EES：热电材料有效质量模型

{"type":"doc","content":[{"type":"image","attrs":{"id":"8a6cb7bc-8df9-4902-acc1-820ad7452855","src":"https://developer.qcloudimg.com/http-save/audit-12559234/c1e49dea7e1d1039f6fb13c02ee68feb.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"a04c8663-079c-4fa1-9b1f-933d36b41c6b","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"99698534-7ddb-4940-9162-5902553bb1be","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Effective Mass Model for Thermoelectrics — A Comprehensive Derivation"}]},{"type":"paragraph","attrs":{"id":"745a755c-3c25-4585-a5fe-2a8fc2400718","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Heo M, Lee KH, Song J, et al. | Energy Environ. Sci., 2026, 19, 3195–3202"}]},{"type":"paragraph","attrs":{"id":"9836e42b-40ad-4af4-bc23-948972d82bdc","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"DOI: 10.1039/d5ee06413a"}]},{"type":"paragraph","attrs":{"id":"7f34dbca-d549-4404-8502-98282f8f15bf","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"c8f8718b-fcae-45b7-b2d3-2951040dcac1","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"d086f7ff-6b75-4472-97d1-16b5919a03af","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"热电材料可将热能直接转化为电能，是废热回收和固态制冷的关键技术。其性能由无量纲热电优值 zT = S²σT/κ 衡量，涉及塞贝克系数 S、电导率 σ 和热导率 κ 三个相互耦合的输运参数。传统上，预测 zT 最大值和最优掺杂浓度需要数值求解费米积分并依赖霍尔效应测量载流子浓度——这些技术门槛限制了大量实验研究者对热电性能的准确评估。"}]},{"type":"paragraph","attrs":{"id":"bf4a17cf-cc18-4f74-b60c-955727014029","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"本文提出了一种基于 |S|（热电势的绝对值）的有效质量模型简化分析方法。其核心思想是：在重掺杂简并半导体中，|S| 是费米能级 EF 的单调函数，因此可以作为掺杂水平的直接描述符。通过将传统 EM 模型中依赖于约化费米能级 η = EF/kBT 的所有输运系数（S, σ, κe）重写为仅依赖于 |S| 的解析函数，最终得到 zT = zT(|S|; B) 的闭式表达式，其中 B 是唯一的材料参数。该方法仅需最常规的热电测量数据（S, σ, κ），无需费米积分数值求解，也无需霍尔效应测量。"}]},{"type":"paragraph","attrs":{"id":"f26105dc-7568-4eb3-aacb-59025794be7c","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"dfff86c0-037e-4900-8176-52899fdf1295","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"e79cabd1-7e3c-445f-8259-7e024be2aa26","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"热电优值 zT 的定义"}]},{"type":"image","attrs":{"id":"5ff9a429-e168-4faf-8c3a-18d92e7a6a29","src":"https://developer.qcloudimg.com/http-save/audit-12559234/666ffb34d3557162f8c2c1dc35a1f538.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"8c2e5e65-3183-4e47-bd48-dc3b2eb2196f","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (1) 热电优值定义"}]},{"type":"paragraph","attrs":{"id":"3416ac7b-cfc3-4c84-8bad-268f0da1ca01","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"zT 的三个构成参数 S、σ 和 κ 随载流子浓度 n 的变化呈现相互制约的三角关系：S 和 σ 随 n 反向变化（S 随 n 增加而减小，σ 随 n 增加而增大），而总热导率 κ = κe κL 中，电子贡献 κe 随 n 增加而增大（通过 Wiedemann-Franz 定律 κe = LσT），晶格贡献 κL 近似与 n 无关。这种三重制约使得 zT 在某一最优掺杂水平达到峰值，偏离该最优值任一方向都会导致 zT 下降。"}]},{"type":"paragraph","attrs":{"id":"c03b5e55-8950-4516-a5b4-5f47fb3a53da","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"070bad1b-bee4-4c85-9159-a7e4e14ad8a2","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}},{"type":"bold"}],"text":"单抛物带模型的基本假设"}]},{"type":"paragraph","attrs":{"id":"e8efe21a-51d1-46fe-8760-0e6ef9e58958","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"有效质量模型建立在单抛物带（Single Parabolic Band, SPB）模型之上，其核心假设包括："}]},{"type":"paragraph","attrs":{"id":"daa12c32-8d79-49f9-ada9-11a7d85b5aee","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(1) 能带为抛物型：E(k) = ħ²k²/2m*，其中 m* 为态密度（DOS）有效质量。"}]},{"type":"paragraph","attrs":{"id":"bee80af8-8b89-44bc-b384-055dbcc9605b","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(2) 载流子服从费米-狄拉克统计，输运性质由费米-狄拉克积分描述。"}]},{"type":"paragraph","attrs":{"id":"3ffc213e-54c9-4ceb-a818-2738352fc41c","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(3) 声学声子散射为主导散射机制：弛豫时间 τ ∝ E^{-1/2}，迁移率前置因子 μ₀ 为常数。"