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Thermoelectric refrigeration technology is an environment-friendly refrigeration technology with broad application prospects. The Peltier effect plays a central role in the thermoelectric refrigeration process, however, the Peltier coefficient is difficult to measure. So in the actual application process, first, the Seebeck coefficient is usually obtained, and then the Peltier coefficient is achieved by the Kelvin's second relation indirectly. It should be noted that the Kelvin's second relation is obtained under linear conditions (Ohm's law, Fourier's law, etc.), while in practice, nonlinear current-voltage relationships (Schottky junction, pn junction, etc.) and nonlinear heat transport relations are common. And quantum effect plays a leading role in the nano-scaled region, then the Peltier effect must consider the influence of nonlinearity, and the applicability of the Kelvin's second relation must also be reconsidered. This paper first summarizes the theoretical derivation of Peltier coefficient and the Kelvin’s second relation by different methods, then discusses the hypothetical conditions used in the derivation process, and points out that the Kelvin’s second relation can be established only under the hypothetical linear conditions. Then, several experimental methods of determining the Peltier coefficient are summarized. It is found that there are still many problems encountered in the measurement of Peltier coefficient, and the Kelvin’s second relation has not been proved accurately by practical experiments. Various side effects (Fourier effect, Thomson effect, Joule effect and Seebeck effect) in the measurement process affect the temperature distribution of the system directly or indirectly, making it difficult to measure Peltier heat. After that, the theoretical work of nonlinear Peltier effect is briefly introduced. In the process of thermal transport and electrical transport on a microscopic scale, quantum effect plays a leading role, and the nonlinear part of the Peltier coefficient gradually emerges. These studies show the cognition of researchers that the Peltier effect gradually changes from linear to nonlinear. The nonlinear Peltier effect not only exists objectively, but also is very important in the practical applications. However, the current research on the nonlinear Peltier effect is still at the theoretical level, and there is almost no experimental work. Finally, we discuss the research strategy and feasible research direction of Peltier effect under nonlinear conditions. An integrated study of the relationship among various heterojunction band structures, interface properties and interface effects is helpful in comprehensively understanding the Peltier effect. With the continuous improvement of experimental conditions and theoretical research, the study of nonlinear Peltier effect is expected to realize a new breakthrough.
