Spins, Effective Spins, Spin Relaxation, Non-Radiative Transitions and all that

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Spins, Effective Spins, Spin Relaxation,
Non-Radiative Transitions and all that

• Marshall Stoneham

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Rate equations for a two-level system

• dNL/dt WULNU -WLUNL
• dNU/dt WLUNL -WULNU
• Let N NL - NU
• Subtract one equation from the other
• dN/dt - (WUL WLU) N -N/t
• Relaxation rate 1/t (WUL WLU)

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Rate equations and better

• Only occupancies NU and NL involved phases and
• wavefunction information are hidden or lost.
• - What interaction causes the transitions
• - Where does the energy go?
• - To which states do Lgt and Ugt correspond? Can
  we really separate them, the energy sink, and the
  interaction?
• - And can we predict 1/t as a number?

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The Spin Hamiltonian 1

• Basic Idea 1 If there are N states with very
  low energy, then we can write an effective
  Hamiltonian for N basis states, with the effects
  of the distant states included by perturbation
  theory.
• Basic Idea 2 We can write this effective
  Hamiltonian in terms of spin operators with
  effective spin S such that N 2S1.
• Note effective spin S does not need to be the
  actual spin of the system (e.g., MgO Co2).

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The Spin Hamiltonian 2 Why bother?

• 1. Spin matrices are well-known and easy to
  manipulate
• 2. The spin Hamiltonian puts everything into a
  format to make comparison of experiment and
  theory easy.
• - It can be used to predict transition energies
  and probabilities so experimenters can get a full
  empirical fit.
• - The key parameters of the fit are what the
  theorists try to predict

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The Spin Hamiltonian 3 Some complications

• - Usually the spin Hamiltonian relates to a
  single defect or spin
• - Equations are often written for an ensemble of
  spins. How do we handle the interactions?
• - Is it true that the magnetic dipoles deduced
  from Zeeman energies are the same as the dipoles
  describing the interactions between spins?
• - The full dipole-dipole interaction includes
  SzSz, SS-, SS and similar terms. What do they
  do?

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Density matrix versions

• Since were talking about a two-level system,
  lets use S as the ensemble average of the spin
  for spins ½ interacting with a heat bath and a
  magnetic field. The density matrix has the form
• ?1 Sz Sx-i Sy ?
• ? ½ ? ?
• ?Sxi Sy 1- Sz ?
• and the equivalent equation of motion is
• - i? ??/?t Hz , ? - i?/t ? - ?o
• where theres a static field and an oscillating
  field
• Hz ??o Sz ½ ??1 S exp (i?t) S- exp
  (-i?t)

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Solve for energy absorption from the field

• The energy absorption can be calculated from
  S.dH/dt. In the steady state, the energy
  absorption is proportional to
• ?1²?o?t / 1 ?1²t² t²(? - ?o)²
• which leads to a maximum energy absorption at
  frequency
• ?m 1/t ? 1 (?1² ?o²)t²

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Resonant and Non-resonant absorption

• Resonant absorption for 1 gtgt t ? (?1² ?o²)
• ?m ? (?1² ?o²)
• or, if ?1 ltlt ?o (usually true) ?m ?o
• Non-resonant absorption for 1 gtgt t ? (?1² ?o²)
• ?m 1/t
• Non-resonant absorption includes dielectric
  relaxation, internal friction and much of the
  early spin lattice relaxation work.

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Energy contour single energy surface

• Note metastable
• and stable
• minima.

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Coordinates versus Normal Modes
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Linear electron-phonon coupling
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What we need to calculateand the Line shape
function G(?)
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• Absorption energy A?B E0EM Emission energy
  C?D E0-EM
• Relaxation energies (cooling transitions) EM and
  EM
• p E0/?? number of accepting phonons
• S0 EM/ ?? Huang-Rhys factor. Strong coupling
  means S0gtgt1
• ? EM/(E0EM) S0/(S0p) determines
  radiative/non-radiative emission

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Typical configuration coordinate diagram for a
absorption and luminescence cycle, showing
optical and cooling transitions
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? is (relaxation energy)/(absorption energy)
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Charge transfer transition between equivalent
sites 1, 2
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Dephasing as cooling occurs
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Accepting and Promoting Modes
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Spin-lattice relaxation of SrF2Tm2 (Sabisky and
Anderson)
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Relaxation routes for excitons and e-h pairs in
alkali halides