}]},{"type":"paragraph","attrs":{"id":"b00d1676-7b79-4bc9-9d87-558d04c8df93","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(4) 材料为重掺杂简并/中等简并半导体，|S| 可作为约化费米能级 η 的单调袋里变量。"}]},{"type":"paragraph","attrs":{"id":"c399893c-81fe-4c0a-822b-e15cb533dc2c","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"尽管真实材料可能偏离这些理想假设（如多带贡献、带非抛物性、电离杂质散射等），但 EM 模型从大量实验数据中提炼出的经验关系使其在广泛的材料体系中仍具有出色的预测能力。此时提取的参数应理解为"有效输运参数"而非严格的能带常数。"}]},{"type":"paragraph","attrs":{"id":"630d5263-aab2-4ecd-a962-4b7eee6c70a3","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"8ddfe1b0-8570-48f1-8c2f-b9ecd1339926","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"约化费米能级 η"}]},{"type":"image","attrs":{"id":"81f05bbc-2a93-4409-bbc4-8d33d75ff391","src":"https://developer.qcloudimg.com/http-save/audit-12559234/cb21605d5809e28b4769adbbfbbc982f.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"9bbb9d66-b9a3-49b9-8dca-d785d418cb6b","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"η 是 EM 模型中最核心的中间变量。所有输运系数（S, σ, κe, n）均可表示为 η 的函数。η 的正负和大小直接反映了体系的简并程度：η ≫ 0（η > 3）对应简并半导体（类金属行为），η ≪ 0（η < -3）对应非简并半导体（类绝缘体），η ≈ 0 对应中等简并。热电材料的最优性能通常出现在 η ≈ -2 到 2 的范围内，对应于 |S| ≈ 100-300 μV/K。"}]},{"type":"paragraph","attrs":{"id":"c53fe171-8977-4eb0-ab47-6150c5fe7367","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"费米-狄拉克积分 Fj(η)"}]},{"type":"image","attrs":{"id":"99028497-666a-4e0a-ae2d-51b101a2a576","src":"https://developer.qcloudimg.com/http-save/audit-12559234/c741b72f49be2595ac8ee49c0d08df55.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"46e23f2a-1128-452b-99d3-926cb7d78374","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"费米积分是 EM 模型数学结构的基石。对于 SPB 体系，所有输运系数均可通过不同阶数的费米积分及其比值表达。两个最重要的极限：在简并极限（η ≫ 0），F_j(η) → η^{j 1}/(j 1)；在非简并极限（η ≪ 0），F_j(η) → Γ(j 1)·e^η。费米积分的比值 F_j(η)/F_{j-1}(η) 出现在 Seebeck 系数、Lorenz 数等关键量的表达式中，是连接 η 和可观测量的桥梁。"}]},{"type":"paragraph","attrs":{"id":"69044c70-116e-496b-b681-f0b7d1ad1bc2","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"8bf35283-e8da-4d0b-b628-86f659aaa7e8","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"载流子浓度 n(η)"}]},{"type":"image","attrs":{"id":"85db4f8d-7994-4718-8ddc-4783bd2beb0a","src":"https://developer.qcloudimg.com/http-save/audit-12559234/3c5b9152859c8cf32b5d7e6125ea1d6d.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"83f70bfd-b335-403e-b05e-84e65f2a88a8","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"在抛物带假设下，载流子浓度 n 由态密度 g(E) ∝ E^{1/2} 在费米-狄拉克分布 f(E) 下的积分给出。h 为普朗克常数，m* 为态密度有效质量。在简并极限（η ≫ 0），n ∝ η^{3/2} ∝ EF^{3/2}，恢复自由电子气的经典结果。在非简并极限（η ≪ 0），n ∝ exp(η) ∝ exp(EF/kBT)，恢复玻尔兹曼统计。"}]},{"type":"paragraph","attrs":{"id":"72a6e917-d059-40fd-a1ac-aad43fbba8cf","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"0ac0cff0-9d92-405e-99b6-a29c99fe112a","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"塞贝克系数 S(η)"}]},{"type":"image","attrs":{"id":"c0b9b9c3-1660-4449-885d-77e74c0bf8d5","src":"https://developer.qcloudimg.com/http-save/audit-12559234/31f0ac03f27992447efd5de103e9099c.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"238a6508-7a59-42c5-8710-41f1adf2d960","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (11) 塞贝克系数"}]},{"type":"paragraph","attrs":{"id":"e3dfb249-6708-4c64-9005-e7084808c45e","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"塞贝克系数 S 是费米面附近态密度不对称性的度量，由 Mott 公式导出。正负号分别对应 p 型和 n 型半导体。S(η) 是单调递减函数：在 η → −∞ 时 |S| → ∞，在 η → ∞ 时 |S| → 0。这一单调性是该方法的基石——给定 |S|，可以唯一地反解出 η，从而将所有依赖于 η 的输运系数重写为 |S| 的函数。"}]},{"type":"paragraph","attrs":{"id":"52ae5167-797a-4622-9647-234f5da6cb7a","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"在简并极限下，S 简化为著名的 Mott 公式："}]},{"type":"image","attrs":{"id":"8a49bcf3-1b7c-4a2a-a121-ebe349625c1d","src":"https://developer.qcloudimg.com/http-save/audit-12559234/31808e5011bfc23e4d6d26f76af2ab16.