<p>Quantum Hall systems having Corbino geometry are expected to have a large Peltier coefficient <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Pi _{rr}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in the quantum Hall plateau region. We present an analytic formula for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\Pi _{rr}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> calculated employing the spectral conductivity obtained based on the self-consistent Born approximation. The coefficient <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\Pi _{rr}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is shown to have a large negative (positive) value just above (below) an integer Landau-level filling, with the absolute value <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(|\Pi _{rr}|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> increasing with decreasing temperature or decreasing disorder, and approaching the saw-tooth shape <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(- (E_{N_\textrm{F} \sigma _\textrm{F}}-\zeta )/e\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow> <msub> <mi>N</mi> <mtext>F</mtext> </msub> <msub> <mi>σ</mi> <mtext>F</mtext> </msub> </mrow> </msub> <mo>-</mo> <mi>ζ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi>e</mi> </mrow> </math></EquationSource> </InlineEquation> in the limit of vanishing disorder, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(E_{N_\textrm{F} \sigma _\textrm{F}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>E</mi> <mrow> <msub> <mi>N</mi> <mtext>F</mtext> </msub> <msub> <mi>σ</mi> <mtext>F</mtext> </msub> </mrow> </msub> </math></EquationSource> </InlineEquation> is the highest occupied Landau level and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\zeta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ζ</mi> </math></EquationSource> </InlineEquation> is the chemical potential. As an initial attempt to experimentally observe the effect of the large <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(|\Pi _{rr}|\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, we measure the electron temperature <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(T_\textrm{out}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mtext>out</mtext> </msub> </math></EquationSource> </InlineEquation> near the outer perimeter of a Corbino disk, applying a radial dc current <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(I_\textrm{dc}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mtext>dc</mtext> </msub> </math></EquationSource> </InlineEquation>. The temperature <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(T_\textrm{out}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mtext>out</mtext> </msub> </math></EquationSource> </InlineEquation> is observed to increase or decrease depending on the direction of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(I_\textrm{dc}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mtext>dc</mtext> </msub> </math></EquationSource> </InlineEquation> and the sign of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(\Pi _{rr}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> as expected from the Peltier effect. Notably, <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(T_\textrm{out}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>T</mi> <mtext>out</mtext> </msub> </math></EquationSource> </InlineEquation> becomes lower than the bath temperature for outward (inward) <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(I_\textrm{dc}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mtext>dc</mtext> </msub> </math></EquationSource> </InlineEquation> in the region where <InlineEquation ID="IEq16"> <EquationSource Format="TEX">\(\Pi _{rr} &lt; 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq17"> <EquationSource Format="TEX">\(\Pi _{rr} &gt; 0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Π</mi> <mrow> <mi mathvariant="italic">rr</mi> </mrow> </msub> <mo>&gt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>).</p>

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Peltier Cooling in Corbino-Geometry Quantum Hall Systems

  • Akira Endo,
  • Yoshiaki Hashimoto

摘要

Quantum Hall systems having Corbino geometry are expected to have a large Peltier coefficient \(\Pi _{rr}\) Π rr in the quantum Hall plateau region. We present an analytic formula for \(\Pi _{rr}\) Π rr calculated employing the spectral conductivity obtained based on the self-consistent Born approximation. The coefficient \(\Pi _{rr}\) Π rr is shown to have a large negative (positive) value just above (below) an integer Landau-level filling, with the absolute value \(|\Pi _{rr}|\) | Π rr | increasing with decreasing temperature or decreasing disorder, and approaching the saw-tooth shape \(- (E_{N_\textrm{F} \sigma _\textrm{F}}-\zeta )/e\) - ( E N F σ F - ζ ) / e in the limit of vanishing disorder, where \(E_{N_\textrm{F} \sigma _\textrm{F}}\) E N F σ F is the highest occupied Landau level and \(\zeta \) ζ is the chemical potential. As an initial attempt to experimentally observe the effect of the large \(|\Pi _{rr}|\) | Π rr | , we measure the electron temperature \(T_\textrm{out}\) T out near the outer perimeter of a Corbino disk, applying a radial dc current \(I_\textrm{dc}\) I dc . The temperature \(T_\textrm{out}\) T out is observed to increase or decrease depending on the direction of \(I_\textrm{dc}\) I dc and the sign of \(\Pi _{rr}\) Π rr as expected from the Peltier effect. Notably, \(T_\textrm{out}\) T out becomes lower than the bath temperature for outward (inward) \(I_\textrm{dc}\) I dc in the region where \(\Pi _{rr} < 0\) Π rr < 0 ( \(\Pi _{rr} > 0\) Π rr > 0 ).