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Calculating scattering rates

AMSET calculates mode dependent scattering rates within the Born approximation using common materials parameters. The differential scattering rate from state nk\mathinner{|n\mathbf{k}\rangle} to state mk+q\mathinner{|m\mathbf{k} + \mathbf{q}\rangle} is calculated using Fermi's golden rule as

τ~nkmk+q1=2πgnm(k,q)2δ(εnkεmk+q), \tilde{\tau}_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1} = \frac{2\pi}{\hbar} \lvert g_{nm}(\mathbf{k}, \mathbf{q}) \rvert^2 \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{m\mathbf{k}+\mathbf{q}}),

where εnk\varepsilon_{n\mathbf{k}} is the energy of state nk\mathinner{|n\mathbf{k}\rangle}, and gnm(k,q)g_{nm}(\mathbf{k}, \mathbf{q}) is the matrix element for scattering from state nk\mathinner{|n\mathbf{k}\rangle} into state mk+q\mathinner{|m\mathbf{k} + \mathbf{q}\rangle}.

Info

Note, this is the expression for elastic scattering. Inelastic scattering contains addition terms, as detailed in the elastic vs inelastic scattering section.

The overall mode-dependent scattering rate is obtained by integrating the scattering rates over the full Brillouin zone. In this section, we report the matrix elements for each scattering mechanism implemented in AMSET. Information on calculating transport properties is given in the transport properties section.

Summary of scattering rates

Mechanism Code Requires Type
Acoustic deformation potential scattering ADP n- and p-type deformation potential, elastic constant Elastic
Ionized impurity scattering IMP static dielectric constant Elastic
Piezoelectric scattering PIE high-frequency dielectric constant, elastic constant, piezoelectric coefficient (e\mathbf{e}) Elastic
Polar optical phonon scattering POP polar optical phonon frequency, static and high-frequency dielectric constants Inelastic

Acoustic deformation potential scattering

The acoustic deformation potential matrix element is given by

gnmad(k,q)=kBT[(Dnk:q^q^)2Qqq^q^+(Dnk:q^q^t1)2Qqq^t1q^t1+(Dnk:q^q^t2)2Qqq^t2q^t2]12ψmk+qψnk, g_{nm}^\mathrm{ad}(\mathbf{k}, \mathbf{q}) = \sqrt{k_\mathrm{B} T} \left[ \frac{(\mathbf{D}_{n\mathbf{k}} : \mathbf{\hat{q}} \otimes \mathbf{\hat{q}})^2} {\mathbf{Q}_\mathbf{q}\mathbf{\hat{q}}\cdot\mathbf{\hat{q}}} + \frac{(\mathbf{D}_{n\mathbf{k}} : \mathbf{\hat{q}} \otimes \mathbf{\hat{q}}_{t_1})^2} {\mathbf{Q}_\mathbf{q}\mathbf{\hat{q}}_{t_1}\cdot\mathbf{\hat{q}}_{t_1}} + \frac{(\mathbf{D}_{n\mathbf{k}} : \mathbf{\hat{q}} \otimes \mathbf{\hat{q}}_{t_2})^2} {\mathbf{Q}_\mathbf{q}\mathbf{\hat{q}}_{t_2}\cdot\mathbf{\hat{q}}_{t_2}} \right]^\frac{1}{2} \mathinner{\langle{\psi_{m\mathbf{k}+\mathbf{q}}|\psi_{n\mathbf{k}}}\rangle},

where Dnk\mathbf{D}_{n\mathbf{k}} is the rank 2 volume deformation potential tensor for state nk\mathinner{|n\mathbf{k}\rangle}, Qq=q^Cq^\mathbf{Q}_\mathbf{q} = \mathbf{\hat{q}} \cdot \mathbf{C} \cdot \mathbf{\hat{q}} is the acoustic tensor in the direction of q\mathbf{q}, C\mathbf{C} is the rank 4 elastic constant tensor, and q^=q/q\mathbf{\hat{q}} = \mathbf{q} / |\mathbf{q}| is the unit scattering direction. The vectors q^t1\mathbf{\hat{q}}_{t_1} and q^t2\mathbf{\hat{q}}_{t_2} are unit vectors perpendicular to q^\mathbf{\hat{q}} and to each other, such that the three form an orthogonal basis. The first term given in square brackets accounts for scattering due to longitudinal phonons, whereas the second two terms describe scattering from transverse modes.

