Reverse heat flow with Peltier-induced thermoinductive effect

The inductive component is the only missing components in thermal circuits unlike their electromagnetic counterparts. Herein, we report an electrically controllable reverse heat flow, which can be regarded as a proper equivalent of the “thermoinductive” effect. The underlying concept is the heating and cooling of the ends of the material by the Peltier effect under an applied ac current; this form a negative temperature gradient in the opposite direction in a controllable manner. We have derived the exact solution indicating that this reverse heat flow occurs universally in solid-state systems, even in conventional metallic Cu, and that it is considerably enhanced by thermoelectric properties (i.e., a large Seebeck coefficient and low thermal conductivity). A local cooling of 25 mK was demonstrated in (Bi,Sb) 2 Te 3 , which was explained by our exact solution. This electrically controlled reverse heat flow is directly applicable to the fabrication of a “thermoinductor” in thermal circuits.

3 evidence for the Peltier-induced thermoinductive effect, achieved using an excellent TE material, (Bi,Sb)2Te3, near room temperature.

Modeling of the Peltier-induced thermoinductive effect
In the rectangular parallelepiped TE material shown in Fig. 1(a), Peltier heating and cooling occur at each interface between the material and the metal electrode under an applied current. Consequently, the direction of the heat flow transmitted as a thermal wave is also reversed with the sign change of the ac current 25,26 . The reversing of the current polarity causes thermal wave collision depending on the position in the material ( Fig. 1(b)). An opposite temperature gradient (i.e., a reverse heat flow) can occur in the material at a certain frequency at which the time for reversing the polarity of the ac current is sufficiently faster than the thermal time constant of the material (see the movie included in the Supplementary Information). We derived an exact solution for the heat conduction equation to reveal the temperature distribution in the material.
The temperature rise caused by the Peltier effect T(x,t) at position x and time t is described as (1 ) (1 ) 0 (1 ) /2 (1 ) /2 where S, T0, κ, and l are the Seebeck coefficient, mean temperature, thermal conductivity, and length of the sample, respectively. J is the current density given by J = J0sinωt (ω is the angular velocity 2f; f is the current frequency). β  (ω/2α) 1/2 is the reciprocal of the thermal diffusion length Dth = (α/πf) 1/2 , where α is the thermal diffusivity of the sample. i is the imaginary number. The imaginary part corresponds to the phase of the thermal wave.
Equation (1) is the exact solution for the one-dimensional unsteady-state heat transfer equation (see Methods for calculation details).
Here, we introduce a characteristic dimensionless parameter F  (l/2) 2 f/2α representing the product of the thermal time constant of the sample (l/2) 2 /α and the current frequency. We proposed that F can classify the behavior of the temperature distribution caused by the Peltier heat. Figure 2 shows the relationship between T(x) or heat flow  (e)). The direction of the temperature gradient is periodically inverted with the polarity reversal of the current ( Fig. 1(b)). As can be seen in Fig. 2(h), Q with the opposite (negative) direction occurs near of the sample center (x ~ 0). This uneven heat flow occurring within the material represents a thermal phase delay against the current. This phenomenon can be interpreted in terms of the thermoinductive effect induced by the Peltier effect.
In addition, our calculation based on the exact solution, Eq. (1), shows that while a reverse heat flow can universally occur in any solid material, it is more prominent in TE materials

Electrical resistance changes caused by thermoinductive effect
We present the analytical model for electrical impedance measurements with a four-probe configuration that can detect temperature changes reflected in the electrical voltage by the Seebeck effect (see Method). The temperature changes caused by the reverse heat flow can be measured as electrical signals on the order of mΩ using TE materials. The measured impedance R can be expressed as follows with the ohmic resistance R0 = ρlv/A: where ρ and lv denote the resistivity and voltage terminal distance. z is the TE figure of merit where z = S 2 /κρ. Here, R1 and R2 are the following functions: cos cosh sin cosh cos sinh sin sinh sin cosh cos sinh 2 (cos cosh ) sin sinh where μ  πF  and νμlv/l are defined as functions depending on F and lv/l. The second term on the right side in Eq. (2) represents the component resulting from the TE effect, which is increased or decreased by the correction terms R1 and R2. R1 and R2 are considered to correspond with the resistance and reactance components of the TE voltage, respectively. 6 Therefore, R1 → 1 and R2 → 0 when F → 0 as a dc limit, and R1 → 0 and R2 → 0 when F → ∞ when an ac current with a sufficiently high frequency is applied. The influence of R1 and R2 can be better observed using high-performance TE materials. Cu requires detecting minute changes on the order of nΩ or less thorough electrical resistivity measurements, which is a challenging task (Fig. S2).