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"68b6b62c-395a-4061-8fc2-11fefed79f4d","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"简并极限下的 Mott 公式"}]},{"type":"paragraph","attrs":{"id":"b1c208de-748a-4e7a-9f46-2d8821cf17e8","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"该式表明 S ∝ m*T/n^{2/3}，即高有效质量、低载流子浓度和高温有利于产生大热电势。"}]},{"type":"paragraph","attrs":{"id":"df84a652-cd87-4aed-8c55-2a11ddf2f332","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"6fcc4925-c0c8-4bc1-b180-077e188cce26","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"电导率 σ(η) 与加权迁移率 μw"}]},{"type":"image","attrs":{"id":"d9396ec2-c4c1-40ac-b818-1d23f03bac93","src":"https://developer.qcloudimg.com/http-save/audit-12559234/b064eba31c4982beff18ca80a341df97.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"79d889ef-5c91-453d-a3aa-5719c8dbc33d","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"10aa1d77-0285-4fa6-9366-725bc387ed01","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"中 μ₀ 为与能带结构无关的迁移率前置因子（在声学声子散射假设下为常数）。为分离材料本征参数和载流子浓度依赖，定义两个关键量："}]},{"type":"image","attrs":{"id":"5814a52e-6717-43f6-b40e-fd2b36641e5d","src":"https://developer.qcloudimg.com/http-save/audit-12559234/2bc66dde8ecbd7de96bacbe2cfcf9466.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"50b12674-313f-43cc-8973-c01f6cf91a2d","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"9da0d7d7-1574-4204-9871-e71ccb674f84","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"μw 将迁移率 μ₀ 和态密度有效质量 m* 合并为一个参数。μw 的物理意义在于：它直接决定了给定 |S|（即给定 η）下的电导率——μw 越大，相同 |S| 下的 σ 越大。μw 是比 σ 或功率因子 PF 更可靠的电子输运品质因子，因为它剥离了依赖于掺杂水平（η）的部分，仅保留反映材料本征电子输运品质的部分。"}]},{"type":"image","attrs":{"id":"024389db-c16e-4578-bc57-ae29babd8cde","src":"https://developer.qcloudimg.com/http-save/audit-12559234/5440f8884b2b7cd42624e038ff3bf731.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"d9881190-7567-4828-9709-3e500faaa5eb","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"2dab82ce-5ead-47cc-a842-5f8348e7b272","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"σE₀ 仅依赖于温度和基本常数，不依赖于材料参数。在 300 K 时，σE₀ 约为 3.02 × 10⁴ Ω⁻¹m⁻¹"}]},{"type":"paragraph","attrs":{"id":"f6688c6a-ae9a-4a06-b1bc-1f32a7fbabc1","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"2142170b-f876-484a-8a8c-546b13a17d71","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"电子热导率 κe 与洛伦兹因子 L"}]},{"type":"image","attrs":{"id":"a96b7c35-3b17-4419-96fc-d5fe127730de","src":"https://developer.qcloudimg.com/http-save/audit-12559234/f8b55580a0316f45d8aee8d1d8ac7330.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"image","attrs":{"id":"bf6f9424-c283-42e1-a7a1-fffeb91c03d0","src":"https://developer.qcloudimg.com/http-save/audit-12559234/425aa841d141ac7852e9c4fd8a5338bb.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"99bf14c9-3b49-4dd3-8d74-3eb3f50c01ea","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"Wiedemann-Franz 定律"}]},{"type":"paragraph","attrs":{"id":"27ff4f7c-e4c2-4826-988d-e870ec919156","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"洛伦兹因子 L 在 SPB 模型中由费米积分比值给出："}]},{"type":"image","attrs":{"id":"4c48adc1-7147-443e-9a9a-3f4372603bb7","src":"https://developer.qcloudimg.com/http-save/audit-12559234/2e2a4fed38000c46ea3f24337a238e96.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"a541a0b1-90fc-40c3-aec2-f60857e4ea03","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"洛伦兹因子的一般表达式"}]},{"type":"paragraph","attrs":{"id":"4db07b81-5ac7-41bc-972f-1762cf789428","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"在简并极限（金属），L → L₀ = (π²/3)(kB/e)² ≈ 2.44 × 10⁻⁸ WΩK⁻²。在非简并极限，L → 2(kB/e)² ≈ 1.48 × 10⁻⁸ WΩK⁻²。