„Termoelektromos jelenségek” változatai közötti eltérés

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9. sor: 9. sor:
  
 
$$V \big|_{I=0}=\underbrace{-\frac{\pi^2 k^2 T}{3e}\cdot \frac{1}{\mathcal{T}}\frac{\partial \mathcal{T}(\varepsilon )}{\partial \varepsilon} \bigg|_{\mu}}_{S} \cdot \Delta T$$
 
$$V \big|_{I=0}=\underbrace{-\frac{\pi^2 k^2 T}{3e}\cdot \frac{1}{\mathcal{T}}\frac{\partial \mathcal{T}(\varepsilon )}{\partial \varepsilon} \bigg|_{\mu}}_{S} \cdot \Delta T$$
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$$\frac{2}{L} \sum (-e) \cdot v_k \cdot f(\varepsilon_k) =  -\frac{2}{h}\int e\cdot f(\varepsilon)\,\mathrm{d} \varepsilon \rightarrow I$$
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$$\frac{2}{L} \sum \varepsilon_k \cdot v_k \cdot f(\varepsilon_k) =  \frac{2}{h}\int \varepsilon \cdot f(\varepsilon)\,\mathrm{d} \varepsilon \rightarrow I_\varepsilon$$
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$$\frac{2}{L} \sum (\varepsilon_k-\mu) \cdot v_k \cdot f(\varepsilon_k) =  \frac{2}{h}\int (\varepsilon-\mu) \cdot f(\varepsilon)\,\mathrm{d} \varepsilon \rightarrow I_Q$$
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$$I_Q=\frac{2}{h} \cdot \int \mathcal{T(\varepsilon)}\cdot (\varepsilon-\mu_1)\cdot \left[f_1(\varepsilon,\mu_1,T_1)-f_2(\varepsilon,\mu_2,T_2)\right]\mathrm{d}\varepsilon$$
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$$I_Q\approx\frac{2}{h}\frac{\pi^2}{6}(kT_1)^2\cdot\mathcal{T}(\mu) -\frac{2}{h}\frac{\pi^2}{6}(kT_2)^2\cdot \mathcal{T}(\mu) =\frac{2}{h}\frac{\pi^2 k^2}{3}\cdot\Delta T\cdot T \cdot \mathcal{T}(\mu)$$
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$$I=G\cdot V;\ \ \ I_Q=G_Q \cdot \Delta T$$
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$$\frac{G_Q}{G}=\mathcal{L}\cdot T;\ \ \ \mathcal{L}=\frac{\pi^2k^2}{3e^2}=2.44\times 10^{-8}\,\mathrm{W\,\Omega\,K^{-2}}$$
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$$\frac{\kappa}{\sigma}=\mathcal{L}\cdot T$$
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$$I_Q=\frac{2}{h} \cdot \int \mathcal{T(\varepsilon)}\cdot (\varepsilon-\mu_1)\cdot \left[f_1(\varepsilon,\mu_1,T)-f_2(\varepsilon,\mu_2,T)\right]\mathrm{d}\varepsilon$$
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$$I_Q\approx\frac{2}{h} \cdot \underbrace{\int_{\mu_2}^{\mu_1} \mathcal{T}(\varepsilon)(\varepsilon-\mu_1)\,\mathrm{d}\varepsilon}_{\sim (eV)^2} +\frac{2}{h}\frac{\pi^2}{6}(kT)^2\left[\overbrace{\underbrace{\left(\mathcal{T}(\varepsilon)(\varepsilon-\mu_1)\right)^\prime |_{\mu_1}}_{\mathcal{T}(\mu_1)}- \underbrace{\left(\mathcal{T}(\varepsilon)(\varepsilon-\mu_1)\right)^\prime|_{\mu_2}}_{\mathcal{T}^\prime (\mu_2)(\mu_2-\mu_1)+\mathcal{T}(\mu_2)}}^{2\mathcal{T}^\prime (\mu)\cdot eV}\right]$$
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$$I_Q\approx \frac{2e}{h}\frac{\pi^2 k^2 T^2}{3}\cdot \frac{\partial \mathcal{T}(\varepsilon)}{\partial \varepsilon}\bigg|_\mu \cdot V$$
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$$I= -\frac{2e^2}{h}\mathcal{T}\cdot V$$
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$$\frac{I_Q}{I}\bigg|_{T_1=T_2}\approx -\frac{\pi^2 k^2 T^2}{3e}\cdot \frac{1}{\mathcal{T}}\frac{\partial \mathcal{T}(\varepsilon)}{\partial \varepsilon}\bigg|_\mu =\Pi=T\cdot S$$
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A lap 2018. február 22., 17:08-kori változata


