Nernst and Ettingshausen effects in the Laughlin geometry

The ideal reversible thermodynamic cycle visualization of the Nernst effect in Laughlin geometry, excluding the kinetic contribution, is proposed. The Ettingshausen effect is also treated in the fashion using the reverse cycle. The corresponding values of the off-diagonal thermoelectric coefficients are expressed through the ratio of the entropy budget per magnetic flux. Our approach enlightens the profound thermodynamic origin of the relation between the Nernst effect and magnetization currents.

Figure 1

The Nernst heat engine in the cylindrical Laughlin's geometry. Electrons are confined on the cylindrical surface in the presence of a magnetic field $H$ applied perpendicularly to the surface. The upper (I) and lower (II) edges are kept at the temperatures $T_1$ and $T_2$. The heat flow is entering through the upper edge ${\dot{Q}}_1$ and it leaves the system via the lower edge ${\dot{Q}}_2$ as shown by the green arrows. The red and blue arrows correspond to the diamagnetic edge currents $J_{1,2}$ that for $T_1\ne T_2$ do not compensate each other and result in finite tangential net Nernst current.

Figure 2

A scheme of the gedanken Nernst heat engine based on a Laughlin cylinder whose edges are immersed in thermal baths of different temperatures. Panel (a) shows the broken circuit regime, were no work is produced, panel (b) shows the active regime, where the work is produced by a current flowing through the bulb. This current is given by the difference of two edge currents. It is compensated by a thermal current flowing through the cylinder in the opposite direction. Panel (c) shows the Ettingshausen heater based on the inverse effect: The current produced by the generator introduces the imbalance of edge currents that is compensated due to the induced temperature imbalance between the two edges. If the lower edge is kept at a constant temperature of a bath, the temperature at the upper edge increases. Panel (d) shows the $\mu$–$N$ diagram for the Nernst heat engine cycle. $\mathit{\alpha }\to \mathit{\beta }$ (isothermal process): the entering heat flow $Q_1$ from the higher edge kept at the constant temperature $T_1$ to the electronic system at the upper edge results in the increase of the kinetic energy of the electrons at the upper edge and the increase of their entropy, so that $Q_1=T_1{S}_1={\mu }_1(-\delta N)$. $\mathit{\beta }\to \mathit{\gamma }$: the work is produced while electrons pass through the load situated between the edges. As the process is adiabatic and reversible, the incoming and outgoing entropy rates ${\dot{S}}_1={\dot{S}}_2=\dot{S}$ [see Eq. (5)]. $\mathit{\gamma }\to \mathit{\delta }$ (isothermal process): the cooler absorbs the heat flow coming from the lower edge $Q_2=T_2{S}_2={\mu }_2(-\delta N)$. $\mathit{\delta }\to \mathit{\alpha }$: corresponds to the adiabatic return to the initial state. The entropy flow $-{\dot{S}}_1=-{\dot{S}}_2=\dot{S}$.