对于实际热电材料，L 在两者之间过渡，可由 |S| 通过经验公式精确估算："}]},{"type":"image","attrs":{"id":"1a5fd917-7961-4457-8e03-a2086576bedc","src":"https://developer.qcloudimg.com/http-save/audit-12559234/bc4abc4c3be1cb2dc266add37bae3260.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"3eb56811-e662-46dd-b958-b690c5a3f41d","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"该经验公式在 |S| > 20 μV/K 范围内精度极高，使得 κL 可以从总 κ 中准确分离：κL = κ − LσT。"}]},{"type":"paragraph","attrs":{"id":"990ef5a3-20db-46b9-ba4c-1f656218b084","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"1d9db761-ed3e-4077-9120-04517fb01758","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}},{"type":"bold"}],"text":"品质因子 B"}]},{"type":"paragraph","attrs":{"id":"da9ef7bd-6a76-4f4d-a4d8-c33ec2221594","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"热电品质因子 B 是决定材料最大 zT 的唯一材料参数。其理论定义为："}]},{"type":"image","attrs":{"id":"081ea065-454c-4cfc-85cc-b5984fdd2914","src":"https://developer.qcloudimg.com/http-save/audit-12559234/a459dd36ebbc2ece4fe0df515c659725.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"2d230aaa-8949-43ad-83fc-63c2d88b7296","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (1) B 的定义"}]},{"type":"paragraph","attrs":{"id":"01fc391a-a1fc-4cfa-bb1e-4df6f27f08e1","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"4a2452f0-ed68-42bd-885d-794c58636620","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"B 的物理意义是"加权电子输运能力与晶格热输运之比"。在 300 K 的实用形式为："}]},{"type":"image","attrs":{"id":"23e52171-5f21-4f70-9b4a-2dfe6d0a1abf","src":"https://developer.qcloudimg.com/http-save/audit-12559234/8e656c4a8c82d4196e4460eb169bc71c.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"95181b7f-ca2a-4b6d-b96c-9a0ad6e98584","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (2) B 的实用表达式"}]},{"type":"paragraph","attrs":{"id":"398a2a3a-9c5f-4ce8-bfc0-113cab3086ac","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"B 越大，zTmax 越高。要实现 zTmax ≈ 1，需要 B ≈ 0.4；zTmax ≈ 2 需要 B ≈ 1.5；zTmax ≈ 3 需要 B ≈ 3.5。在 300 K 和 κL ≈ 1 W/mK 的典型条件下，B = 0.4 对应 μw ≈ 600 cm²/Vs。"}]},{"type":"paragraph","attrs":{"id":"beae8123-955a-4509-abd2-395d24c3a622","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"40eb49a2-9e48-4096-a0f2-8dfcd2dc6004","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"40e7175e-1035-4158-9c87-09b9dea58d60","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"关键步骤："},{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}},{"type":"bold"}],"text":"变量代换 η → |S|"}]},{"type":"paragraph","attrs":{"id":"ad8b730b-e57e-4ed3-ab0f-2584fe770f8d","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"以上推导表明，σ(η) 和 κe(η) 均通过费米积分依赖于 η。而 S(η) 是 η 的单调函数。因此，可以通过以下逻辑链将 zT 重写为仅依赖于 |S| 的函数："}]},{"type":"paragraph","attrs":{"id":"7123bd5f-9723-4c1e-9bf2-c3f7f070bcdb","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 1：给定 |S|，通过 S(η) 的反函数得到 η = S^{-1}(|S|)。由于 S(η) 严格单调，反函数唯一存在。"}]},{"type":"paragraph","attrs":{"id":"4896f75d-1cda-4bdb-b565-123932dd1ab4","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 2：将 η 代入 σ(η) 和 κe(η)，得到 σ(|S|) 和 κe(|S|)。"}]},{"type":"paragraph","attrs":{"id":"21b3500a-361d-43e7-a402-c3ff7654a3c6","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 3：代入 zT = S²σT/κ，得到 zT = zT(|S|; B)。"}]},{"type":"paragraph","attrs":{"id":"840273c9-22db-424b-9037-10b6ae3c537b","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"在实用层面，作者通过数值拟合将费米积分比值直接近似为 |S| 的解析函数，绕过了显式求解 η 的步骤。定义公共分式 f(|S|)："}]},{"type":"image","attrs":{"id":"8733ac99-709c-4ef2-9233-bfa8188f0b2a","src":"https://developer.qcloudimg.com/http-save/audit-12559234/50884c98b4359030b828265da6fa3e60.