Discussion
To investigate the conditions under which the reverse heat flow appears, we discussed the heat flow behavior in the frequency domain. In this study, a reverse heat flow is generated by utilizing the material's thermal inertia (the product of the thermal resistivity and volumetric heat capacity) and the phase delay caused by the external current reversal with periodic modulation. Therefore, the behavior of Q(F) can be separable into the resistorcapacitor (RC) component that originates from the thermal inertia and the oscillation component that is induced by the current reversal (Fig. 4).  Fig. 4). This oscillation component substantially contributes 8 to Q(F) for F > 0.1, causing a significant reverse heat flow at F ~ 1.
In addition, even after Q(F) becomes negative once, Q(F) oscillates with increasing F while repeating its sign inversion. Because the value of F at which the reverse heat flow appears corresponded to that in the case where Q(F) = 0, the first negative Q appeared in the region of π/8 < F < 9π/8. Thereafter, the sign of Q inverts in the π cycle, which corresponds to the cycle of the current's reversing polarity. Further, we find that when μ = (l/2)/Dth = (2n−1)π/2, (n = 1, 2, ...), the sign of Q is inverted. Therefore, we infer that the reverse heat flow in materials can be controlled by the ratio of the sample length and Dth. This thermoinductive effect is a higher-order thermal response due to the Peltier effect, which cannot be obtained in a conventional lumped-parameter model. The authors declare no competing interests.

Methods
Experimental setup. Impedance was measured using a polycrystalline p-type (Bi,Sb)2Te3 material. The dimensions of the sample were 15 mm × 4 mm × 1 mm as shown in Fig. S3 (see Supplemental information). To capture the Peltier heat generated at the edge of the sample effectively, Au electrodes were fabricated by sputtering. Upon confirming that the adhesion between the sample and the lead wires was insufficient, the measured dc voltage was found to have a large deviation. To suppress the heat conduction of the lead wire, thin Au wires ( = 30 μm) were connected to the sample using Ag paste. To isolate the sample thermally, the sample was suspended with sufficiently long Au wires. The sample space was covered with a radiation shield to reduce heat loss due to thermal radiation. The measurement apparatus was assembled in a vacuum chamber, and measurement was performed under high vacuum conditions (10 −3 Pa or less) to suppress the influence of heat convection. The impedance was measured using a commercial impedance analyzer (HIOKI, IM3590).
Coaxial cables were used to connect the case, and measurements were performed using the four-terminal pair method. A schematic of the measurement setup in this study is shown in Fig. S3. Although we had to ensure that the Joule heat did not exceed the Peltier heat, the applied current dependency was measured in advance, and the measurement was performed with an rms ac current of 10 mA, which did not affect the results in this study.
Complete derivation of the exact solution. The analysis model in this study is described below. We considered the case where the electrical impedance measurement is performed on a rectangular parallelepiped sample with a sample length of l and a cross-sectional area A using the four-terminal pair method with a voltage terminal distance of lv. The Seebeck coefficient, resistivity, thermal conductivity, and thermal diffusivity of the material are denoted as S, ρ, κ, and α respectively. These physical parameters are considered temperature independent. The temperature distribution at any position x and time t is described as T(x,t).
When an ac current with a current density J = J0sint is applied, T(x,t) can be obtained by solving the following one-dimensional unsteady-state heat transfer equation 28 : According to the method of separation of variables, the general solution of Eq. (4) is described as where C1 and C2 are arbitrary constants. From Eq. (5), it can be understood that T(x,t) behaves as a thermal wave. The imaginary part corresponds to the phase of the thermal wave.
For simplicity, we assume that only Peltier heat QPeltier = SJT0 occurs at both ends of the sample x = ±l/2, where T0 is the mean temperature of the sample. Assuming that all the Peltier heat flows into the sample, the effects of other heat losses, such as those by convection and radiation, are not considered in this model. Then, the boundary condition is described as Using T(x,t) obtained from Eq. (1), the voltage measured between ±lv/2 is expressed as follows: The first term on the right side represents the ohmic voltage between lv, and the second term represents the Seebeck voltage. In actual measurements, the temperature rise caused by the Peltier effect is considered small (~ 0.1 K), so the integral part can be replaced with A heat flow Q = −AdT(x,t)/dx is described as follows: The real part of Q can be rewritten as follows: cos cosh cos cosh sin sinh sin sinh Re( ) cos cosh sin sinh The imaginary part of Q corresponds to the phase of the heat flow.
In the experimental setup used in this study, only small temperature changes of less than 0.1 K occurred in the TE material. To increase this effect, the current value can be increased by using a TE material with high zT0. However, our model ignores the influence of Joule heating; therefore, if the current value is increased, the effect of Joule heating cannot be ignored. In such a case, a more complicated analysis is required.