  • Abbreviation: APD
  • Type: Elastic
  • References: 3, 2, 1
  • Requires: deformation_potential, elastic_constant

Ionized impurity scattering

The ionized impurity matrix element is given by

gnmii(k,q)=nii1/2Zeq^ϵsq^ψmk+qψnkq2+β2, g_{nm}^\mathrm{ii}(\mathbf{k}, \mathbf{q}) = \frac{n_\mathrm{ii}^{1/2} Z e }{\mathbf{\hat{q}} \cdot \boldsymbol{\epsilon}_\mathrm{s} \cdot \mathbf{\hat{q}}} \frac{\mathinner{\langle{\psi_{m\mathbf{k}+\mathbf{q}}|\psi_{n\mathbf{k}}}\rangle}} {\left | \mathbf{q} \right | ^2 + \beta^2},

where ϵs\boldsymbol{\epsilon}_\mathrm{s} is the static dielectric constant, niin_\mathrm{ii} is the concentration of ionized impurities (i.e., nholes+nelectronsn_\mathrm{holes} + n_\mathrm{electrons}), and β\beta is the inverse screening length, defined as

β2=e2ϵskBTdεVD(ε)f(1f), \beta^2 = \frac{e^2}{\epsilon_\mathrm{s} k_\mathrm{B} T} \int \frac{\mathrm{d}\varepsilon}{V}\,D(\varepsilon) f(1-f),

where VV is the unit cell volume, DD is the density of states, and ff is the Fermi–Dirac distribution given in the transport properties section.

  • Abbreviation: IMP
  • Type: Elastic
  • References: 4, 1
  • Requires: static_dielectric

Piezoelectric scattering

The piezoelectric differential scattering rate is given by

gnmpi(k,q)=[kBTd2q^ϵsq^]12ψmk+qψnkq, g_{nm}^\mathrm{pi}(\mathbf{k}, \mathbf{q}) = \left [ \frac{k_\mathrm{B} T d^2}{\mathbf{\hat{q}}\cdot\boldsymbol{\epsilon}_\mathrm{s}\cdot\mathbf{\hat{q}}} \right ] ^\frac{1}{2} \frac{\mathinner{\langle{\psi_{m\mathbf{k}+\mathbf{q}}|\psi_{n\mathbf{k}}}\rangle}} {\left | \mathbf{q} \right |},

where dd is the dimensionless piezoelectric coefficient.

  • Abbreviation: PIE
  • Type: Elastic
  • References: 1
  • Requires: piezoelectric_coefficient, static_dielectric

Polar optical phonon scattering

The polar optical phonon differential scattering rate is given by

gnmpo(k,q)=[ωpo2(1q^ϵq^1q^ϵsq^)]12ψmk+qψnkq, g_{nm}^\mathrm{po}(\mathbf{k}, \mathbf{q}) = \left [ \frac{\hbar \omega_\mathrm{po}}{2} \left (\frac{1}{\mathbf{\hat{q}}\cdot\boldsymbol{\epsilon}_\infty\cdot\mathbf{\hat{q}}} - \frac{1}{\mathbf{\hat{q}}\cdot\boldsymbol{\epsilon}_\mathrm{s}\cdot\mathbf{\hat{q}}}\right) \right ] ^\frac{1}{2} \frac{\mathinner{\langle{\psi_{m\mathbf{k}+\mathbf{q}}|\psi_{n\mathbf{k}}}\rangle}} {\left | \mathbf{q} \right |},

where ωpo\omega_\mathrm{po} is the polar optical phonon frequency, and ϵ\boldsymbol{\epsilon}_\infty is the high-frequency dielectric constant tensor.

  • Abbreviation: POP
  • Type: Inelastic
  • References: 5, 6, 1
  • Requires: pop_frequency, static_dielectric, high_frequency_dielectric

Elastic vs inelastic scattering

AMSET treats elastic and inelastic scattering mechanisms separately.