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"d5ab3214-c78b-4b9a-8f8d-1ad534d38d3d","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (5) 加权迁移率 μw 的 |S| 表达式"}]},{"type":"image","attrs":{"id":"5eea4ed8-a445-49a9-b1ad-cd688c6858bd","src":"https://developer.qcloudimg.com/http-save/audit-12559234/6c50ff7a7a54d5ac26ebc891013a74dd.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"6067f4e4-ab84-41f1-a3ea-ef6b4b8d102a","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (6) 电导率 σ 的 |S| 表达式"}]},{"type":"paragraph","attrs":{"id":"f7a82d59-6c46-4aac-b770-dc2eea72299c","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (5) 和 Eq. (6) 互为逆关系。Eq. (5) 从 σ 和 |S| 计算 μw（剥离掺杂依赖），Eq. (6) 从 μw 和 |S| 预测 σ（评估 μw 改善对 zT 的影响）。"}]},{"type":"paragraph","attrs":{"id":"ac4faae7-4044-44ca-b617-d6ba8ad25aa0","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"2a1b9ded-d13c-4fdd-806d-17240b4ba6fe","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"zT 的 |S| 表达式 — 核心结果"}]},{"type":"image","attrs":{"id":"2c1a7dd8-7d38-4c5f-a833-0ca487b5499b","src":"https://developer.qcloudimg.com/http-save/audit-12559234/63f2a5180a7987e43ef33ab6d5687fb6.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"bfbb49dd-7814-4d9f-a53c-fa872150a338","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (7)zT 的热电势表达式"}]},{"type":"paragraph","attrs":{"id":"efee2102-b529-42ec-8471-d4d852778d7b","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"分子 S² ∝ η² 代表热电驱动力；分母第一项 (1/B)×f(|S|) 代表电子输运的电阻；分母第二项 1.34×[1.5 exp(−|S|/116)] 代表电子热导的损耗（其中 1.5 exp(−|S|/116) 即 L 的经验表达式）。在扩散热电势主导且双极效应可忽略的条件下最为可靠。对于窄带隙材料在高温下的应用，计算得到的 zT 应视为近似描述符。"}]},{"type":"paragraph","attrs":{"id":"0671de35-b2f8-488d-8613-9a40b1b7db14","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"ebc9de82-0573-4c2e-b4fc-96d814c07429","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"有效质量 m* 的提取"}]},{"type":"image","attrs":{"id":"2b1f2609-a80e-402f-926d-209859c69fb1","src":"https://developer.qcloudimg.com/http-save/audit-12559234/8317cd72bd9ae10c59c8887b34f6bb77.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"2bc8f2bf-6e57-44c0-a990-8adb93c16228","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"Eq. (8) 有效质量的提取"}]},{"type":"paragraph","attrs":{"id":"e1343c93-95f5-4325-9e55-59fa90b39d10","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"虽然 EM 模型的核心分析（zT 预测和优化）不需要显式知道 m*，但若需要区分电子结构效应和散射效应对输运的影响，可以利用霍尔效应测量（RH 为霍尔系数）提取 m*。在理想 SPB 体系中，m*/me 是一个与掺杂浓度和温度无关的材料常数。但在实际材料中，m*/me 随掺杂、温度和合金化而变化，反映了能带结构的变化（如带收敛效应）或主导散射机制的变化。"}]},{"type":"paragraph","attrs":{"id":"0117dc91-ca5a-4120-8009-46e965ab6a8a","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"e6cae396-67ee-462e-b267-5ea1e3b1ade2","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"ea2aac66-cdcf-49ea-8e90-ecb71af7e60a","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"5bbfbef3-26d0-4491-a034-eae63af4cf9f","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}},{"type":"bold"}],"text":"基于 EM 模型的数据分析流程"}]},{"type":"paragraph","attrs":{"id":"ff42484d-d0b8-4454-a8c0-72e173af3774","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"标准分析步骤"}]},{"type":"paragraph","attrs":{"id":"ff2d0dac-214e-4257-a073-eb0cc10fcebe","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"EM 模型提供了一套仅需 S、σ 和 κ 即可完成的热电性能分析流程："}]},{"type":"paragraph","attrs":{"id":"04286769-5511-4f36-b62e-35370e3c2bdf","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 1 — 计算 κL：由 Eq. (4) 从 |S| 估算 L，再由 κL = κ − LσT 分离晶格热导率。"}]},{"type":"paragraph","attrs":{"id":"b0c262de-45fe-4e30-89e3-96e87d6f2c4e","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 2 — 计算 μw：由 Eq. (5) 从 σ 和 |S| 计算加权迁移率。"}]},{"type":"paragraph","attrs":{"id":"37fb27e6-f1f5-42db-b983-16bd94fe469d","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 3 — 计算 B：由 Eq. (2) 从 μw 和 κL 计算品质因子。"