\[I=\frac{2 e}{h} \cdot \int \mathcal{T(\varepsilon)}\cdot \left[f_1(\varepsilon,\mu_1,T_1)-f_2(\varepsilon,\mu_2,T_2)\right]\mathrm{d}\varepsilon\]
\[\int_{-\infty}^\infty H(\varepsilon)\cdot f(\varepsilon,\mu,T)\,\mathrm{d}\varepsilon = \int_{-\infty}^\mu H(\varepsilon)\,\mathrm{d}\varepsilon + \frac{\pi^2}{6}(kT)^2 H^\prime(\mu) + \mathrm{O} \left(\frac{kT}{\mu}\right)^4\]
\[I\approx\frac{2 e}{h} \cdot \int_{\mu_2}^{\mu_1} \mathcal{T}(\varepsilon)\,\mathrm{d}\varepsilon +\frac{2 e}{h}\frac{\pi^2}{6}(kT_1)^2 \mathcal{T}^\prime(\mu_1)-\frac{2 e}{h}\frac{\pi^2}{6}(kT_2)^2 \mathcal{T}^\prime(\mu_2)\approx \frac{2 e}{h} \cdot eV \cdot\bar{\mathcal{T}}(\varepsilon)+\frac{2 e}{h}\frac{\pi^2 k^2}{3}\cdot\Delta T\cdot T \cdot\mathcal{T}^\prime(\mu)\]
\[\Delta T=T_1-T_2;\ \ \ T=\frac{T_1+T_2}{2};\ \ \ \mu=\frac{\mu_1+\mu_2}{2}\]
\[V \big|_{I=0}=\underbrace{-\frac{\pi^2 k^2 T}{3e}\cdot \frac{1}{\mathcal{T}}\frac{\partial \mathcal{T}(\varepsilon )}{\partial \varepsilon} \bigg|_{\mu}}_{S} \cdot \Delta T\]
\[\frac{2}{L} \sum (-e) \cdot v_k \cdot f(\varepsilon_k) = -\frac{2}{h}\int e\cdot f(\varepsilon)\,\mathrm{d} \varepsilon \rightarrow I\]


\[\frac{2}{L} \sum \varepsilon_k \cdot v_k \cdot f(\varepsilon_k) = \frac{2}{h}\int \varepsilon \cdot f(\varepsilon)\,\mathrm{d} \varepsilon \rightarrow I_\varepsilon\]
\[\frac{2}{L} \sum (\varepsilon_k-\mu) \cdot v_k \cdot f(\varepsilon_k) = \frac{2}{h}\int (\varepsilon-\mu) \cdot f(\varepsilon)\,\mathrm{d} \varepsilon \rightarrow I_Q\]
\[I_Q=\frac{2}{h} \cdot \int \mathcal{T(\varepsilon)}\cdot (\varepsilon-\mu_1)\cdot \left[f_1(\varepsilon,\mu_1,T_1)-f_2(\varepsilon,\mu_2,T_2)\right]\mathrm{d}\varepsilon\]
\[I_Q\approx\frac{2}{h}\frac{\pi^2}{6}(kT_1)^2\cdot\mathcal{T}(\mu) -\frac{2}{h}\frac{\pi^2}{6}(kT_2)^2\cdot \mathcal{T}(\mu) =\frac{2}{h}\frac{\pi^2 k^2}{3}\cdot\Delta T\cdot T \cdot \mathcal{T}(\mu)\]
\[I=G\cdot V;\ \ \ I_Q=G_Q \cdot \Delta T\]
\[\frac{G_Q}{G}=\mathcal{L}\cdot T;\ \ \ \mathcal{L}=\frac{\pi^2k^2}{3e^2}=2.44\times 10^{-8}\,\mathrm{W\,\Omega\,K^{-2}}\]
\[\frac{\kappa}{\sigma}=\mathcal{L}\cdot T\]


\[I_Q=\frac{2}{h} \cdot \int \mathcal{T(\varepsilon)}\cdot (\varepsilon-\mu_1)\cdot \left[f_1(\varepsilon,\mu_1,T)-f_2(\varepsilon,\mu_2,T)\right]\mathrm{d}\varepsilon\]


\[I_Q\approx\frac{2}{h} \cdot \underbrace{\int_{\mu_2}^{\mu_1} \mathcal{T}(\varepsilon)(\varepsilon-\mu_1)\,\mathrm{d}\varepsilon}_{\sim (eV)^2} +\frac{2}{h}\frac{\pi^2}{6}(kT)^2\left[\overbrace{\underbrace{\left(\mathcal{T}(\varepsilon)(\varepsilon-\mu_1)\right)^\prime |_{\mu_1}}_{\mathcal{T}(\mu_1)}- \underbrace{\left(\mathcal{T}(\varepsilon)(\varepsilon-\mu_1)\right)^\prime|_{\mu_2}}_{\mathcal{T}^\prime (\mu_2)(\mu_2-\mu_1)+\mathcal{T}(\mu_2)}}^{2\mathcal{T}^\prime (\mu)\cdot eV}\right]\]
\[I_Q\approx \frac{2e}{h}\frac{\pi^2 k^2 T^2}{3}\cdot \frac{\partial \mathcal{T}(\varepsilon)}{\partial \varepsilon}\bigg|_\mu \cdot V\]
\[I= -\frac{2e^2}{h}\mathcal{T}\cdot V\]
\[\frac{I_Q}{I}\bigg|_{T_1=T_2}\approx -\frac{\pi^2 k^2 T^2}{3e}\cdot \frac{1}{\mathcal{T}}\frac{\partial \mathcal{T}(\varepsilon)}{\partial \varepsilon}\bigg|_\mu =\Pi=T\cdot S\]


















A lap eredeti címe: „https://fizipedia.bme.hu/index.php?title=Termoelektromos_jelenségek&oldid=22060”