Thermal Conductivity

Thermal conductivity

  • The ability of material to resist flow of heat

Thermal resistivity:

  • the reciprocal of thermal conductivity
  • heat can be transmitted through a crystal by phonons, photons, free electrons or holes and electron holes pair
  • single phonon cannot be used describe a deviation from equilibrium in one region
  • phonons be localized in space within distance

Phonon mean free path

Thermal conductivity:

  • the constant of proportionality between a temperature gradient and the rate of energy flow per unit area Q
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  • any distribution of phonons, it may define as nominal mean free path
  • in equilibrium, situation Λ decreases rapidly with increasing phonon energy

Anharmonic effects

Example

Thermal expansion

Thermal conductivity

  • phonon collide with each other and these collisions limit thermal conductivity due to the flow of phonons
  • consider the potential energy of 1D monoatomic
  • forces between only nearest neighbors
  • the two terms come from the two springs that are attached to the nth atom
  • 3D crystal, Hooke's law may appear
  • non-negligible springs between any one atom and its nearest, non-nearest neighbors
  • forces are linear in the displacements
  • normal mode decomposition of the atomic motion is exact
    phonons can't interact with another

Process

Normal Process

  • consider three phonon process with wave vectors k1 and k2 collide and create wave vector k3

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The conservation energy:
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  • these phonons can interact and annihilate one another resulting a phonon with wave vector k3
  • This process does not alter the direction of energy flow, it can't contribute to thermal resistance of crystal

Umklapp Process

  • K≠0
  • provide thermal resistance to phonon flow
  • energy flow for k1 and k2 was the right, the energy flow for k3 to the left

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The conservation law:
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  • Let say TA an arbitrary point is picked then corresponds to k1
  • Using this point as origin, the ω vs k curves
  • The intersection of solid and dashed curves is required solution
  • This particular value have phonon of wave vector k3 and energy ℏω3 that satisfies
  • The conservation laws are fulfilled with K=0 is an N-process

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  • consider an arbitrary but larger, value for k1 and proceed in same manner as before
  • Value k2 is obtained with circle that satisfies the conservation law
  • The intersection is outside of the first Brillouin zone
  • The addition of a reciprocal lattice vector, -2π/a
  • The point marked with an x and k3
  • This phonon with wave vector k3, the conservation laws using K≠0 are satisfied
  • The flow energy reversed in direction
  • So, this is a U-process

Thermal conductivity due to defects

Defects of one sort or another also are important in determining the thermal conductivity as mentioned.

Defects

point defects

dislocations

rough crystal surfaces

disorder due to alloying

different isotopes of a given chemical species

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Example of isotopic disorder

  • Normal germanium is 37% 74Ge, with the rest consisting of isotopes with mass 70, 72, 73, and 76.
  • A sample that is isotopically enriched to 94% 74Ge
  • Thermal conductivity is increased considerably at low temperatures
  • Crystals that are normally isotopically pure have a steeper rise in 𝜅ℓ at low temperatures than those with several isotopes.