}]},{"type":"paragraph","attrs":{"id":"c57bfee5-c53a-44d1-8b93-d56a9b3ce99c","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 4 — 预测 zTmax：利用 Eq. (7) 绘制 zT(|S|) 曲线，读取峰值。"}]},{"type":"paragraph","attrs":{"id":"7633e973-3243-41a8-9e9e-40defc987951","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"步骤 5 — 确定最优 |S|：zT 峰值对应的 |S| 即最优热电势。若当前 |S| 偏离此值，则需调整掺杂水平。"}]},{"type":"paragraph","attrs":{"id":"7bc7fcd7-8515-4907-b08f-8be3ba2c3f16","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"bb6e1eb0-66f7-4d1b-a328-518642a403eb","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"热电势 |S|"}]},{"type":"paragraph","attrs":{"id":"c5260a90-329d-4928-9322-b26e6a22f66a","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"|S| 的绝对值大小直接指示材料的掺杂状态："}]},{"type":"paragraph","attrs":{"id":"96d2eff3-1c6a-44a2-ba52-0f5d0a6c2034","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"|S| > 300 μV/K：材料处于欠掺杂状态，应增加载流子浓度以提升 σ。"}]},{"type":"paragraph","attrs":{"id":"e4edbe1a-d1ee-4ffe-88f1-5f452a0a0de9","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"|S| < 200 μV/K：材料处于过掺杂状态，应降低载流子浓度以提升 |S|。"}]},{"type":"paragraph","attrs":{"id":"23619eff-baa0-4949-81f8-ca7796cae02a","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"|S| ∝ T（金属/简并半导体行为）：表明 EM 模型参数可靠。"}]},{"type":"paragraph","attrs":{"id":"c5bd28e0-ebf6-4be5-92de-82ec3159c6e2","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"|S| 随 T 升高而降低：表明双极传导开始，EM 模型预测的可靠性下降。"}]},{"type":"paragraph","attrs":{"id":"b3be57d9-a7ce-4b9e-8656-4335da7d961b","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"|S| 出现峰值（|S|max 在 Tmax）：可估算热学带隙 Eg ≈ 2|S|maxTmax。"}]},{"type":"paragraph","attrs":{"id":"e403d634-7b00-4706-82b8-d2e09da0651b","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"becbdb38-3752-486e-aa7d-fa7782ab9588","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"加权迁移率 μw"}]},{"type":"paragraph","attrs":{"id":"c830924d-25b8-4266-bf39-21f7ef11e4a0","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"μw 的大小和温度依赖性提供了丰富的输运机制信息："}]},{"type":"paragraph","attrs":{"id":"b7212ab2-7397-4c89-9365-0426b5b3efb7","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"μw 的绝对值：μw ≈ 600 cm²/Vs 是室温下实现 zTmax ≈ 1 的典型门槛（κL ≈ 1 W/mK）。"}]},{"type":"paragraph","attrs":{"id":"d84cd18c-881d-4b15-8266-9ac11e808122","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"μw ∝ T^{-3/2}：声子散射主导，符合 SPB 模型的基本假设。"}]},{"type":"paragraph","attrs":{"id":"50440fbf-cc35-4c1f-bb91-d284670367d9","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"μw 随 T 升高而升高：非声子散射（晶界势垒、电离杂质）主导。例如，Mg₃(Sb,Bi)₂ 中 1 μm 晶粒尺寸的样品表现出随 T 升高的 μw，归因于晶界势垒散射。"}]},{"type":"paragraph","attrs":{"id":"575956b4-d4e0-49ad-b566-8e6b9aa2c6c9","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"μw 在固定温度下随不同样品变化 < 20%：差异主要来自掺杂水平，而非本征输运性质的变化。"}]},{"type":"paragraph","attrs":{"id":"91ab021b-562f-4497-9a35-aec0ad2ccb3d","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"μw 在固定温度下随不同样品变化 > 50%：差异来自能带结构或散射机制的根本变化。"}]},{"type":"paragraph","attrs":{"id":"f2cafbd8-f93d-48f3-8d72-af7d4f04d654","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"345c7968-61e4-4710-9219-1d4979a14b6e","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"功率因子优化"}]},{"type":"image","attrs":{"id":"71dab065-e74d-409d-b5fd-70f0e93ea90c","src":"https://developer.qcloudimg.com/http-save/audit-12559234/a1b1a58657fbdd95e798c33c59748b81.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"0ce4a3ff-5513-4c87-bd4f-8a8720530f68","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"功率因子 PF = S²σ 在 |S| ≈ 167 μV/K 处达到峰值，与材料无关。然而，zT 的峰值出现在 |S| ≈ 200-250 μV/K（取决于 B 值）。这意味着仅基于 PF 的优化会系统性导致过掺杂，使 zT 低于理论最大值达 25%（对于 B ≈ 0.4 的材料）。这是因为 PF 优化忽略了 κe 的贡献——在 |S| 较低（过掺杂）时，σ 虽然大，但 κe 也大，导致 zT 被抑制。"