Stress arising from a difference in thermal expansion. For a thin film on a substrate it is estimated as σ = EΔαΔT/(1 − ν); if the film's expansion coefficient is larger, the film is in tension after cooling.
Japanese name: 熱応力
Thermal stress appears in 5 practice questions across 3 topics, including Thin films, surfaces and nanomaterials, Thermal properties of materials, High-temperature materials and creep.
If a body is heated uniformly and is free to expand, no stress arises. Thermal stress appears when expansion or contraction is prevented, in one of the following cases: (1) materials with different coefficients of linear thermal expansion are joined, (2) there is a temperature difference within the body, or (3) the body is constrained from outside. A thin film on a substrate is the classic example of (1), and the surface of quenched glass of (2). Heating a bar fixed at both ends by ΔT produces a compressive stress of magnitude EαΔT. Because thermal stress arises without any external load, it is easily overlooked.
A thin film on a sufficiently thick substrate is constrained equally in both in-plane directions, so the biaxial modulus E/(1 − ν) is used and the stress is estimated as σ = EΔαΔT/(1 − ν). For example, cooling a copper film on a silicon substrate (E = 120 GPa, ν = 0.34, α = 17×10⁻⁶ K⁻¹) from 400 °C to 25 °C would give a tensile stress of about 1 GPa if it stayed elastic. Real copper films yield partway through, and the stress often ends up at a few hundred MPa. The room-temperature stress of a film deposited at high temperature is frequently set by plastic deformation during cooling.
Residual stress in a thin film is not only thermal stress. Sputtered films tend to develop compressive stress as energetic particles are driven into them, and evaporated films tend to develop tensile stress as islands join together; such stress created during deposition is called intrinsic stress. The measured residual stress is the sum of thermal and intrinsic stress, so trying to explain all of it from the difference between the deposition and room temperatures does not work. The usual approach is to measure the stress while changing the temperature and to separate out the part that varies in proportion to temperature as the thermal stress.
The standard way to obtain the stress in a film on a wafer is from the radius of curvature R of the wafer's bow, using the Stoney equation σ = Es hs²/{6(1 − νs) hf R} (Es and νs are the substrate's Young's modulus and Poisson's ratio, and hs and hf the substrate and film thicknesses). Thermal stress causes hillocks in aluminum interconnects, thermal fatigue of solder joints, spallation of thermal barrier coatings and cracking of substrates in power semiconductors. Countermeasures include choosing materials with similar expansion coefficients, inserting a layer with an intermediate coefficient, and heating and cooling slowly to keep temperature differences small.
Answer: C. Tensile, about 220 MPa
Δα = 23×10⁻⁶ − 2.6×10⁻⁶ = 20.4×10⁻⁶ K⁻¹ and ΔT = 100 K, so σ = 70×10⁹ × 20.4×10⁻⁶ × 100/(1 − 0.35) ≈ 2.2×10⁸ Pa, about 220 MPa. On cooling, aluminum tries to shrink more than silicon, but constrained by the substrate it cannot, so the film is pulled into tension. Calling it compressive gets the sign wrong. About 143 MPa forgets to divide by (1 − ν). The film is in a biaxial stress state, constrained simultaneously in both in-plane directions, so the biaxial modulus E/(1 − ν) must be used. Thermal stresses like this cause stress migration and voids in interconnects.
A thin film is in a biaxial stress state, constrained in both in-plane directions at once, so it deforms less readily than under uniaxial stress. The biaxial modulus E/(1 − ν) expresses this. For a bar constrained in one direction only, the stress magnitude is EαΔT.
After cooling, the film is in tension if its linear expansion coefficient is larger than the substrate's and in compression if it is smaller. On heating the sign reverses. Intrinsic stress also adds to it, so the actual sign is confirmed by measurements such as wafer bow.
Thermal stress is the name of the stress caused by differences in expansion or temperature. Thermal shock is the cracking that occurs when rapid heating or cooling creates large temperature differences within a body and the resulting thermal stress breaks it; it is one manifestation of thermal stress.
The sign of thermal stress can be judged this way: if the film's thermal expansion coefficient is larger than the substrate's, the film is in tension after cooling; if smaller, in compression.
A thermal oxide film (SiO₂, linear thermal expansion coeffi…For this reason, parts that combine metals with polymers or ceramics develop thermal stress and warping from the difference in expansion when the temperature changes.
Which ordering of linear thermal expansion coefficients nea…On quenching, only the surface tries to contract first and is held back by the interior, creating a tensile thermal stress σ ≈ EαΔT/(1 − ν) at the surface.
Red-hot quartz glass (fused silica) rarely cracks when drop…The lower the thermal conductivity and the thicker the coating, the larger the temperature difference, but thicker coatings spall more easily under thermal stress, so the thickness is limited.
A heat flux of 1.0 MW/m² passes uniformly through the thick…Coefficient of linear thermal expansion Young's modulus Fracture toughness Thermal barrier coating Creep Nickel-based superalloy Critical nucleus radius Ostwald ripening Quantum size effect Stranski–Krastanov growth