Thermoelectric Effect

Thermoelectric eﬀect

Manish Anand

Department of Physics, Bihar National College Patna

Manish Anand (Department of Physics) Thermoelectric eﬀect 1 / 8 Introduction

(i) The term “thermoelectric effect” consists of three seperately identified effect: the Seebeck effect, Peltier effect and Thomson effect. (ii) These effects provide a way of direct conversion of temperature differences to electric voltage and vice versa via a thermocouple. (iii) When temperaure difference is maintained between the junction of two different metal, an electric potential is developed. It is known as Seebeck effect. (iv) In Peltier effect, heat is released at one junction and absorbed at the other junction when electric potential is developed. (v) The Thomson effect describes the heating or cooling of a current-carrying conductor when temperature gradient is maintained.

Manish Anand (Department of Physics) Thermoelectric eﬀect 2 / 8 Derivation of thermoelectric equations

1 To understand these effects precisely, we need to do the mathematics of heat and charge flow. 2 As heat flow is primarily driven by temperature gradient, and electric current by is dictated by the variations in the chemical potential of the electrons. We can write the basic relation by just considering these two facts. 3 In deriving the basic thermolectric equations, we also assume that heat and charge flow are function of first derivative of temperature and electric potential.

Manish Anand (Department of Physics) Thermoelectric eﬀect 3 / 8 Some Mathematics

Considering the above facts, we can write following equations for heat ﬂux q and charge density j dT dφ q = f , µ (1) 1 dx dx and dT dφ j = f , µ . (2) 2 dx dx

Here T is the temperature and φµ is the chemical potential per unit electronic charge. To make things simpler, we consider the system to be one-dimensional.

Manish Anand (Department of Physics) Thermoelectric effect 4 / 8 We now perform Taylor series expansion and retain up to first order term. dT dφ q = A + A µ (3) 11 dx 12 dx dT dφ j = A + A µ (4) 21 dx 22 dx These four coefficients will normally need to be determined experimentally for a given material at a given temperature. By convention, the four coefficients are rewritten in terms of four other as: dT dφ q = −(κ + PSσ) − Pσ µ (5) dx dx dT dφ j = −Sσ − σ µ (6) dx dx Here κ is the heat conductivity, σ: the electrical conductivity, S: the Seebeck coefficient and P is the Peltier coefficient of the material.

Manish Anand (Department of Physics) Thermoelectric eﬀect 5 / 8 As the Seebeck eﬀect corresponds to the case when there is no current (j = 0). In this case, Eq. (6) reduces to

dφ dT µ = −S (7) dx dx So, Eq. (7) describes the Seebeck eﬀect. It is often convenient to express the heat ﬂux density q in terms of the current density. Using Eq. (6) in Eq. (5), we obtain following form of q.

dT q = −κ + Pj (8) dx In the absence of current, Eq. (8) is the well-known Fourier’s law of heat conduction.

Manish Anand (Department of Physics) Thermoelectric effect 6 / 8 As, the total energy jE flowing through the system is the sum of the thermal heat flux and the energy carried along by the electrons. So

jE = q + jφµ (9)

In the case of constant energy flow, the same energy flows out of a piece dx of the system as flows into it. Otherwise the negative x-derivative of dE the energy flux density gives the net energy accumulation dt per unit volume as dE dj dq dφ = − E = − − j µ (10) dt dx dx dx Using Eq. (6) and Eq. (8) in the above relation, we obtain

dE d dT j2 dT = κ + − K j. (11) dt dx dx σ dx

Manish Anand (Department of Physics) Thermoelectric effect 7 / 8 Here dP K = − S (12) dT Eq. (12) is the Thomson effect. K is known as the Kelvin (Thomson) coefficient.

Manish Anand (Department of Physics) Thermoelectric eﬀect 8 / 8