}]},{"type":"paragraph","attrs":{"id":"811253ca-4886-457a-ae91-fa6306f5126f","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"44a216c7-5fa0-4a50-bdde-c33b6b34fbce","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"d57b6361-c094-4970-8ccc-d3a625135b11","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"28a674a6-9501-497c-bbd6-b8029c76a2e8","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"模型验证与应用案例"}]},{"type":"image","attrs":{"id":"2cfcded3-6d87-4ee5-95e6-fb6bceebc0fa","src":"https://developer.qcloudimg.com/http-save/audit-12559234/99208ec1a2c1007523203f84b195440a.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"c0fc9be3-9991-4c83-b5fe-eb3321fb4528","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图1 EM 模型框架：(a) 不同 B 值下的 zT(|S|) 曲线族；(b) B-zTmax 对应关系；(c) 标准分析流程示意"}]},{"type":"paragraph","attrs":{"id":"b8f4fb42-3e40-4ecb-8e06-a0c2d3c4fffd","textAlign":"justify","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图1 综合展示了 EM 模型的核心框架。(a) zT 作为 |S| 的函数，不同 B 值对应不同的 zT 曲线族。每条曲线的峰值（橙色圆点）给出该 B 值下的理论 zTmax 和最优 |S|。(b) 将 (a) 中的峰值点映射为 B-zTmax 曲线，建立了品质因子与最大优值之间的直接对应关系。(c) 展示了从 S/σ/κ 三参数出发，依次计算 L → κL → μw → B → zTmax 的完整分析流程。"}]},{"type":"image","attrs":{"id":"9ad78cc3-479b-418c-ba9a-db0577c73e65","src":"https://developer.qcloudimg.com/http-save/audit-12559234/ee986b377c6f0df682f8f75a1afab6db.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"dfd8ce24-50c0-430c-b6da-8d98f137acc0","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图2 (a) 功率因子 PF 与 |S| 的关系；(b) B-zTmax 曲线与实验数据对比"}]},{"type":"paragraph","attrs":{"id":"7664e395-802d-4e1f-bdd8-4fefee82148b","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图2(a) 展示了功率因子 PF 作为 |S| 的函数。PF 在 |S| ≈ 167 μV/K 处达到峰值，与材料无关——这解释了为什么仅基于 PF 的优化往往导致过掺杂。图2(b) 将多种热电材料的实验数据与理论 B-zTmax 曲线对比，验证了 EM 模型在 Bi₂Te₃ 基、PbTe 基、Mg₃Sb₂ 基、half-Heusler 等多种材料体系中的普适性。"}]},{"type":"image","attrs":{"id":"00fa5d29-131b-475b-97fc-5c121759dcaf","src":"https://developer.qcloudimg.com/http-save/audit-12559234/80342c26ee54dc89a7c9f65b12f2a14c.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"a445f218-bd53-4563-97d6-e50097a977e1","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图3 μw 的分析应用：(a) (Bi,Sb)₂Te₃ 中带收敛对 μw 的增强；(b) Mg₃(Sb,Bi)₂ 中晶粒尺寸对 μw 的影响"}]},{"type":"paragraph","attrs":{"id":"838c36e5-4ff0-4ebe-9c22-9f109a4c9df7","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图3 展示了 μw 在识别输运机制变化中的诊断能力。(a) (Bi₀.₂₅Sb₀.₇₅)₂Te₃ 的 μw 比 Bi₂Te₃ 高 50% 以上，归因于 Bi₂Te₃-Sb₂Te₃ 合金化引起的带收敛效应——多个能带汇聚增大了 md*，而 μw ∝ (md*/me)^{3/2}。(b) Mg₃(Sb,Bi)₂ 中，晶粒尺寸从 1 μm 增大到 30 μm 降低了晶界密度，使 μw 升高；Nb 掺杂进一步抑制了晶界势垒，μw 甚至高于大晶粒样品。"}]},{"type":"image","attrs":{"id":"984f3bef-8cd8-4212-b797-415f18e571c7","src":"https://developer.qcloudimg.com/http-save/audit-12559234/0201773c063174496780c70eb77f127f.webp","extension":"","align":"center","alt":"","showAlt":false,"href":"","boxShadow":"","width":"","aspectRatio":0,"status":"success","showText":true,"isPercentage":false,"percentage":0,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"267f8fc3-d021-45cf-934b-69e1648a9883","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图4 基于 EM 模型的 zT 预测：(a) n 型 Mg₃(Sb,Bi)₂；(b) p 型 (Bi,Sb)₂Te₃；(c) n 型 PbTe；(d) p 型 SnSe"}]},{"type":"paragraph","attrs":{"id":"6c3fa4d1-0afa-407a-86af-654ee4c439db","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"图4 展示了 EM 模型在多种代表性热电材料中对 zT 的预测精度。仅需 S、σ 和 κ 的实验数据（无需霍尔效应测量），Eq. (7) 即可准确再现 zT 随 |S| 的变化趋势，并预测最大可达 zT 和最优掺杂水平。在 Mg₃(Sb,Bi)₂、PbTe、SnSe 等体系中，预测值与实测值高度一致，验证了 EM 模型的广泛适用性。"}]},{"type":"paragraph","attrs":{"id":"d42df94b-602c-4bee-82e7-2f0c53f0fc0f","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"45a66b11-409a-4665-bddf-46e332807190","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}},{"type":"bold"}],"text":"讨论："}]},{"type":"paragraph","attrs":{"id":"a49e8f25-53c6-4fe2-aac4-a8fba5ed6eb6","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"适用条件"}]},{"type":"paragraph","attrs":{"id":"9f1edcfe-68e6-457f-a012-d5d6217b9372","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(1) 扩散热电势主导：假设热电输运由载流子扩散驱动，声子曳引效应可忽略。