Article Special Topic: Mechanical Behaviour of Micro- and Nano-Scale Materials

Science China Technological Sciences

, Volume 57, Issue 4, pp 692-698

First online:

Density-functional-theory formulation of classical and quantum Hooke’s law

  • Hao HuAffiliated withDepartment of Materials Science and Engineering, University of Utah
  • , Feng LiuAffiliated withDepartment of Materials Science and Engineering, University of Utah Email author 

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A fundamental property of solid materials is their stress state. Stress state of a solid or thin film material has profound effects on its thermodynamic stability and physical and chemical properties. The classical mechanical stress (σ M) originates from lattice strain (ɛ), following Hooke’s law: σ M=, where C is elastic constant matrix. Recently, a new concept of quantum electronic stress (σ QE) is introduced to elucidate the extrinsic electronic effects on the stress state of solids and thin films, which follows a quantum analog of classical Hooke’s law: σ QE=Ξ(Δn), where Ξ is the deformation potential of electronic states and Δn is the variation of electron density. Here, we present mathematical derivation of both the classical and quantum Hooke’s law from density functional theory. We further discuss the physical origin of quantum electronic stress, arising purely from electronic excitation and perturbation in the absence of lattice strain (ɛ=0), and its relation to the degeneracy pressure of electrons in solid and their interaction with the lattice.


stress in the solid quantum electronic stress quantum Hooke’s law density functional theory