Various carriers (electrons, magnons, phonons, and polarons) and their interactions substantially affect the Seebeck coefficient. The Seebeck coefficient can be decomposed with two contributions, α S = α S,pres + α S,trans, where α S,pres is the sum of contributions to the carrier-induced entropy change, i.e., α S,pres = α S,mix + α S,spin + α S,vib (α S,mix : entropy-of-mixing, α S,spin : spin entropy, and α S,vib : vibrational entropy). The other contribution α S,trans is the net energy transferred in moving a carrier divided by qT (q : carrier charge). The electron's contributions to the Seebeck coefficient are mostly in α S,pres. The α S,mix is usually dominant in lightly doped semiconductors. The change of the entropy-of-mixing upon adding an electron to a system is the so-called Heikes formula α α S , m i x = 1 q ∂ ∂ S m i x ∂ ∂ N = k B q ln ⁡ ⁡ ( 1 − − f e o f e o ) , where f e = N / N a is the ratio of electrons to sites (carrier concentration). Using the chemical potential (μ), the thermal energy (k B T) and the Fermi function, above equation can be expressed in an alternative form, α S,mix = (k B / q)[(E e − μ)/(k B T)]. Extending the Seebeck effect to spins, a ferromagnetic alloy can be a good example. The contribution to the Seebeck coefficient that results from electrons' presence altering the systems spin entropy is given by α S,spin = Δ S spin / q = (k B / q)ln[(2 s + 1)/(2 s 0 +1)], where s 0 and s are net spins of the magnetic site in the absence and presence of the carrier, respectively. Many vibrational effects with electrons also contribute to the Seebeck coefficient. The softening of the vibrational frequencies produces a change of the vibrational entropy is one of examples. The vibrational entropy is the negative derivative of the free energy, i.e., S v i b = − − ∂ ∂ F m i x ∂ ∂ T = 3 N k B T ∫ ∫ 0 ω ω { ℏ ℏ ω ω 2 k B T coth ⁡ ⁡ ( ℏ ℏ ω ω 2 k B T ) − − ln ⁡ ⁡ [ 2 sinh ⁡ ⁡ ( ℏ ℏ ω ω 2 k B T ) ] } D p ( ω ω ) d ω ω , where D p (ω) is the phonon density-of-states for the structure. For the high-temperature limit and series expansions of the hyperbolic functions, the above is simplified as α S,vib = (Δ S vib / q) = (k B / q)Σ i (-Δ ω i / ω i).