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Carrier Scattering at Low Electric Fields

Book cover Semiconductor Physics

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Abstract

Carrier scattering, originating from deviations from ideal lattice periodicity, acts as a damping process for carrier motion. Both elastic and inelastic scattering involve a large variety of scattering centers. Carriers are scattered by acoustic and optical phonons, at neutral or charged impurities, at interfaces, and at other scattering centers. Most scattering events are elastic, changing only the momentum of a carrier but not its energy. Inelastic scattering involves optical phonons and intervalley scattering; in these processes carriers lose much of their energy to the lattice.

At low electric fields, many elastic scattering events precede an inelastic event. The dominating type of scattering changes with lattice temperature. Usually, ionized-impurity scattering prevails at low temperatures and scattering at phonons at high temperatures. The type of carrier scattering determines the relaxation time and with it the carrier mobility.

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Notes

  1. 1.

    These are electric fields F ≪ v rms/μ, where carriers gain only a small fraction of k T between scattering events; this means F ≪ 5×104 V/cm for a typical semiconductor with m n = 0.05 m 0 and μ = 1000 cm2/(Vs).

  2. 2.

    An error up to 20% can occur when applying Eq. 3 because of nonlinearities and interaction of different scattering events as shown by Rode and Knight (1971).

  3. 3.

    The deformation potential is defined as the change in bandgap energy per unit strain and is typically on the order of 10 eV. For a listing, see Table 1. It should be noted that deformation potentials are generally tensors with components Ξ ij , relating the shift of the band edge δE c to the components of the strain tensor e ij (see chapter “Elasticity and Phonons): \( \delta E=\sum_{ij}{\varXi}_{ij}\;{e}_{ij} \); Eq. (9) hence provides only an average in anisotropic media.

  4. 4.

    The experimentally observed exponent of T is −1.67 for Ge (Conwell 1952) and not −1.5. The exponent of T for Si is still larger (≅ 2.4). Inserting actual values for Si (c l = 15.6·1010 N/m2, m n = 0.2 m 0, and Ξ = 9.5 eV), one obtains μ n = 5900 cm2/Vs, a value that is larger by a factor of ~4 than the measured μ n = 1500 cm2/Vs at 300 K.

  5. 5.

    K 2 can be expressed as the ratio of the mechanical to the total work in a piezoelectrical material: \( {K}^2=\left({e}_{\mathrm{pz}}^2/{c}_l\right)/\left(\varepsilon\;{\varepsilon}_0+{e}_{\mathrm{pz}}^2/{c}_l\right) \), with e pz the piezoelectric constant (which is on the order of 10−5 As/cm2), and c l the longitudinal elastic constant (relating the tension T to the stress S and the electric field F as T = c l  S − e pz F).

  6. 6.

    Here, \( {F}_{\mathrm{opt}}=1+\frac{2}{\beta} \ln \left(\beta +1\right)+1/\left(\beta +1\right) \), with β = (2|k|L D)2; L D is the Debye length given in Eqs. 94 of chapter “Interaction of Light With Solids” and 49 of chapter “Crystal Interfaces.”

  7. 7.

    More sophisticated estimates are given for thin metal layers (e.g., Fuchs 1938, Sondheimer 1952), where surface effects are less complex. These yield results on the same order of magnitude as those given here.

  8. 8.

    See Fig. 8b, which shows two equivalent transitions: one requires 2·0.8 π/a = 1.6 π/a in the extended E(k) diagram, but actually one needs only a momentum transfer of 2·0.2 π/a = 0.4 π/a.

  9. 9.

    Umklapp processes were introduced for scattering in metals, in which changes in the magnitude of the momentum after scattering from one to another side of the near-spherical Fermi surface are even smaller.

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Authors and Affiliations

  1. Institut für Festkörperphysik, Technische Universität Berlin, Hardenbergstr, 36 10623, Berlin, Germany

    Udo W. Pohl

  2. 138 Moorings Park Rd. O 301, Naples, FL, USA

    Karl W. Böer

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Böer, K.W., Pohl, U.W. (2016). Carrier Scattering at Low Electric Fields. In: Semiconductor Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-06540-3_23-1

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  1. Latest

    Carrier Scattering at Low Electric Fields
    Published:
    17 June 2022

    DOI: https://doi.org/10.1007/978-3-319-06540-3_23-4

  2. Carrier Scattering at Low Electric Fields
    Published:
    27 February 2020

    DOI: https://doi.org/10.1007/978-3-319-06540-3_23-3

  3. Carrier Scattering at Low Electric Fields
    Published:
    27 September 2017

    DOI: https://doi.org/10.1007/978-3-319-06540-3_23-2

  4. Original

    Carrier Scattering at Low Electric Fields
    Published:
    11 February 2017

    DOI: https://doi.org/10.1007/978-3-319-06540-3_23-1

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