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Investigating Hot-Carrier Effects using the Backward Monte Carlo Method

5 Electron-Electron Scattering

In the previous chapter we presented a method to investigate the high-energy tail of the energy distribution. One particular scattering mechanism and its implications on the high energy tail of the energy distribution function is controversially discussed in the literature, namely electron-electron scattering (EES) [5, 10, 14, 15, 31, 32, 57, 79, 80, 103, 107] [P2].

One can distinguish between selfconsistent models which assume the actual or an approximate non-equilibrium distribution for the partner electrons, and non-selfconsistent models which assume an equilibrium distribution for the partner electrons. The latter approach is suitable to describe the interaction of channel hot electrons with a reservoir of cold electrons in the drain region of a MOSFET. This case is studied in the present thesis [P5].

Scattering due to a perturbation potential can be treated by Fermi’s Golden rule, see Section 2.3. The essential term of Fermi’s Golden rule is the matrix element of the perturbation potential.

5.1 Matrix Element

The perturbation potential for electron-electron scattering is caused by two-body screened Coulomb interaction. Assuming a three-dimensional electron gas (3DEG), the perturbation potential reads [11]:

(5.1) \{begin}{align} U_s(\rv _1, \rv _2) = \frac {e^2}{4\pi \epsilon _s}\frac {\e ^{-\beta _s\abs {\rv _1 - \rv _2}}} {\abs {\rv _1 - \rv _2}}, \{end}{align}

where \( \rv _1 \) and \( \rv _2 \) are the spatial coordinates of the electrons and \( 1/\beta _s \) is the screening length also known as the Debye length , defined as [102]:

(5.2) \{begin}{align} \beta _s^2 = \frac {e^2\,n}{\epsilon _s\,k_B T}\,. \label {eq:beta_s} \{end}{align}

The electronic states for the electron-electron scattering are assumed to be plane-waves.

(5.3) \{begin}{align} \ket {\kv _1,\kv _2} = \frac {1}{\Omega }\; \e ^{\imag \kv _1\cdot \rv _1}\; \e ^{\imag \kv _2\cdot \rv _2} \{end}{align}

With these assumptions, the matrix element (2.24) can be written as:

(5.4) \{begin}{multline} M = \bra {\kv _1’,\kv _2’} U_s \ket {\kv _1,\kv _2} \\= \frac {e^2}{4\pi \epsilon _s}\frac {1}{\Omega ^2} \int \limits _\Omega \d ^3r_1\int \limits _\Omega d^3 r_2\: \e ^{-\imag (\kv _1’\cdot \rv _1 + \kv _2’\cdot \rv _2)} \frac {\e ^{-\beta _s\abs {\rv _1 - \rv _2}}} {\abs {\rv _1 - \rv _2}} \e ^{\imag (\kv _1\cdot \rv _1 + \kv _2\cdot \rv _2)} \label {eq:matrix-ees} \{end}{multline}

We define the spatial distance between the electrons as a new integration variable:

(5.5) \{begin}{align} \uv = \rv _1 - \rv _2 , \{end}{align}

In (5.4) the following substitutions are made:

(5.6) \{begin}{align} \rv _1 = \rv _2 + \uv ,\qquad d^3r_1 = d^3 u . \{end}{align}

This leads to following expression for the matrix element:

(5.7) \{begin}{align} \bra {\kv _1’,\kv _2’} U_s \ket {\kv _1,\kv _2} = \frac {e^2}{\epsilon _s}\frac {1}{\Omega ^2} \int \limits _\Omega \frac {\e ^{-\beta _s u}}{4\pi \, u} \e ^{\imag (\kv _1 - \kv _1’)\cdot \uv } d^3 u \int \limits _\Omega \e ^{\imag (\kv _1 + \kv _2 - \kv _1’ - \kv _2’)\cdot \rv _2}\; d^3 r_2 . \label {eq:int-matrix} \{end}{align}

The first integral represents the Fourier transform of the screened Coulomb potential, see Appendix A.3:

(5.8) \{begin}{align} \int \limits _{R^3} \frac {\e ^{-\beta _s u}}{4\pi \, u}\; \e ^{\imag (\kv _1 - \kv _1’)\cdot \uv }\; d^3 u = \frac {1}{\abs {\kv _1 -\kv _1’}^2 + \beta _s^2}, \label {eq:int-scr-pot} \{end}{align}

whereas the second integral results in a Kronecker-delta

(5.9) \{begin}{align} \int \limits _\Omega \e ^{\imag (\kv _1 + \kv _2 - \kv _1’ - \kv _2’)\cdot \rv _2’}\; d^3 r_2 = \Omega \; \delta _{\kv _1 + \kv _2, \kv _1’ + \kv _2} \„ \label {eq:int-moment} \{end}{align}

which describes conservation of the total momentum

(5.10) \{begin}{align} \kv _1 + \kv _2 = \kv _1’ + \kv _2’ \,. \{end}{align}

Due to the finite normalization volume \( \Omega \) all wave vectors are discrete. Substituting (5.8) and (5.9) into (5.7) leads to the following expression for the matrix element [P5]:

(5.11) \{begin}{align} \bra {\kv _1’,\kv _2’} U_s \ket {\kv _1,\kv _2} = \frac {e^2}{\epsilon _s\Omega } \frac {\delta _{\kv _1 + \kv _2, \kv _1’ + \kv _2’}}{ \abs {\kv _1 -\kv _1’}^2 + \beta _s^2} \label {eq:ees-matrix} \{end}{align}

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