在 T < 100 K 时，声子曳引可能显著，EM 模型参数的物理意义变得模糊。"}]},{"type":"paragraph","attrs":{"id":"d6487793-1092-4ae3-9985-676c7bc9b0eb","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(2) 双极效应可忽略：在窄带隙材料的高温区，少数载流子的激发导致 S 被部分抵消、κ 增加（双极热导），此时提取的 κL 和 B 被低估。"}]},{"type":"paragraph","attrs":{"id":"53b6c42f-a93d-499b-87ad-b211687a72e8","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(3) |S| > 20 μV/K：μw 的解析表达式在该范围内精度最高。"}]},{"type":"paragraph","attrs":{"id":"15bf2af8-a17f-4a2e-af19-78f5ec8e5fb6","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"(4) 重掺杂简并/中等简并半导体：|S| 作为 η 的单调袋里变量最为可靠。"}]},{"type":"paragraph","attrs":{"id":"ebf44a66-1d31-438d-9601-bf11f8ecd1f5","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"有效参数的解释"}]},{"type":"paragraph","attrs":{"id":"fc1b120a-8192-4a7b-be85-f21df9c0ae93","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"当体系偏离理想 SPB 行为（如多带贡献、带非抛物性、非声学声子散射）时，EM 模型提取的 μw、m* 和 B 应理解为"有效输运参数"而非严格的能带常数。这些有效参数仍然具有实用价值——它们能够正确预测 zT 的优化方向和最大可达值，但不应过度解读其微观物理含义。"}]},{"type":"paragraph","attrs":{"id":"c97a9d91-0dad-4057-8173-d25fe5422577","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"bold"}],"text":"与霍尔效应方法的对比"}]},{"type":"paragraph","attrs":{"id":"264d9ce6-7f21-4b67-aae6-c93c93012578","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"传统 EM 模型需要霍尔效应测量来确定 n 和 m*。本文的 |S| 基方法消除了这一要求，但霍尔效应测量在以下场景中仍有价值：(1) 区分电子结构效应（m* 变化）和散射效应（μ₀ 变化）对 μw 的贡献；(2) 验证单带假设的合理性（通过霍尔因子 rH 的温度依赖性）；(3) 在多相或非均匀材料中提供独立的载流子浓度信息。"}]},{"type":"paragraph","attrs":{"id":"096595c0-9d52-467d-9b0c-6c228b2af47c","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"740a9f8f-97a1-47c5-840c-b5f6576fa668","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false}},{"type":"paragraph","attrs":{"id":"5f0f3303-269a-4cb1-840c-fb802574af32","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}},{"type":"bold"}],"text":"核心结论"}]},{"type":"paragraph","attrs":{"id":"f111f58b-b41e-4faa-a339-6dc5138b5598","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"1. 本文建立了基于 |S| 的有效质量模型分析方法，将传统上需要数值求解费米积分和霍尔效应测量的热电性能评估，转化为仅需常规热电测量（S, σ, κ）即可完成的简洁流程。"}]},{"type":"paragraph","attrs":{"id":"d1d22b79-7461-48d1-8d6b-20805bc0c0e7","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"2. 模型的核心数学技巧在于：利用 S(η) 的单调性，将 zT 表达式中所有依赖于 η 的项通过数值拟合转换为 |S| 的解析函数，最终得到 zT = zT(|S|; B)。"}]},{"type":"paragraph","attrs":{"id":"4a7b64b3-8f6f-4083-9540-c02f7b6f0abf","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"3. 加权迁移率 μw 是比 σ 或 PF 更可靠的电子输运品质因子，其温度依赖性可识别主导散射机制（声子 vs 晶界 vs 电离杂质）。"}]},{"type":"paragraph","attrs":{"id":"3dac4639-d9fe-4a88-993b-6a89f16c39df","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"4. 品质因子 B 直接决定了材料的理论最大 zT：B = 0.4 → zTmax ≈ 1，B = 1.5 → zTmax ≈ 2，B = 3.5 → zTmax ≈ 3。"}]},{"type":"paragraph","attrs":{"id":"905fbb9b-7d5c-41b6-a117-f485c8d8ffae","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"5. 热电势 |S| 本身即是最优掺杂水平的诊断工具：|S| > 300 μV/K 指示欠掺杂，|S| < 200 μV/K 指示过掺杂。功率因子优化会系统性导致过掺杂，使 zT 低于理论最大值达 25%。"}]},{"type":"paragraph","attrs":{"id":"eeafebe8-f36d-42ed-bbc0-e17145543ffa","textAlign":"inherit","indent":0,"color":null,"background":null,"isHoverDragHandle":false},"content":[{"type":"text","marks":[{"type":"textStyle","attrs":{"color":"","background":""}}],"text":"6. 该框架已在多种热电材料体系（Bi₂Te₃ 基、PbTe 基、Mg₃Sb₂ 基、SnSe、half-Heusler、方钴矿等）中得到验证，适用于从实验合成到第一性原理计算再到数据驱动材料筛选的完整研究链条。"}]}]}","createTime":1782811032,"ext":{"closeTextLink":0,"comment_ban":0,"description":"","focusRead":0},"favNum":0,"html":"","isOriginal":0,"likeNum":0,