Research ArticlePHYSICS

Observation of Poiseuille flow of phonons in black phosphorus

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Science Advances  22 Jun 2018:
Vol. 4, no. 6, eaat3374
DOI: 10.1126/sciadv.aat3374
  • Fig. 1 Crystal structure and resistivity.

    (A) Side view of the crystal structure of BP. (B) Top view of single-layer BP. (C) Temperature dependence of the electrical resistivity ρ of BP along the a and c axes in a logarithmic scale. The upper inset shows an activated behavior of ρ for T > 20 K, where solid lines represent an Arrhenius behavior. The lower inset shows the ratio ρac, which quantifies charge transport anisotropy. At low temperatures, VRH governs, as shown in the bottom panels where ρ is plotted against T−1/2 (D), T−1/3 (E), and T−1/4(F).

  • Fig. 2 Thermal conductivity.

    (A) Temperature dependence of the thermal conductivity along the two high-symmetry axes for the samples with a comparable thickness (solid circles). The a-axis thermal conductivity measurement is extended down to 0.1 K using the thicker sample (open circles). We compared our data to three sets of previous reports (10, 12, 13). We also show the thermal conductivity of P-doped Si obtained by using the same experimental setup. Top inset illustrates a schematic of the measurement setup for the thermal conductivity. Bottom inset shows κ divided by T3 as a function of temperature. By plotting the thermal conductivity as a function of T3, one can see how the ballistic regime, where κ ∝ T3, evolves to the Poiseuille regime where κ changes faster than T3. Such a behavior can be seen in BP (B) and in the literature data of Bi (20) (C), but it is absent in P-doped Si (D). (E) Schematic representation of Poiseuille flow: the temperature dependence of mean free path of normal scattering lN (dashed-dotted line) and resistive scattering lR (solid line) and the sample dimension d in the three regimes of thermal transport. Poiseuille flow of phonons takes place in the limited temperature range between ballistic and diffusive transport when the inequality of lNdlR is fulfilled. In the center of samples, the phonon mean free path (dashed line) becomes lphd2/lN in these conditions. The three small panels represent three distinct scattering mechanisms. Normal scattering between two phonons does not lead to any loss of momentum and does not degrade the heat current. In contrast, Umklapp scattering and collision with impurities lead to a loss in heat current and amplify thermal resistance. In any finite-size sample, phonons are also scattered by the boundaries.

  • Fig. 3 Thickness dependence of thermal conductivity.

    Temperature dependence of the thermal conductivity in samples with different thicknesses along the a axis (A) and along the c axis (B). We also show the data from the thin BP flakes (14). Thermal conductivity shows a thickness dependence in the whole temperature range. The extracted phonon mean free path lph in samples with different thicknesses along the a axis (C) and along the c axis (D). The local maximum and minimum are present in all samples. The insets of the (C) and (D) show plots of lph versus T in a logarithmic scale for BP including the thicker sample and for P-doped Si, respectively. (E) The effective phonon mean free path of Bi for various average diameters from (20).

  • Fig. 4 Thickness dependence of the Knudsen minimum.

    (A) Schematic illustration of the Knudsen minimum adapted from (32). The diffuse boundary scattering rate is effectively increased by normal scattering when the mean free path of normal scattering lN represented by the dashed circle is comparable with the sample dimension d, producing a local minimum in the effective phonon mean free path. With increasing thickness, the minimum shifts to lower temperatures since the fraction of phonons that suffer numerous normal scattering (phonons in the red region) is larger in the thicker sample. The phonon mean free path normalized by the value at the Knudsen minimum in samples with different thicknesses along the a axis (B) and the c axis (C) and Bi adapted from (20) (D).

  • Fig. 5 T−1 dependence of thermal conductivity and relative decrease of thermal conductivity with thickness.

    (A) κ as a function of T−1 in samples with different thicknesses. Note the gradual decrease in slope with thickness, as shown in the inset. The temperature dependence of the relative decrease in κ, κratio, when the thickness decreases from 20(30) μm to 6(15) μm along the a(c) axis (B). We defined κratio as (κ(t1) − κ(t2))/κ(t2) × t2/(t1t2), where t1,2 denotes different thicknesses. One expects this to become of the order of unity in the ballistic regime and zero in the high temperature when heat transport is purely diffusive. These features are seen both in Bi (20) (when the average diameter decreases from 0.51 to 0.26 cm) and in Ge (34) (when the width was decreased from 4 to 1 mm). In BP samples of a typical size of 6 to 140 μm, κratio persists (while slowly decreasing) in the diffusive regime.

  • Fig. 6 Phonon dispersions.

    Calculated phonon dispersions of BP (A) and Si (D). (B) and (E) show the blow-up in the region below 100 cm−1. (C) and (F) show the Brillouin zones along with the high-symmetry points. The out-of-plane direction in BP corresponds to the path along Γ-Y. (G) Calculated PDOS of BP and Si. (H) PDOS below 100 cm−1. The acoustic modes of BP are less dispersive than that of Si, leading to higher PDOS in BP in the low-frequency region. At low temperatures, BP should have much more phase space for the momentum-conserving phonon scattering processes among the linearly dispersive acoustic modes that are relevant for Poiseuille heat flow.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaat3374/DC1

    Supplementary Text

    table S1. Size of samples.

    table S2. Phonon velocity along the high-symmetry axes.

    fig. S1. Thermal resistance of BP samples and manganin.

    fig. S2. Reproducibility of thermal conductivity data.

    fig. S3. Specific heat.

    fig. S4. Phonon dispersions at low energies.

    References (4655)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text
    • table S1. Size of samples.
    • table S2. Phonon velocity along the high-symmetry axes.
    • fig. S1. Thermal resistance of BP samples and manganin.
    • fig. S2. Reproducibility of thermal conductivity data.
    • fig. S3. Specific heat.
    • fig. S4. Phonon dispersions at low energies.
    • References (46–55)

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