Inelastic

The differential scattering rate for inelastic processes is calculated as

τnkmk+q1=2πgnm(k,q)2×[(npo+1fmk+q)δ(εnkεmk+qωpo)(npo+fmk+q)δ(εnkεmk+q+ωpo)], \begin{aligned} \tau_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1} = \frac{2\pi}{\hbar} \lvert g_{nm}(\mathbf{k}, \mathbf{q}) \rvert^2 \times [ &{} (n_\mathrm{po} + 1 - f_{m\mathbf{k} + \mathbf{q}}) \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{m\mathbf{k}+\mathbf{q}} - \hbar\omega_\mathrm{po}) \\ &{} (n_\mathrm{po} + f_{m\mathbf{k} + \mathbf{q}}) \delta(\varepsilon_{n\mathbf{k}} - \varepsilon_{m\mathbf{k}+\mathbf{q}} + \hbar\omega_\mathrm{po})], \end{aligned}

where ωpo\omega_\mathrm{po} is an effective phonon frequency, npo=1/[exp(ωpo/kBT)1]n_\mathrm{po} = 1 / [\exp (\hbar \omega_\mathrm{po} / k_\mathrm{B} T) - 1] denotes the Bose–Einstein distribution of phonons, and the ωpo-\hbar \omega_\mathrm{po} and +ωpo+\hbar \omega_\mathrm{po} terms correspond to scattering by phonon absorption and emission, respectively.

The overall inelastic scattering rate for state nk\mathinner{|n\mathbf{k}\rangle} is calculated as

τnk1=md3qΩτnkmk+q1 \tau^{-1}_{n\mathbf{k}} = \sum_m \int \frac{\mathrm{d}^3q}{\Omega} \tau_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1}

where Ω\Omega is the volume of the Brillouin zone.

Elastic

Elastic rates are calculated using the momentum relaxation time approximation (MRTA), given by

τ~nk1=md3qΩ[1vnkvmk+qvnk2]τ~nkmk+q1 \tilde{\tau}^{-1}_{n\mathbf{k}} = \sum_m \int \frac{\mathrm{d}^3q}{\Omega} \left [ 1 - \frac{\mathbf{v}_{n\mathbf{k}} \cdot \mathbf{v}_{m\mathbf{k} + \mathbf{q}}}{\lvert \mathbf{v}_{n\mathbf{k}} \rvert^2} \right ] \tilde{\tau}_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1}

where τ~nkmk+q1\tilde{\tau}_{n\mathbf{k}\rightarrow m\mathbf{k}+\mathbf{q}}^{-1} is the elastic differential scattering rate defined at the top of this page and vnk\mathbf{v}_{n\mathbf{k}} is the group velocity of state nk\mathinner{|n\mathbf{k}\rangle}.

Overlap integral

In the Born approximation, the scattering rate equations depend on the wavefunction overlap ψmkψnk\mathinner{\langle{\psi_{m\mathbf{k}^\prime}|\psi_{n\mathbf{k}}}\rangle}. AMSET uses pawpyseed to obtain wavefunction coefficients including PAW core regions from the pseudo wavefunction coefficients written by VASP. The wavefunctions coefficient are linearly interpolated onto the mesh used to calculate scattering rates.

Brillouin zone integration

All scattering rates depend on the Dirac delta function δ\delta, which imposes conservation of energy. Due to finite k-point sampling and numerical noise, it is unlikely that this condition will ever be satisfied exactly. Furthermore, many scattering rates have a 1/q21 / \lvert\mathbf{q}\rvert ^2 dependence which requires an extremely dense k-point mesh to achieve convergence.

To account for this, AMSET employs a modified tetrahedron integration scheme. AMSET first identifies a constant energy surface by computing tetrahedral cross sections using the tetrahedron method. Next, the constant energy surface is resampled using an ultra-fine mesh of k-points generated using the quadpy numerical integration package. The wavefunction coefficients and group velocities are reinterpolated into the ultra-fine mesh using linear interpolation and the matrix elements are calculated directly. This methodology allows for significantly faster convergence than the regular tetrahedron method.

The methodology for combining rates from multiple scattering mechanisms is given in the transport properties section.


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  2. Shockley, W. & others. Electrons and holes in semiconductors: with applications to transistor electronics. (van Nostrand New York, 1950). 

  3. Bardeen, J. & Shockley, W. Deformation potentials and mobilities in non-polar crystals. Phys. Rev. 80, 72-80 (1950). 

  4. Dingle, R. B. XCIV. Scattering of electrons and holes by charged donors and acceptors in semiconductors. London, Edinburgh, Dublin Philos. Mag. J. Sci. 46, 831-840 (1955). 

  5. Fröhlich, H. Electrons in lattice fields. Adv. Phys. 3, 325–361 (1954). 

  6. Conwell, E. M. High Field Transport in Semiconductors. Academic Press, New York (1967).