The Seebeck coefficient (also known as thermopower, thermoelectric power, and thermoelectric sensitivity) of a material is a measure of the magnitude of an induced thermoelectric voltage in response to a temperature difference across that material, as induced by the Seebeck effect. The SI unit of the Seebeck coefficient is volts per kelvin (V/K), although it is more often given in microvolts per kelvin (μV/K). The use of materials with a high Seebeck coefficient is one of many important factors for the efficient behaviour of thermoelectric generators and thermoelectric coolers. More information about high-performance thermoelectric materials can be found in the Thermoelectric materials article. In thermocouples the Seebeck effect is used to measure temperatures, and for accuracy it is desirable to use materials with a Seebeck coefficient that is stable over time. Physically, the magnitude and sign of the Seebeck coefficient can be approximately understood as being given by the entropy per unit charge carried by electrical currents in the material. It may be positive or negative. In conductors that can be understood in terms of independently moving, nearly-free charge carriers, the Seebeck coefficient is negative for negatively charged carriers (such as electrons), and positive for positively charged carriers (such as electron holes).
Definition
One way to define the Seebeck coefficient is the voltage built up when a small temperature gradient is applied to a material, and when the material has come to a steady state where the current density is zero everywhere. If the temperature difference ΔT between the two ends of a material is small, then the Seebeck coefficient of a material is defined as:
S = − Δ V Δ T {\displaystyle S=-{\Delta V \over \Delta T}}
where ΔV is the thermoelectric voltage seen at the terminals. (See below for more on the signs of ΔV and ΔT.) Note that the voltage shift expressed by the Seebeck effect cannot be measured directly, since the measured voltage (by attaching a voltmeter) contains an additional voltage contribution, due to the temperature gradient and Seebeck effect in the measurement leads. The voltmeter voltage is always dependent on relative Seebeck coefficients among the various materials involved. Most generally and technically, the Seebeck coefficient is defined in terms of the portion of electric current driven by temperature gradients, as in the vector differential equation
J = − σ ∇ V − σ S ∇ T {\displaystyle \mathbf {J} =-\sigma {\boldsymbol {\nabla }}V-\sigma S{\boldsymbol {\nabla }}T}
where J {\displaystyle \scriptstyle \mathbf {J} } is the current density, σ {\displaystyle \scriptstyle \sigma } is the electrical conductivity, ∇ V {\displaystyle \scriptstyle {\boldsymbol {\nabla }}V} is the voltage gradient, and ∇ T {\displaystyle \scriptstyle {\boldsymbol {\nabla }}T} is the temperature gradient. The zero-current, steady state special case described above has J = 0 {\displaystyle \scriptstyle \mathbf {J} =0} , which implies that the two electrical conductivity terms have cancelled out and so ∇ V = − S ∇ T . {\displaystyle {\boldsymbol {\nabla }}V=-S{\boldsymbol {\nabla }}T.}
Sign convention The sign is made explicit in the following expression:
S = − V l e f t − V r i g h t T l e f t − T r i g h t {\displaystyle S=-{\frac {V_{\rm {left}}-V_{\rm {right}}}{T_{\rm {left}}-T_{\rm {right}}}}}
Thus, if S is positive, the end with the higher temperature has the lower voltage, and vice versa. The voltage gradient in the material will point against the temperature gradient. The Seebeck effect is generally dominated by the contribution from charge carrier diffusion (see below) which tends to push charge carriers towards the cold side of the material until a compensating voltage has built up. As a result, in p-type semiconductors (which have only positive mobile charges, electron holes), S is positive. Likewise, in n-type semiconductors (which have only negative mobile charges, electrons), S is negative. In most conductors, however, the charge carriers exhibit both hole-like and electron-like behaviour and the sign of S usually depends on which of them predominates.
Relationship to other thermoelectric coefficients
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