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Q 热量(J) $ \bar {l} $ 电子平均自由程(m) $ {Q}_{\rm{P}} $ Peltier热(J) G 吉布斯自由能(J) $ {Q}_{\rm{T}} $ Thomson热(J) H 焓(J) I 电流(A) S 熵(J/K) U 电压(V) Δq 形成静电电压的电荷差值 i 电流密度(A/m2) a 电子热容系数(J/K) E 电场强度(V/m) N 电子总数 T 绝对温度(K) A 热电臂的横截面积(m2) Ji 通量, 代指热通量或电通量等 K 热导系数(W/K) Xj 引起迁移的广义力或动力 R 电热导率(S/m) Lij 唯象系数 $ {P}_{\rm{J}} $ Joule热功率密度(W/m3) J 粒子流密度 $ {P}_{\rm{P}} $ Peltier热功率密度(W/m3) q 热流密度(W/m2) m 电子有效质量(kg) S 熵流密度(W/(K·m2)) w 能量流密度(W/m2) 希腊字母 W 能量通量(W) $ \varPi $ Peltier系数(V) Is 熵流(W/K) $ \alpha $ Seebeck系数(V/K) E 电子能量(eV) $ {\sigma }_{\rm{T}} $ Thomson系数(V/K) $ {E}_{\rm{F}} $ 费米能级(eV) φ 接触电势(V) $ {E}_{\rm{c}} $ 导带底(eV) $ \widetilde {\mu } $ 电化学势, 参照文献[19,20], 代表材料费米能级到真空能级的能量差(eV) $ {E}_{{\rm{v}}} $ 价带顶(eV) μ 化学势, 参照文献[19,20], 代表费米能级到导带底的能量差(eV) $ \Delta {E}_{\rm{tn}} $ 在半导体中相对于导带底, 输运电子的平均能量(eV) ε 电子动能(eV) $ \Delta {E}_{\rm{m}} $ 在金属中相对于费米能级, 输运电子的平均能量(eV) $ \gamma $ 金属之间电子转移数目相关的系数 n 载流子浓度 ΘV 特征温度(K) r 散射因子 κ 热导率(W/(m·K)) D 扩散系数(m2/s) Γ 热转移系数(W/K) u 迁移率(m2/(V·s)) $ \sigma $ 电导率(S/m) k 玻尔兹曼常数 $ \tau $ 弛豫时间(s) h 普朗克常数 $ {\tau }_{i} $ 非弹性弛豫时间(s) $ \hslash $ 约化普朗克常数 $ \tau \left(E\right)$ 描述分布函数如何弛豫的特征时间(s) 
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理论 假设条件 存在问题 疑问 经典热力学推导 三种热电效应可逆 忽略Joule热和Fourier热 Peltier效应线性可逆(Peltier系数与
电流无关)Joule热和Fourier
热不可忽略Peltier系数真
的与电流无
关吗?不可逆热力学推导 控制不可逆过程的宏观规律可以用线性形式 Onsager倒易关系 非线性关系大量存在(如pn结中的电流-电压关系) Onsager倒易关系只适用于线性条件 能带理论
推导爱因斯坦关系式 界面处由于费米能级的不连续性导致的电子跃迁放热归结为Joule热 爱因斯坦关系式是一种线性关系 界面处电子跃迁引起的热量变化似乎和Peltier效应更为类似 吉布斯函数推导 热力学平衡态 有温度梯度存在时则不能将其考虑成热力学平衡态 DownLoad:
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研究对象 测试温度/K Seebeck系数/mV·K–1 $ \alpha T $/mV Peltier系数/mV 文献 金属 Cu/Bi热电偶 300 $ {\alpha }_{\rm{Bi}}-{\alpha }_{\rm{Cu}} $ 0.0548 16.4 $ {\varPi }_{\rm{Bi}}-{\varPi }_{\rm{Cu}} $ 16 [10] 镍铝/镍铬合金热电偶 310 $ {\alpha }_{\rm{ch}}-{\alpha }_{\rm{al}} $ 0.0406 12.6 $ {\varPi }_{\rm{ch}}-{\varPi }_{\rm{al}} $ 27.2±1.3 [16] Co/Au热电偶 295 $ {\alpha }_{\rm{Au}}-{\alpha }_{\rm{Co}} $ 0.0329 9.7 $ {\varPi }_{\rm{Au}}-{\varPi }_{\rm{Co}} $ 27 [12] 半导体 p型硅的固液界面 1683 — — — $ {\varPi }_{\text{固}}-{\varPi }_{\text{液}} $ 190±25% [11] 碲化铋半导体 300 $ {\alpha }_{\rm{p}}-{\alpha }_{\rm{n}} $ 0.420 126 $ {\varPi }_{\rm{p}}-{\varPi }_{\rm{n}} $ 124±0.7 [14] 薄膜 悬浮的Ni-ett 300 ${\alpha _{ {\rm{poly} }\left( { {\rm{Ni} }\text-{\rm{ett} } } \right)} }$ –0.079 –23.5 ${\varPi _{{\rm{poly}}\left( {{\rm{Ni}} \text- {\rm{ett}}} \right)}} $ –21.6 [30] 悬浮的磁性薄膜 300 $ {\alpha }_{\rm{Fe}} $ 0.0062 1.86 $ {\varPi }_{\rm{Fe}} $ 1.81 [15] 300 $ {\alpha }_{\rm{Ni}} $ –0.015 –4.50 $ {\varPi }_{\rm{Ni}}$ 3.89 [15] 300 ${\alpha }_{ {\rm{Ni} }\text-{\rm{Fe} } }$ –0.023 –6.90 ${\varPi }_{ {\rm{Ni} }\text-{\rm{Fe} } }$ –6.50 [15] DownLoad:
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研究人员 研究对象 主要结论 Kulik[34] 金属膜 $\varPi =\dfrac{ {\text{π} }^{2}{T}^{2} }{2{\rm{e} }\mu }+\dfrac{3{\text{π} }^{2}{\rm{e} }\tau {\tau }_{\rm{i} } }{m}{\left|{{E} }\right|}^{2}$, 在低温时, Peltier效应的线性部分随温度的降低而逐渐消失, 非线性部分将变为主导 López等[23] 量子点接触 $\varPi ={\varPi }_{1}\left[1+\dfrac{1}{ {\sigma }_{11} }\left(\dfrac{ {R}_{111} }{ {R}_{11} }-\dfrac{ {\sigma }_{111} }{ {\sigma }_{11} }\right)I+\cdots \right]$, Peltier系数的非线性部分由非线性传导系数与线性传导系数的相对强度以及导体的热特性和电特性之间的差值给出 Bogachek等[24] 量子点接触 $\varPi =T\dfrac{\partial {I}_{\rm{s} } }{\partial I}$, Peltier效应可能受到外加电压、垂直磁场或两者共同作用影响而呈现非线性 Çipiloğlu等[36] 量子点接触 非线性Peltier效应最低阶是三阶的 Zebarjadi等[21] 掺杂InGaAs半导体 $ \varPi ={\varPi }_{1}+{\varPi }_{3}{{{i}}}^{2} $, 三阶Peltier系数正比于电子有效质量, 反比于载流子浓度的平方, 在77 K时非线性Peltier效应将使制冷性能提高700% Sadeghian等[38] InAs1–xSbx半导体 300 K下非线性系统的最大冷却量是线性系统的两倍以上 DownLoad:
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