solutions of the form where p is the polarisation vector, satisfying the eigenvalue equation 2 w p ikrwt, t pe, u r t sn c snt k s k n p This is of the same form as the solutions to the discrete harmonic mode model in the long wavelength limit. t.
Cubic system For the cubic system, we have only three independent components C 11 c xxxx c yyyy c zzzz, C 12 c xxyy c yyzz c zzxx, C 44 c xyxy c yzyz c zxzx.
Cubic semiconductors Material C 11 (GPa) C 12 (GPa) C 44 (GPa) Ref C 1079 124 579 [1] Si 166 64 80 [2] Ge 126 44 68 [3] 3C-SiC 290 235 55 [4] BN* 820 190 480 [5] GaN* 239 159 155 [6] AlAs 120 57 59 [7] GaAs 119 53 60 [8] InAs 83 45 40 [9] *zinc blende
Waves in elastic media
Waves in cubic, elastic media For a cubic system, the eigenvalue equation 2 w p t sn c snt k s k n p t. reduces to 2 w p x C 11 k C C k k p C C k k p, 12 2 x C 44 44 k x 2 y C y y 44 k 2 z p 12 x 44 x z z etc.
Waves in cubic, elastic media For a longitudinal wave along the k x, k y or k z direction (i.e. the [100] direction), we would then have and hence v L C11 w 1/ 2 k x C11 100., 1/ 2
Waves in cubic, elastic media For a transverse wave along the k y direction polarised in the x direction, we have w C44 1/ 2 k y and similarly for the k x and k z directions. Hence v T C44 1/ 2 100.
Strain in elastic media
Strain in cubic systems bulk modulus Previously, we saw that the bulk modulus is defined as the resistance of a material to compression. In terms of the pressure P and volume V, the bulk modulus B was given as B V P V.
Strain in cubic systems bulk modulus Alternatively, we may describe the bulk modulus in terms of internal energy. For a hydrostatic pressure P, the change in energy with volume is given by U V P. Substituting this into the previous expression B V 2 U 2 V.
Strain in cubic systems dilation We may calculate this by defining the dilation V, V where V is the change in the crystal volume V. For a uniform dilation, this is given by D 3 D xx yy zz.
Strain in cubic systems dilation Now, we require the energy density, which in the primitive cell is given by U 1 2 nst stn so we shall now put V. For the uniform dilation, this expression reduces to s c tn, U 1 2 n c nn nn.
Strain in cubic systems cubical dilation For a cubic crystal, we therefore have U 1 2 11 12 D 6 C 2C. We now differentiate with respect to volume to obtain 2 U 2 6 2 C 2C, 11 12 2 2 D where, since is the initial volume, it is taken to be constant.
Strain in cubic systems cubical dilation The change in is given by so D, 1 D. Making this substitution, we have 2 U 2 1 2 2 11 12 2 D D 6 C 2C.
Strain in cubic systems cubical dilation Hence However, since 2 U 2 B 1 3 C 2C. 11 2 U 2 So, multiplying the RHS by -1 to obtain a positive solution, B 1 3, 12 C 11 2C 12.
Strain In a cubic system, the strain is given by s s s s s s xx yy zz xy yz zx C C C 0 0 0 11 12 12 C C C 12 11 12 0 0 0 C C C 12 12 11 0 0 0 C 0 0 0 44 0 0 C 0 0 0 0 44 0 C 0 0 0 0 0 44 xx yy zz xy yz zx.
Strain in cubic systems shear modulus We saw previously that the shear modulus G is defined as the ratio of shear stress to shear strain. Hence s xy G xy. From the previous matrix equation for the stress, for a cubic system, we therefore have G C 44.
Strain in cubic systems shear modulus In practice, this result may not be accurate due to the deformation of the material and some kind of averaging scheme may be required. Two schemes used are the Voigt average, giving G V and the Reuss average G R C 5 4C 11 3C 44 C 44 C 12 5 11 C 12 C44 3C C. 11 12
Young s modulus and Poisson s ratio From the linear theory of elasticity, we have the relations Y for Young s modulus and v 9BG 3B G 3B 2 2G, 3B G. for Poisson s ratio. Hence, if we know how to choose G we can approximate these quantities in terms of the elastic stiffness constants C ij.
Cubic semiconductors (references) References: [1] McSkimin, H. J. and P. Andreatch, J. Appl. Phys. 43, 2944-2948 (1972) [2] McSkimin, H. J., J. Appl. Phys. 24, 988-997 (1953) [3] Nikanorov S. P. and B. K. Kardashev, Moscow, "Nauka" Publ. House (1985) [4] Gmelins Handbuch der Anorganischen Chemie, 8th edition, Silicium, Part B, Weinheim, Verlag Chemie, GmbH (1959) [5] Grimsditch, M., E.S. Zouboulis, J. Appl. Phys. 76, 832-834 (1994) [6] Wright, A.F., J. Appl. Phys. 82. 2833-2839 (1997) [7] S.Adachi, J. Appl. Phys. 58, R1-R29 (1985). [8] Burenkov, Yu. A., Yu. M. Burdukov, S. Yu. Davidov, and S. P. Nikanorov, Sov. Phys. Solid State 15, 1175-1177 (1973) [9] Burenkov, Yu. A., S. Yu. Davydov, and S. P. Nikanorov, Sov. Phys. Solid State 17, 1446-1447 (1975)